Download Draft Essential Mathematics Subject Outline

DRAFT FOR ONLINE CONSULTATION
Essential Mathematics
Subject Outline
Stage 1 and Stage 2
Version 1.0 – Draft for Online Consultation
Ref: A374762
11 March 2015 – 17 April 2015
CONTENTS
Introduction ................................................................................................................... 3
Subject Description .............................................................................................. 3
Mathematical Options ........................................................................................... 4
Capabilities ........................................................................................................... 5
Aboriginal and Torres Strait Islander Knowledge, Cultures, and Perspectives ...... 7
SACE Numeracy Requirement ............................................................................. 8
Stage 1 Essential Mathematics
Learning Scope and Requirements ............................................................................. 10
Learning Requirements ...................................................................................... 10
Content ............................................................................................................... 10
Assessment Scope and Requirements ........................................................................ 37
Evidence of Learning .......................................................................................... 37
Assessment Design Criteria................................................................................ 38
School Assessment ............................................................................................ 39
Performance Standards ...................................................................................... 41
Assessment Integrity .......................................................................................... 43
Support Materials ........................................................................................................ 43
Subject-specific Advice ....................................................................................... 43
Advice on Ethical Study and Research ............................................................... 43
Stage 2 Essential Mathematics
Learning Scope and Requirements ............................................................................. 45
Learning Requirements ...................................................................................... 45
Content ............................................................................................................... 45
Assessment Scope and Requirements ........................................................................ 64
Evidence of Learning .......................................................................................... 64
Assessment Design Criteria................................................................................ 65
School Assessment ............................................................................................ 66
External Assessment .......................................................................................... 67
Performance Standards ...................................................................................... 69
Assessment Integrity .......................................................................................... 71
Support Materials ........................................................................................................ 71
Subject-specific Advice ....................................................................................... 71
Advice on Ethical Study and Research ............................................................... 71
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
2 of 71
INTRODUCTION
SUBJECT DESCRIPTION
Essential Mathematics is a 10-credit subject or a 20-credit subject at Stage 1, and a
20-credit subject at Stage 2.
Essential Mathematics offers senior secondary students the opportunity to extend
their mathematical skills in ways that apply to practical problem solving in everyday
and workplace contexts. Students apply their mathematics to diverse settings,
including everyday calculations, financial management, business applications,
measurement and geometry, and statistics in social contexts.
In Essential Mathematics there is an emphasis on developing students’
computational skills and expanding their ability to apply their mathematical skills in
flexible and resourceful ways.
This subject is intended for students planning to pursue a career in a range of trades
or vocations.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
3 of 71
MATHEMATICAL OPTIONS
The diagram below represents the possible mathematical options that students might
select as Stage 1 and Stage 2 subjects.
Solid arrows indicate the mathematical options that lead to completion of each
subject at Stage 2. Dotted arrows indicate a pathway that may provide sufficient
preparation for an alternative Stage 2 mathematics subject.
SM
MM
GM
EM
Specialist Mathematics
Mathematical Methods
General Mathematics
Essential Mathematics
Year 10
10
and
10A optional*
10
10
Stage 1
SM**
MM
GM
EM
Semester 1
SM**
MM
GM
EM
Semester 2
Stage 2
Semester 1
SM**
MM
GM
EM
Semester 2
Notes: * Although it is advantageous for students to study Australian Curriculum 10 and 10A
in Year 10, the 10A curriculum per se is not a prerequisite for the study of Specialist
Mathematics and Mathematical Methods. The essential aspects of 10A are included
into the curriculum for Specialist Mathematics and Mathematical Methods.
** Mathematical Methods can be studied as a single subject; however, Specialist
Mathematics is designed to be studied together with Mathematical Methods.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
4 of 71
CAPABILITIES
The capabilities connect student learning within and across subjects in a range of
contexts. They include essential knowledge and skills that enable people to act in
effective and successful ways.
The SACE identifies seven capabilities. They are:
 literacy
 numeracy
 information and communication technology capability
 critical and creative thinking
 personal and social capability
 ethical understanding
 intercultural understanding.
