Download Mathematics Methods

T A S M A N I A N
Q U A L I F I C A T I O N S
Mathematics Methods
A U T H O R I T Y
MTM315XXX, TQA Level 3, Size Value = 15
THE COURSE DOCUMENT
This document contains the following sections:
RATIONALE ................................................................................................................................................. 2
AIMS ............................................................................................................................................................ 2
COURSE SIZE AND COMPLEXITY ............................................................................................................ 2
ACCESS ...................................................................................................................................................... 3
PATHWAYS ................................................................................................................................................. 3
RESOURCES .............................................................................................................................................. 3
LEARNING OUTCOMES ............................................................................................................................. 3
COURSE CONTENT ................................................................................................................................... 3
FUNCTIONS AND GRAPHS
................................................................................................................... 3
CIRCULAR (TRIGONOMETRIC) FUNCTIONS ............................................................................................ 4
CALCULUS ......................................................................................................................................... 5
STATISTICS AND PROBABILITY ............................................................................................................. 5
ASSESSMENT ............................................................................................................................................ 6
QUALITY ASSURANCE PROCESSES ..................................................................................................... 6
COURSE CRITERIA............................................................................................................................. 6
STANDARDS ...................................................................................................................................... 7
QUALIFICATIONS AVAILABLE ............................................................................................................. 15
AWARD REQUIREMENTS .................................................................................................................. 15
COURSE EVALUATION ............................................................................................................................ 15
EXPECTATIONS DEFINED BY NATIONAL STANDARDS ...................................................................... 16
ACCREDITATION ...................................................................................................................................... 23
VERSION HISTORY .................................................................................................................................. 23
Copyright for part(s) of this course may be held by organisations other than the TQA
Version X
2
Mathematics Methods
TQA Level 3
RATIONALE
Mathematics is an immense, multi-faceted field of knowledge with a rich history, a dynamic present and
an exciting future. It impacts upon the daily life of people everywhere and helps them to understand the
world in which they live and work. Mathematics has always been and continues to be a vital component of
a soundly based general education.
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 real-world 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.
The major themes of TQA level 3 Mathematics Methods are function study, calculus and statistics. They
include as necessary prerequisites studies of algebra, functions and their graphs, and probability. These
are developed systematically, with increasing levels of sophistication and complexity. Calculus is essential
for developing an understanding of the physical world because many of the laws of science are
relationships involving rates of change. Statistics is used to describe and analyse phenomena involving
uncertainty and variation. For these reasons this subject provides a foundation for further studies in
disciplines in which mathematics and statistics have important roles, including engineering, the sciences,
economics, health and social sciences.
AIMS
Mathematics Methods aims to develop students’:
•
understanding of concepts and techniques drawn from algebra, the study of functions, calculus,
probability and statistics
•
ability to solve applied problems using concepts and techniques drawn from algebra, functions
study, calculus, probability and statistics
•
reasoning in mathematical and statistical contexts and interpretation of mathematical and
statistical information including ascertaining the reasonableness of solutions to problems
•
capacity to communicate in a concise and systematic manner using appropriate mathematical
and statistical language
•
capacity to choose and use technology appropriately and efficiently.
COURSE SIZE AND COMPLEXITY
This course has a complexity level of TQA level 3.
At TQA level 3, the student is expected to acquire a combination of theoretical and/or technical and
factual knowledge and skills and use judgment when varying procedures to deal with unusual or
unexpected aspects that may arise. Some skills in organising self and others are expected. TQA level 3 is
a standard suitable to prepare students 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.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
3
Mathematics Methods
TQA Level 3
ACCESS
It is highly recommended that students attempting this course will have previously successfully completed
TQA level 2 Mathematics Methods – Foundation.
Students may choose to attempt TQA level 3 Mathematics Specialised either concurrently with, or on
completion of, this course. Mathematics Methods is designed for students, whose future pathways may
involve mathematics and statistics and their applications in a range of disciplines at the tertiary level,
including engineering, the sciences, economics, health and social sciences.
PATHWAYS
Students may choose to attempt TQA level 3 Mathematics Specialised either concurrently with, or on
completion of, this course. Mathematics Methods is designed for students, whose future pathways may
involve mathematics and statistics and their applications in a range of disciplines at the tertiary level,
including engineering, the sciences, economics, health and social sciences.
RESOURCES
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.
Students should have access to graphics calculators and become proficient in their use. Graphics
calculators can be used in all aspects of this course in the development of concepts and as a tool for
solving problems. Where feasible, students should have some experience with CAS technology.
The use of computers is recommended as an aid to students’ learning and mathematical development. A
range of packages is appropriate and, in particular, spreadsheets should be used.
LEARNING OUTCOMES
Through studying this course, students will:
•
understand the concepts and techniques in function study, calculus, probability and statistics
•
solve problems in function study, calculus, probability and statistics
•
apply reasoning skills in function study, calculus, probability and statistics
•
interpret and evaluate mathematical and statistical information and ascertain the reasonableness
of solutions to problems
•
communicate their arguments and strategies when solving problems.
COURSE CONTENT
FUNCTIONS AND GRAPHS
This area of study will include:
•
graphs of polynomial functions in factored form with linear factors, including repeated factors
•
review of factorisation of polynomials and its use in curve sketching and determining the nature of
stationary points
•
graphs of y = x for n = -2, -1, ½, 1, 2 and 3, graphs of y = a and y = logax, where a = 2, 10 and e,
including domain, range and (where relevant) asymptotic behaviour
•
logarithm laws, including change of base
•
solution of simple exponential and logarithmic equations
n
Tasmanian Qualifications Authority
Version X
x
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
4
Mathematics Methods
TQA Level 3
•
graphs derived from others, using translation, reflection in x-axis, reflection in y-axis, dilation from
axes and combinations of these (resultant graphs with asymptotes that are not parallel to one of
the Cartesian axes are excluded)
•
basic composition of functions
•
equations and graphs of circle and semi-circle (x – a) + (y – b) = r
•
informal consideration of one-to-one and many-to-one functions, conditions for existence of
inverse functions
•
finding inverses of functions such as:
2
2
y=
y= a(x+b) +c
x
y = ae +b + b
2
2
a
+c
x +b
y = aln(x+b) + b
•
graphs of inverses derived from graphs of original functions
•
recognition of the general form of possible models for data presented in graphical or tabular form,
using polynomial, power, circular (trigonometric), exponential and logarithmic functions.
CIRCULAR (TRIGONOMETRIC) FUNCTIONS
This area of study will include:
•
revision of radians: definition, conversion between radians and degrees
•
revision of unit circle
o
definition of sine, cosine and tangent
o
special relationships sin x + cos x = 1 and that -1 ≤ sin x ≤ 1, -1 ≤ cos x ≤ 1
o
special values; eg, sin(0) = 0, cos(π) = -l
o
symmetry properties, such as sin(π ± x), cos(π ± x), sin(2π ± x), cos(2π ± x),
o
tan(π ± x), tan(2π ± x), sin(π/2 ± x), cos(π/2 ± x), tan(π/2 ± x);
2
2
•
exact values of sine, cosine and tangent of integer multiples of
•
the identity tan x =
•
applications of the angle sum and difference identities
€
π πππ
6, 4, 3, 2
sin x
cos x
o
sin(A + B), sin(A – B), cos(A + B), cos(A – B), tan(A + B), tan(A – B)
o
sin2A, cos2A, tan2A
•
simple illustrations of the application of circular (trigonometric) functions; e.g. tidal heights, sound
waves, biorhythms, ovulation cycles, temperature fluctuations during a day
•
recognition and interpretation of period and amplitude
•
graphs of:
y= asin(bx+c) + d,
•
y = acos(bx+c) + d,
y = atan(bx+c) + d
inverse trigonometric functions to enable the solution of trigonometric equations of the form:
sin(ax+b) = c, cos(ax+b) = c and tan(ax+b) = c in a given domain
•
and also of the form:
sin(ax+b) = k cos(ax+b), in a given domain.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
5
Mathematics Methods
TQA Level 3
CALCULUS
This area of study will include:
•
deducing the graph of the gradient function, including its domain, from the graph of a function
•
limit theorems made plausible
•
first principles approach to gradient functions f(x) = x , x and other simple polynomials
•
rules for derivatives of x , e , lnx, sinx, cosx, tanx, and linear combinations of these functions
(formal derivation not examined)
•
product, quotient and chain rules; applications involving simple manipulations
•
applications of differentiation to curve sketching, stationary points (turning points and points of
inflection with zero gradient), tests for distinguishing turning points including use of the second
derivative, graph of the first derivative and ‘change of sign of derivative test’
•
equations of tangents, maximum/minimum problems and rates of change including numerical
evaluation of derivatives
•
informal treatment of the fundamental theorem of calculus
•
definite and indefinite integrals of x and (ax + b) , e +b, sin(ax+b), cos(ax+b) and linear
-1
-1
combinations of these functions – in the case of x and (ax + b) an informal discussion of the
use of absolute value is appropriate
•
properties of integrals
•
integration by recognition that
•
application of integration to calculating the area of a region under a curve and simple cases of
areas between curves.
2
n
3
x
n
n
ax
d
[ f ( x )] = f '( x ) = g ( x ) implies that
dx
∫ g ( x ) dx = f ( x ) + c
STATISTICS AND PROBABILITY
This area of study will include:
•
•
•
•
discrete random variables:
o
concept of a random variable
o
construction of a discrete probability distribution
o
calculation and interpretation of the expected value and variance of a discrete
random variable
o
property that, for many random variables, approximately 95 per cent of the
distribution is within two standard deviations of the mean
binomial distribution:
o
application of the binomial distribution to the number of successes in a fixed number,
n, of Bernoulli trials with probability p of success
o
the effect of the parameters n and p on the graph of the probability function
o
calculation of probabilities
o
formulas for the expectation and variance of a binomial random variable and their use
hypergeometric distribution:
o
application of the hypergeometric distribution to sampling without replacement
o
calculation of probabilities
o
formulae for the expectation and variance of a hypergeometric random variable and
their use
normal distribution:
o
the normal curve as the limit of the histogram using examples such as weights and
heights of people (for large samples)
o
the effect of the mean and variance on the shape of the normal distribution
o
use calculators or computer packages to calculate probabilities and quantiles (for
example: quartiles) for normal distributions
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
6
Mathematics Methods
TQA Level 3
ASSESSMENT
Criterion-based assessment is a form of outcomes assessment that identifies the extent of student
achievement at an appropriate end-point of study. Although assessment – as part of the learning program
- is continuous, much of it is formative, and is done to help students identify what they need to do to attain
the maximum benefit from their study of the course. Therefore, assessment for summative reporting to the
Tasmanian Qualifications Authority should focus on what both teacher and student understand to reflect
end-point achievement.