Literacy
In this subject students develop their literacy capability by, for example:
 communicating mathematical reasoning and ideas for different purposes,
using appropriate language and representations, such as symbols, equations,
tables, and graphs
 interpreting and responding to appropriate mathematical language and
representations
 analysing information and explaining mathematical results.
Mathematics provides a specialised language to describe and analyse phenomena. It
provides a rich context for students to extend their ability to read, write, visualise, and
talk about situations that involve investigating and solving problems.
Students apply and extend their literacy skills and strategies by using verbal, graphic,
numerical, and symbolic forms of representing problems and displaying statistical
information. Students learn to communicate their findings in different ways, using
different systems of representation.
Numeracy
Being numerate is essential for participating in contemporary society. Students need
to reason, calculate, and communicate to solve problems. Through the study of
mathematics, they understand and use skills, concepts, and technologies in a range
of contexts that can be applied to:
 using measurement in this physical world
 gathering, representing, interpreting, and analysing data
 using spatial sense and geometric reasoning
 investigating chance processes
 using number, number patterns, and relationships between numbers
 working with graphical, statistical and algebraic representations, and other
mathematical models.
Information and communication technology capability
In this subject students develop their information and communication technology
(ICT) capability by, for example:
 understanding the role of electronic technology when using mathematics
 making informed decisions about the use of electronic technology
 understanding the mathematics involved in computations carried out using
technologies, so that reasonable interpretations can be made of the results.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
5 of 71
Students extend their skills in using technology effectively and processing large
amounts of quantitative information.
Students use ICT to extend their theoretical mathematical understanding and apply
mathematical knowledge to a range of problems. They use software relevant for
study and/or workplace contexts. This may include tools for statistical analysis,
algorithm generation, data representation and manipulation, and complex calculation.
They use digital tools to make connections between mathematical theory, practice
and application; for example, to use data, address problems, and operate systems in
particular situations.
Critical and creative thinking
In this subject students develop critical and creative thinking by, for example:
 building confidence in applying knowledge and problem-solving skills in a
range of mathematical contexts
 developing mathematical reasoning skills to think logically and make sense of
the world
 understanding how to make and test projections from mathematical models
 interpreting results and drawing appropriate conclusions
 reflecting on the effectiveness of mathematical models, including the
recognition of assumptions, strengths, and limitations
 using mathematics to solve practical problems and as a tool for learning
 making connections between concrete, pictorial, symbolic, verbal, written, and
mental representations of mathematical ideas
 thinking abstractly, making and testing conjectures, and explaining processes.
Problem-solving in mathematics builds students’ depth of conceptual understanding
and supports development of critical and creative thinking. Learning through
problem-solving helps students when they encounter new situations. They develop
their creative and critical thinking capability by listening, discussing, conjecturing, and
testing different strategies. They learn the importance of self-correction in building
their conceptual understanding and mathematical skills.
Personal and social capability
In this subject students develop their personal and social capability by, for example:
 arriving at a sense of self as a capable and confident user of mathematics
through expressing and presenting ideas in a variety of ways
 appreciating the usefulness of mathematical skills for life and career
opportunities and achievements
 understanding the contribution of mathematics and mathematicians to society.
The elements of personal and social competence relevant to mathematics include
the application of mathematical skills for informed decision-making, active citizenship,
and effective self-management. Students build their personal and social competence
in mathematics through setting and monitoring personal and academic goals, taking
initiative, and building adaptability, communication, and teamwork.
Students use mathematics as a tool to solve problems they encounter in their
personal and working lives. They acquire a repertoire of strategies and build the
confidence needed to:
 meet the challenges and innovations of a changing world
 be the designers and innovators of the future, and leaders in their fields.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
6 of 71
Ethical understanding
In this subject students develop their ethical understanding by, for example:
 gaining knowledge and understanding of ways in which mathematics can be
used to support an argument or point of view
 examining critically ways in which the media present particular information
and perspectives
 sharing their learning and valuing the skills of others
 considering the social consequences of making decisions based on
mathematical results
 acknowledging and learning from errors rather than denying findings and/or
evidence.
Areas of ethical understanding relevant to mathematics include issues associated
with ethical decision-making and working collaboratively as part of students’
mathematically related explorations. They develop ethical understanding in
mathematics through considering social responsibility in ethical dilemmas that may
arise when solving problems in personal, social, community, and/or workplace
contexts.