The standard of achievement each student 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 student demonstrates any achievement against a criterion less than
the standard specified for the ‘C’ rating.
A ‘z’ notation is to be used where a student provides no evidence of achievement at all.
Providers offering this course must participate in quality assurance processes specified by the Tasmanian
Qualifications Authority to ensure provider validity and comparability of standards across all awards.
Further information on quality assurance processes, as well as on assessment, is available in the TQA
Senior Secondary Handbook or on the website at http://www.tqa.tas.gov.au
Internal assessment of all criteria will be made by the provider. Providers will report the student’s rating for
each criterion to the Tasmanian Qualifications Authority.
The Tasmanian Qualifications Authority will supervise the external assessment of designated criteria (*).
The ratings obtained from the external assessments will be used in addition to those provided from the
provider to determine the final award.
QUALITY ASSURANCE PROCESSES
The following processes will be facilitated by the TQA to ensure there is:
• a match between the standards of achievement specified in the course and the standards
demonstrated by students
• community confidence in the integrity and meaning of the qualifications.
Process - the Authority 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.
COURSE CRITERIA
The assessment for Mathematics Methods will be based on the degree to which a student can:
1. communicate mathematical ideas and information
2. demonstrate mathematical reasoning and strategy in problem solving situations
3. plan, organise and complete mathematical tasks
4. *demonstrate an understanding of polynomial, hyperbolic, exponential and logarithmic functions
5. *demonstrate an understanding of circular functions
6. *use differential calculus in the study of functions
7. *use integral calculus in the study of functions
8. *demonstrate an understanding of binomial, hypergeometric and normal probability distributions
* = externally assessed
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
7
Mathematics Methods
TQA Level 3
STANDARDS
C RITERION 1
C OMMUNICATE M ATHEMATICAL I DEAS AND I NFORMATION
Rating ‘C’
A student:
•
presents work that conveys the line of reasoning that has been followed between question and
answer
•
generally presents work that follows mathematical conventions and correctly uses mathematical
symbols correctly
•
presents work with the final answer apparent
•
presents work that shows some attention to detail
•
generally presents the final answer with correct units when required
•
presents graphs that convey meaning
•
adds a diagram to a solution when prompted.
Rating ‘B’
A student:
•
presents work that clearly conveys the line of reasoning that has been followed between question
and answer
•
generally presents work that follows mathematical conventions and uses mathematical symbols
correctly
•
presents work with the final answer clearly identified
•
presents work that shows attention to detail
•
consistently presents the final answer with correct units when required
•
presents detailed graphs
•
adds a diagram to supplement a solution.
Rating ‘A’
A student:
•
presents work that clearly conveys the line of reasoning that has been followed between question
and answer, including suitable justification and explanation of methods and processes used
•
consistently presents work that follows mathematical conventions and uses mathematical
symbols correctly
•
presents work with the final answer clearly identified and articulated in terms of the question
where necessary
•
presents work that shows a high level of attention to detail
•
consistently presents the final answer with correct units when required
•
presents graphs that are meticulous in detail
•
adds a detailed diagram to supplement a solution.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
8
Mathematics Methods
TQA Level 3
C RITERION 2
D EMONSTRATE M ATHEMATICAL R EASONING AND S TRATEGY IN P ROBLEM S OLVING
S ITUATIONS
Rating ‘C’
A student:
•
identifies an appropriate strategy to solve familiar problems
•
describes solutions to routine problems
•
identifies inappropriate solutions to routine problems
•
identifies and describes limitations of simple models
•
uses calculator techniques to solve familiar problems.
Rating ‘B’
A student:
•
identifies and follows an appropriate strategy to solve familiar problems where several may exist
•
interprets solutions to routine and simple non routine problems
•
describes the reasonableness of the results and solutions to routine and simple, non-routine
problems
•
identifies and explains the limitations of familiar models
•
chooses to use calculator techniques when appropriate.
Rating ‘A’
A student:
•
identifies multiple strategies and follows an appropriate strategy to solve unfamiliar problems
where several may exist
•
interprets solutions to routine and non routine problems in a variety of contexts
•
explains the reasonableness of the results and solutions to routine and non-routine problems in a
variety of contexts
•
identifies and explains the limitations of familiar and more complex models
•
explores calculator techniques in unfamiliar contexts.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
9
Mathematics Methods
TQA Level 3
C RITERION 3
P LAN , O RGANISE AND C OMPLETE M ATHEMATICAL T ASKS
Rating ‘C’
A student:
•
uses planning tools to achieve objectives within proposed times
•
divides a task into sub-tasks as directed
•
selects from a range of strategies and formulae to successfully complete routine problems
•
monitors progress towards meeting goals and timelines
•
meets timelines and addresses most elements of the required task.
Rating ‘B’
A student:
•
selects and uses planning tools and strategies to achieve and manage activities within proposed
times
•
divides a task into appropriate sub-tasks
•
selects from a range of strategies and formulae to successfully complete routine problems and
more complex problems
•
monitors and analyses progress towards meeting goals and timelines, and plans future actions
•
meets specified timelines and addresses all required task elements.
Rating ‘A’
A student:
•
evaluates, selects and uses planning tools and strategies to achieve and manage activities within
proposed times
•
assists others to divide a task into sub-tasks
•
chooses strategies and formulae to successfully completes routine and more complex problems
•
monitors and critically evaluates goals and timelines, and plans effective future actions
•
meets specified timelines and addresses all required elements of the task with a high degree of
accuracy.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
10
Mathematics Methods
TQA Level 3
C RITERION 4
D EMONSTRATE AN U NDERSTANDING OF P OLYNOMIAL , H YPERBOLIC , E XPONENTIAL AND
L OGARITHMIC F UNCTIONS
Rating ‘C’
A student:
•
recognises and sketches graphs of the standard functions of this course, that is of y= xn for n= 2
2
2
x
2,-1,1, 2, 3, (x – a) + (y – b) = r , y = a (for a = 2, 10 or e) or y = logax (for a = 2, 10 or e)
•
understands the difference between: one-to-one and many-to one and many-to-many relations
•
understands the difference between a relation and a function, and if given the graph of a relation,
can define the domain and range of that relation and state whether the relation is a function
•
understands the conditions for the existence of inverse functions
•
can graph a polynomial function of the form y=a(bx +c) +d where n= 2, 3
•
can graph functions of the form y=a(bx +c) +d where n= 1, 2
•
given the graph of a simple function, can draw a graph of the inverse function
•
can algebraically find the inverse of simple polynomial functions
•
sketches the graph of y = aln(bx+c)+d
•
uses the logarithm laws for products, quotients and powers to solve simple logarithmic equations
•
determines simple composite functions.
n
-n
Rating ‘B’
In addition to the standards for a C rating, a student:
•
writes the equation of a function that is either a simple translation or reflection of any standard
function of this course
•
sketches a graph of a function and uses it to determine whether or not an inverse exists, giving
reasons
•
given the graph of a more complex function, can draw a graph of its inverse
•
can algebraically find the inverse of hyperbolic functions
•
determines more complex composite functions.
Rating ‘A’
In addition to the standards for a C and B rating, a student:
•
writes the equation of a function that is a translation, reflection and/or dilation of any standard
function of this course
•
can algebraically find the inverse of exponential and logarithmic functions
•
solves more complex logarithmic equations.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
11
Mathematics Methods
TQA Level 3
C RITERION 5
D EMONSTRATE AN U NDERSTANDING OF C IRCULAR F UNCTIONS
Rating ‘C’
A student:
•
identifies and uses correct definitions, special relationships and symmetrical properties of the
trigonometric ratios in solving problems
•
calculates exact values in both radians and degrees
•
can graph trigonometric functions of the form y = asin(bx), y = acos(bx) and y = atan(bx)
•
solves trigonometric equations of the form sin(ax) = b, cos(ax) = b and tan(ax) = b in a given
domain
•
finds the equations of trigonometric functions of the form y = asin(x)+b, y = acos(x)+b and
y = atan(x)+b from graphical or worded information
•
uses the angle sum and difference identities in routine problems
•
draws conclusions about real world applications of circular functions.
Rating ‘B’
In addition to the standards for a C rating, a student:
•
can graph trigonometrical functions of the form y = asin(bx+c), y = acos(bx+c) and y = atan(bx+c)
•
solves trigonometric equations of the form sin(ax+b) = c, cos(ax+b) = c and tan(ax+b) = c in a
given domain
•
finds the equations of functions of the form y = asin(bx)+c, y = acos(bx)+c and y = atan(bx)+c
from graphical or worded information.
Rating ‘A’
In addition to the standards for a C and B rating, a student:
•
can graph trigonometric functions of the form y = asin(bx+c)+d, y = acos(bx+c)+d and y =
atan(bx+c)+d
•
solves trigonometric equations of the form sin(ax+b) = kcos(ax+b) in a given domain
•
finds the equations of functions of the form y = asin(bx+c)+d, y = acos(bx+c)+d and y =
atan(bx+c)+d from graphical or worded information
•
applies the angle sum and difference identities in more complex problems.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
12
Mathematics Methods
TQA Level 3
C RITERION 6
U SE D IFFERENTIAL C ALCULUS IN THE S TUDY OF F UNCTIONS
Rating ‘C’
A student:
•
sketches gradient functions for continuous functions and states the domain
•
applies differentiation rules to differentiate x , e , lnx, sinx, cosx, tanx and linear combinations
thereof
•
finds the equations of the tangent and/or normal to a curve given the x coordinate of a point on
the curve
•
finds and classifies stationary points of known functions.
n
x
Rating ‘B’
In addition to the standards for a C rating, a student:
•
applies differentiation rules to differentiate products, quotients and simple composite functions
•
differentiates a simple linear or quadratic expression from first principles
•
finds the equations of the tangent and/or normal to a curve given its slope
•
finds and justify stationary points of known functions and interpret the results.