Intercultural understanding
In this subject students develop their intercultural understanding by, for example:
 understanding mathematics as a body of knowledge that uses universal
symbols that have their origins in many cultures
 understanding how mathematics assists individuals, groups, and societies to
operate successfully across cultures in the global, knowledge-based
economy.
Mathematics is a shared language that crosses borders and cultures, and is
understood and used globally.
Students read about, represent, view, listen to, and discuss mathematical ideas.
They become aware of the historical threads from different cultures that have led to
the current bodies of mathematical knowledge. These opportunities allow students to
create links between their own language and ideas and the formal language and
symbols of mathematics.
ABORIGINAL AND TORRES STRAIT ISLANDER KNOWLEDGE, CULTURES, AND
PERSPECTIVES
In partnership with Aboriginal and Torres Strait Islander communities, and schools
and school sectors, the SACE Board of South Australia supports the development of
high-quality learning and assessment design that respects the diverse knowledge,
cultures, and perspectives of Indigenous Australians.
The SACE Board encourages teachers to include Aboriginal and Torres Strait
Islander knowledge and perspectives in the design, delivery, and assessment of
teaching and learning programs by:
 providing opportunities in SACE subjects for students to learn about
Aboriginal and Torres Strait Islander histories, cultures, and contemporary
experiences
 recognising and respecting the significant contribution of Aboriginal and
Torres Strait Islander peoples to Australian society
 drawing students’ attention to the value of Aboriginal and Torres Strait
Islander knowledge and perspectives from the past and the present
 promoting the use of culturally appropriate protocols when engaging with and
learning from Aboriginal and Torres Strait Islander peoples and communities.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
7 of 71
SACE NUMERACY REQUIREMENT
Completion of 10 or 20 credits of Stage 1 Essential Mathematics with a C grade or
better, or 20 credits of Stage 2 Essential Mathematics with a C- grade or better, will
meet the numeracy requirement of the SACE.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
8 of 71
Stage 1
Essential Mathematics
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
9 of 71
LEARNING SCOPE AND REQUIREMENTS
The learning requirements summarise the knowledge, skills, and understanding that
students are expected to develop and demonstrate through their learning in this
subject.
In this subject, students are expected to:
1. understand mathematical concepts, demonstrate mathematical skills, and apply
mathematical techniques
2. develop skills in gathering, representing, analysing, and interpreting data relevant
to everyday situations in a variety of contexts
3. use numeracy skills to investigate and solve practical problems in familiar and
some unfamiliar everyday contexts
4. interpret results, draw conclusions, and reflect on the reasonableness of solutions
in context
5. make discerning use of electronic technology
6. communicate mathematically and present mathematical information in a variety of
ways.
CONTENT
Stage 1 Essential Mathematics may be studied as a 10-credit subject or a 20-credit
subject.
In Stage 1 Essential Mathematics students extend their mathematical skills in ways
that apply to practical problem solving in everyday and workplace contexts. A
problem-based approach is integral to the development of mathematical skills and
associated key ideas in this subject.
Topics studied cover a range of applications of mathematics, including: general
calculation, measurement and geometry, money management, and statistics.
Throughout Essential Mathematics there is an emphasis on extending students’
computational skills and expanding their ability to apply their mathematical skills in
flexible and resourceful ways.
Stage 1 Essential Mathematics consists of the following list of six topics:
Topic 1: Calculations, Time, and Ratio
Topic 2: Earning and Spending
Topic 3: Geometry
Topic 4: Data in Context
Topic 5: Measurement
Topic 6: Investing
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
10 of 71
Programming
For a 10-credit subject students study three topics from the list.
For a 20-credit subject students study all six topics from the list.
The topics selected can be sequenced and structured to suit individual cohorts of
students. The suggested order of the topics provided in the list is a guide only.
Each topic consists of a number of subtopics. These are presented in the subject
outline in two columns as a series of ‘key questions and key concepts’ side by side
with ‘considerations for developing teaching and learning strategies’.
The ‘key questions and key concepts’ cover the prescribed content for teaching,
learning, and assessment in this subject. The ‘considerations for developing teaching
and learning strategies’ are provided as a guide only.