Rating ‘A’
In addition to the standards for a C and B rating, a student:
•
sketches gradient functions for simple discontinuous functions and states the domain
•
applies differentiation rules to differentiate functions that involve combinations of the
differentiation rules
•
finds the equation of a curve given its slope (x, y) and a point on the curve
•
establishes a function in one variable using given information, finds and justifies its stationary
points and interprets the results.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
13
Mathematics Methods
TQA Level 3
C RITERION 7
U SE I NTEGRAL C ALCULUS IN THE S TUDY OF F UNCTIONS
Rating ‘C’
A student:
•
determines the indefinite integral of a polynomial function in expanded form and verifies by
differentiation
•
determines the indefinite integral of (ax+b) , where n is a positive integer, and e +b, and verifies
by differentiation
•
sketches simple polynomial and simple exponential functions, and calculates the area between
the graph and the x-axis by evaluating one definite integral
•
determines the distance travelled by an object moving in a straight line in one direction by
integrating a simple polynomial velocity function in time t
•
evaluates the area between the graphs of two polynomial functions with two points of intersection
that are given.
n
ax
Rating ‘B’
In addition to the standards for a C rating, a student:
•
integrates polynomial functions in factorised form
•
determines the indefinite integral of (ax+b) , where n is any integer, e +b, sin(ax+b), cos (ax+b)
or a linear combination of these functions
•
sketches more complex polynomial and exponential functions and simple trigonometric functions
and calculates the area between the graph and the x-axis by evaluating one or more definite
integrals
•
determines the distance travelled by an object moving in a straight line with one or two changes in
direction by integrating a polynomial velocity function in time t
•
evaluates the area between the graphs of two polynomial functions with two points of intersection
that are easily determined.
n
ax
Rating ‘A’
In addition to the standards for a C and B rating, a student:
•
sketches hyperbolic or truncus functions and calculates the area between the graph and the xaxis by evaluating one or more definite integrals
•
evaluates the area between the graphs of any two functions with two or more points of
intersection that are easily determined.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
14
Mathematics Methods
TQA Level 3
C RITERION 8
D EMONSTRATE AN U NDERSTANDING OF B INOMIAL , H YPERGEOMETRIC AND N ORMAL
P ROBABILITY D ISTRIBUTIONS
Rating ‘C’
A student:
•
identifies and defines randomness, discrete variables and continuous variables
•
constructs discrete probability distributions using tree diagrams and tables
•
calculates expected value of a discrete random variable
•
describes the characteristics of normal, binomial and hypergeometric distributions and determines
which distribution is appropriate in a given situation
•
calculates a binomial probability for a single outcome
•
calculates expected value and variance of a binomial random variable
•
determines probabilities in a normal distribution.
Rating ‘B’
In addition to the standards for a C rating, a student:
•
calculates the variance or standard deviation of a discrete random variable
•
calculates a binomial probability for multiple outcomes
•
calculates a hypergeometric probability for a single outcome
•
calculates expected value and variance of a hypergeometric random variable
•
determines quantities in a normal distribution.
Rating ‘A’
In addition to the standards for a C and B rating, a student:
•
calculates a hypergeometric probability for multiple outcomes
•
uses a binomial approximation for a hypergeometric distribution with justification
•
determines mean and standard deviation in a normal distribution.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
15
Mathematics Methods
TQA Level 3
QUALIFICATIONS AVAILABLE
Mathematics Methods,TQA 3 (with the award of):
EXCEPTIONAL ACHIEVEMENT
HIGH ACHIEVEMENT
COMMENDABLE ACHIEVEMENT
SATISFACTORY ACHIEVEMENT
PRELIMINARY ACHIEVEMENT
AWARD REQUIREMENTS
The final award will be determined by the Tasmanian Qualifications Authority from the 12 rating (7
ratings from the internal assessment and 5 ratings from the external assessment).
EXCEPTIONAL ACHIEVEMENT (EA)
11 ‘A’, 2 ‘B’ ratings (4 ‘A’, 1 ‘B’ from external assessment)
HIGH ACHIEVEMENT (HA)
5 ‘A’, 5 ‘B’, 3 ‘C’ ratings (2 ‘A’, 2 ‘B’ and 1 ‘C’ from external assessment)
COMMENDABLE ACHIEVEMENT (CA)
7‘B’, 5 ‘C’ ratings (2 ‘B’, 2 ‘C’ from external assessment)
SATISFACTORY ACHIEVEMENT (SA)
11 ‘C’ ratings (3 ‘C’ from external assessment)
PRELIMINARY ACHIEVEMENT (PA)
6 ‘C’ ratings
A student 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.
COURSE EVALUATION
Courses are accredited for a specific period of time (up to five years) and they are evaluated in the
year prior to the expiry of accreditation.
As well, anyone may request a review of a particular aspect of an accredited course throughout the
period of accreditation. Such requests for amendment will be considered in terms of the likely
improvements to the outcomes for students and the possible consequences for delivery of the
course.
The TQA can evaluate the need and appropriateness of an accredited course at any point throughout
the period of accreditation.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
16
Mathematics Methods
TQA Level 3
EXPECTATIONS DEFINED BY NATIONAL STANDARDS IN CONTENT STATEMENTS
DEVELOPED BY ACARA
The statements in this section, taken from Australian Senior Secondary Curriculum: Mathematical
Methods 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.
Unit 1 – Topic 1: Functions and Graphs
Inverse proportion:
•
examine examples of inverse proportion (ACMMM012)
•
recognise features of the graphs of y =
1
a
and y =
, including their hyperbolic shapes, and
x
x −b
their asymptotes (ACMMM013).
Powers and polynomials:
n
•
recognise features of the graphs of y=x for n N, n=−1 and n=½, including shape, and behaviour
as x→∞ and x→−∞ (ACMMM014)
•
identify the coefficients and the degree of a polynomial (ACMMM015)
•
expand quadratic and cubic polynomials from factors (ACMMM016)
•
recognise features of the graphs of y=x , y=a(x−b) +c and y=k(x−a)(x−b)(x−c), including shape,
intercepts and behaviour as x→∞ and x→−∞ (ACMMM017)
•
factorise cubic polynomials in cases where a linear factor is easily obtained (ACMMM018)
•
solve cubic equations using technology, and algebraically in cases where a linear factor is easily
obtained (ACMMM019).
3
3
Graphs of relations:
2
2
2
2
2
2
•
recognise features of the graphs of x +y =r and (x−a) +(y−b) =r , including their circular shapes,
their centres and their radii (ACMMM020)
•
recognise features of the graph of y =x including its parabolic shape and its axis of symmetry
(ACMMM021).
2
Functions:
•
understand the concept of a function as a mapping between sets, and as a rule or a formula that
defines one variable quantity in terms of another (ACMMM022)
•
use function notation, domain and range, independent and dependent variables (ACMMM023)
•
understand the concept of the graph of a function (ACMMM024)
•
examine translations and the graphs of y=f(x)+a and y=f(x+b) (ACMMM025)
•
examine dilations and the graphs of y=cf(x) and y=f(kx) (ACMMM026)
•
recognise the distinction between functions and relations, and the vertical line test (ACMMM027).
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
17
Mathematics Methods
TQA Level 3
Unit 1 – Topic 2: Trigonometric Functions
Circular measure and radian measure:
•
define and use radian measure and understand its relationship with degree measure
(ACMMM032).
Trigonometric functions:
•
understand the unit circle definition of cosθ, sinθ and tanθ and periodicity using radians
(ACMMM034)
•
recognise the exact values of sinθ, cosθ and tanθ at integer multiples of
•
recognise the graphs of y=sinx, y=cosx, and y=tanx on extended domains (ACMMM036)
•
examine amplitude changes and the graphs of y=asinx and y=acosx (ACMMM037)
•
examine period changes and the graphs of y=sinbx, y=cosbx, and y=tan bx (ACMMM038)
•
examine phase changes and the graphs of y=sin(x+c), y=cos(x+c) and (ACMMM039)
•
!
"
π$
π%
y=tan (x+c) and the relationships sin # x + & =cosx and cos $ x − ' =sinx (ACMMM040)
2%
2&
"
#
•
prove and apply the angle sum and difference identities (ACMMM041)
•
identify contexts suitable for modelling by trigonometric functions and use them to solve practical
problems (ACMMM042)
•
solve equations involving trigonometric functions using technology, and algebraically in simple
cases (ACMMM043).
π
π
and
(ACMMM035)
6
4
Unit 1 – Topic 3: Counting and Probability
Combinations:
•
recognise the numbers
()
n
r
n
as binomial coefficients, (as coefficients in the expansion of (x+y) )
(ACMMM047)
Unit 2 – Topic 1: Exponential Functions
Indices and the index laws:
•
review indices (including fractional indices) and the index laws (ACMMM061)
•
use radicals and convert to and from fractional indices (ACMMM062)
•
understand and use scientific notation and significant figures (ACMMM063).
Exponential functions:
•
establish and use the algebraic properties of exponential functions (ACMMM064)
•
recognise the qualitative features of the graph of y=a (a>0) including asymptotes, and of its
translations (y=ax+b and y=ax+c) (ACMMM065)
•
identify contexts suitable for modelling by exponential functions and use them to solve practical
problems (ACMMM066)
•
solve equations involving exponential functions using technology, and algebraically in simple
cases (ACMMM067).
x
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
18
Mathematics Methods
TQA Level 3
Unit 2 – Topic 3: Introduction to Differential Calculus
Rates of change:
•
interpret the difference quotient
f ( x + h )− f ( x )
as the average rate of change of a function f
h
(ACMMM077)
•
use the Leibniz notation δx and δy for changes or increments in the variables x and y
(ACMMM078)
•
use the notation δyδx for the difference quotient
•
interpret the ratios
f ( x + h )− f ( x )
where y=f(x) (ACMMM079)
h
f ( x + h )− f ( x )
and δyδx as the slope or gradient of a chord or secant of the
h
graph of y=f(x) (ACMMM080).
The concept of the derivative:
f ( x + h )− f ( x )
as h→0 as an informal
h
introduction to the concept of a limit (ACMMM081)
•
examine the behaviour of the difference quotient
•
define the derivative f'(x) as limh→0
•
use the Leibniz notation for the derivative:
f ( x + h )− f ( x )
(ACMMM082)
h
dy
dy
=limδx→0δyδx and the correspondence
=f'(x)
dx
dx
where y=f(x) (ACMMM083)
•
interpret the derivative as the instantaneous rate of change (ACMMM084)
•
interpret the derivative as the slope or gradient of a tangent line of the graph of y=f(x)
(ACMMM085).
Computation of derivatives:
•
estimate numerically the value of a derivative, for simple power functions (ACMMM086)
•
examine examples of variable rates of change of non-linear functions (ACMMM087)
•
establish the formula
d
n
( x n ) = nx n −1 for positive integers n by expanding (x+h) or by factorising
dx
n
n
(x+h) −x (ACMMM088).
Properties of derivatives:
•
understand the concept of the derivative as a function (ACMMM089)
•
recognise and use linearity properties of the derivative (ACMMM090)
•
calculate derivatives of polynomials and other linear combinations of power functions
(ACMMM091).
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
19
Mathematics Methods
TQA Level 3
Applications of derivatives:
•
find instantaneous rates of change (ACMMM092)
•
find the slope of a tangent and the equation of the tangent (ACMMM093)
•
construct and interpret position-time graphs, with velocity as the slope of the tangent
(ACMMM094)
•
sketch curves associated with simple polynomials; find stationary points, and local and global
maxima and minima; and examine behaviour as x→∞ and x→−∞ (ACMMM095)
•
solve optimisation problems arising in a variety of contexts involving simple polynomials on finite
interval domains (ACMMM096).
Anti-derivatives:
•
calculate anti-derivatives of polynomial functions and apply to solving simple problems involving
motion in a straight line (ACMMM097).
Unit 3 – Topic 1: Further Differentiation and Applications
Exponential functions:
a h −1
as h→0 using technology, for various values of a >0 (ACMMM098)
h
•
estimate the limit of
•
recognise that e is the unique number a for which the above limit is 1 (ACMMM099)
•
establish and use the formula
•
use exponential functions and their derivatives to solve practical problems (ACMMM101).
d
= (e x ) = e x (ACMMM100)
dx
Trigonometric functions:
d
d
(sinx)=cosx, and
(cosx)=−sinx by numerical estimations of the
dx
dx
limits and informal proofs based on geometric constructions (ACMMM102)
•
establish the formulas
•
use trigonometric functions and their derivatives to solve practical problems (ACMMM103).
Differentiation rules:
•
understand and use the product and quotient rules (ACMMM104)
•
understand the notion of composition of functions and use the chain rule for determining the
derivatives of composite functions (ACMMM105)
•
apply the product, quotient and chain rule to differentiate functions such as xe , tanx,
x
−x
xsinx, e sinx and f(ax+b) (ACMMM106).
Tasmanian Qualifications Authority
Version X
1
xn
,
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
20
Mathematics Methods
TQA Level 3
The second derivative and applications of differentiation:
dy
×δx to estimate the change in the dependent variable y
dx
•
use the increments formula: δy
•
resulting from changes in the independent variable x (ACMMM107)
•
understand the concept of the second derivative as the rate of change of the first derivative
function (ACMMM108)
•
recognise acceleration as the second derivative of position with respect to time (ACMMM109)
•
understand and use the second derivative test for finding local maxima and minima (ACMMM111)
•
sketch the graph of a function using first and second derivatives to locate stationary points and
points of inflection (ACMMM112)
•
solve optimisation problems from a wide variety of fields using first and second derivatives
(ACMMM113).
Unit 3 – Topic 2: Integrals
Anti-differentiation:
•
recognise anti-differentiation as the reverse of differentiation (ACMMM114)
•
use the notation ∫f(x)dx for anti-derivatives or indefinite integrals (ACMMM115)
•
establish and use the formula ∫x dx=
•
establish and use the formula ∫e dx=e +c (ACMMM117)
•
establish and use the formulas ∫sinxdx=−cosx+c and ∫cosxdx=sinx+c (ACMMM118)
•
recognise and use linearity of anti-differentiation (ACMMM119)
•
determine indefinite integrals of the form ∫f(ax+b)dx (ACMMM120)
•
identify families of curves with the same derivative function (ACMMM121)
•
determine f(x), given f' (x) and an initial condition f(a)=b (ACMMM122)
•
determine displacement given velocity in linear motion problems (ACMMM123).
n
x
1 n +1
x + c for n≠−1 (ACMMM116)
n +1
x
Definite integrals:
•
examine the area problem, and use sums of the form ∑if(xi) δxi to estimate the area under the
curve y=f(x) (ACMMM124)
•
interpret the definite integral
•
recognise the definite integral
•
interpret
•
recognise and use the additivity and linearity of definite integrals (ACMMM128).
∫
a
b
∫
a
b
f ( x )dx as area under the curve y=f(x) if f(x)>0 (ACMMM125)
∫
a
b
f ( x )dx as a limit of sums of the form ∑if(xi) δxi (ACMMM126)
f ( x )dx as a sum of signed areas (ACMMM127)
Fundamental theorem:
•
understand the concept of the signed area function F(x)=
•
understand and use the theorem: F'(x)=
•
understand the formula
∫
a
b
∫
x
x
f (t )dt (ACMMM129)
x
d
( ∫ f (t )dt ) =f(x) (......) (ACMMM130)
dx x
f ( x )dx =F(b)−F(a) and use it to calculate definite integrals
(ACMMM131).
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
21
Mathematics Methods
TQA Level 3
Applications of integration:
•
calculate the area under a curve (ACMMM132)
•
calculate the area between curves in simple cases (ACMMM134)
•
determine positions given acceleration and initial values of position and velocity (ACMMM135).
Unit 3 – Topic 3: Discrete Random Variables
General discrete random variables:
•
understand the concepts of a discrete random variable and its associated probability function, and
their use in modelling data (ACMMM136)
•
use relative frequencies obtained from data to obtain point estimates of probabilities associated
with a discrete random variable (ACMMM137)
•
recognise uniform discrete random variables and use them to model random phenomena with
equally likely outcomes (ACMMM138)
•
examine simple examples of non-uniform discrete random variables (ACMMM139)
•
recognise the mean or expected value of a discrete random variable as a measurement of centre,
and evaluate it in simple cases (ACMMM140)
•
recognise the variance and standard deviation of a discrete random variable as a measures of
spread, and evaluate them in simple cases (ACMMM141)
•
use discrete random variables and associated probabilities to solve practical problems
(ACMMM142).
Bernoulli distributions:
•
use a Bernoulli random variable as a model for two-outcome situations (ACMMM143)
•
identify contexts suitable for modelling by Bernoulli random variables (ACMMM144)
•
recognise the mean p and variance p(1−p) of the Bernoulli distribution with parameter p
(ACMMM145)
•
use Bernoulli random variables and associated probabilities to model data and solve practical
problems (ACMMM146).
Binomial distributions:
•
understand the concepts of Bernoulli trials and the concept of a binomial random variable as the
number of ‘successes’ in n independent Bernoulli trials, with the same probability of success p in
each trial (ACMMM147)
•
identify contexts suitable for modelling by binomial random variables (ACMMM148)
•
determine and use the probabilities P ( X = r ) =
( )p
n
r
r
(1− p ) n −r associated with the binomial
distribution with parameters n and p ; note the mean np and variance np (1− p ) of a binomial
distribution (ACMMM149)
•
use binomial distributions and associated probabilities to solve practical problems (ACMMM150).
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
22
Mathematics Methods
TQA Level 3
Unit 4 – Topic 1: The Logarithmic Function
Logarithmic functions:
x
•
define logarithms as indices: a =b is equivalent to x=logab , i.e. alogab=b (ACMMM151)
•
establish and use the algebraic properties of logarithms (ACMMM152)
•
recognise the inverse relationship between logarithms and exponentials: y=a is equivalent to
x=logay (ACMMM153)
•
interpret and use logarithmic scales such as decibels in acoustics, the Richter Scale for
earthquake magnitude, octaves in music, pH in chemistry (ACMMM154)
•
solve equations involving indices using logarithms (ACMMM155)
•
recognise the qualitative features of the graph of y=logax (a>1) including asymptotes, and of its
translations y=logax+b and y=loga(x+c) (ACMMM156)
•
solve simple equations involving logarithmic functions algebraically and graphically (ACMMM157)
•
identify contexts suitable for modelling by logarithmic functions and use them to solve practical
problems (ACMMM158).
x
Calculus of logarithmic functions:
•
define the natural logarithm lnx=logex (ACMMM159)
•
recognise and use the inverse relationship of the functions y=ex and y=lnx (ACMMM160)
•
establish and use the formula
establish and use the formula
•
d
1
(ln x )= (ACMMM161)
dx
x
1
∫ x dx = ln x + c , for x>0 (ACMMM162)
use logarithmic functions and their derivatives to solve practical problems (ACMMM163).
Unit 4 - Topic 2: Continuous Random Variables and the Normal Distribution
General continuous random variables:
•
use relative frequencies and histograms obtained from data to estimate probabilities associated
with a continuous random variable (ACMMM164)
•
understand the concepts of a probability density function, cumulative distribution function, and
probabilities associated with a continuous random variable given by integrals; examine simple
types of continuous random variables and use them in appropriate contexts (ACMMM165)
•
recognise the expected value, variance and standard deviation of a continuous random variable
and evaluate them in simple cases (ACMMM166)
•
understand the effects of linear changes of scale and origin on the mean and the standard
deviation (ACMMM167).
Normal distributions:
•
identify contexts such as naturally occurring variation that are suitable for modelling by normal
random variables (ACMMM168)
•
recognise features of the graph of the probability density function of the normal distribution with
mean µ and standard deviation σ and the use of the standard normal distribution (ACMMM169)
•
calculate probabilities and quantiles associated with a given normal distribution using technology,
and use these to solve practical problems (ACMMM170).
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013
23
Mathematics Methods
TQA Level 3
ACCREDITATION
st
The accreditation period for this course is from 1 January 2014 until xxxxxxx
VERSION HISTORY
st
Version 1 – Accredited 1 October 2008 for use in 2009 – 2013.
Version 2 - xxxx 2013. Inclusion of new criterion that directly relates to the ACARA Mathematical
Methods standards of Reasoning and Communication; links made to content statements in ACARA
Mathematical Methods.
Tasmanian Qualifications Authority
Version X
Period of Accreditation: 1/1/2014 – XXXX
Date of Publishing: 27 May 2013