A problem-based approach is integral to the development of the computational
models and associated key concepts in each topic. Through key questions students
deepen their understanding of concepts and processes that relate to the
mathematical models required to address the problems posed.
The ‘considerations for developing teaching and learning strategies’ present
guidelines for sequencing the development of ideas. They also give an indication of
the depth of treatment and emphases required.
In the ‘considerations for developing teaching and learning strategies’, the term trade
is used to suggest a context in a generic sense to cover a range of industry areas
and occupations such as automotive, building and construction, electrical,
hairdressing, hospitality, nursing and community services, plumbing, and retail.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
11 of 71
Topic 1: Calculations, Time, and Ratio
Students extend their proficiency with calculations required for everyday living.
Computational skills are practised within contexts that are relevant to the students’
interests. To develop a better understanding of the mathematical processes involved, the
initial focus of the learning in this topic is through carrying out calculations by hand. The
discerning use of electronic technology is introduced to enable more complex problems to
be solved efficiently.
Subtopic 1.1: Calculations
Key Questions and Key Concepts
How can basic mathematics skills help us in
our everyday lives?
 solve practical problems requiring basic
number operations
 apply arithmetic operations according to
their correct order
How can we determine the reasonableness
of our calculations?
 use leading-digit approximation to obtain
estimates of calculations
 apply approximation strategies
 ascertain the reasonableness of answers
to arithmetic calculations
 check results of calculations for accuracy
How can a calculator be used to carry out
multi-step calculations?
How many decimal places are appropriate in
an answer?
 recognise the significance of place value
after the decimal point
 evaluate decimal fractions to the required
number of decimal places
 round up or round down numbers to the
required number of decimal places
 use scientific notation to display small and
large numbers
 multiplication and division by multiples of
10
Considerations for Developing Teaching and
Learning Strategies
In everyday contexts, students solve a range of problems
using addition, subtraction, multiplication, and division.
These problems include fractions, decimals, finding the
square root of positive whole numbers that are perfect
squares, and writing numbers in index form and
expanding indices to find the basic numeral. The focus is
on carrying out these computations without the use of
electronic technology. For certain trades students need
to be familiar with the use of long division. Problems
could include checking the calculations for an electricity
bill, determining the cost of a shopping list, or building an
object. Revision of the order of operations for solving
problems when more than one mathematical procedure
is involved in the solution may be useful.
Discuss the importance of questioning and detecting
answers that are not reasonable. Investigate strategies
that enable appropriate approximations for answers to
arithmetic calculations. Use leading digit approximations
(e.g. 41 x 21 is approximately 40 x 20), sensible rounding
of figures (e.g. estimate a shopping bill where the cost of
a number of groceries purchased are rounded to the
nearest $5 or $10 figure or grouped into approximately
these sums) or using the distributive law (23 X 7 = 20 X 7
+ 3 X 7). Students become discerning about the
reasonableness of their answer in the context of the
problem.
Discuss the discerning use of technology.
Students become familiar with using their calculator to
enter and calculate multi-step arithmetic problems.
These calculations include the use of technology to solve
problems involving fractions, decimals, finding the square
root of positive whole numbers, and carrying out
computations with indices.
Consider the place value of figures after a decimal point.
Use technology to convert fractions to decimals, and to
round to a required number of decimal places.
Emphasise the rules for rounding numbers both up and
down, including to tens, tenths, and hundredths. Provide
practical examples to reinforce the need for rounding of
values to specified decimal places. Review the use of
scientific notation to display very small and very large
numbers and multiplication and division by multiples of
ten.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
12 of 71
Subtopic 1.2: Time and Rates
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
Estimate and calculate time:
 use units of time, conversions between
units, fractional, digital, and decimal
representations
 represent time using 12-hour and 24-hour
clocks
 calculate time intervals, such as time
between, time ahead, time behind
Discuss units of time (why is a metric system not used
for time?) and the different representations of time
including fractional, digital, and decimal. Carry out
conversions between a 12-hour clock and a 24-hour
clock, and use practical examples to calculate time
intervals between a start and finish time (e.g. bus or train
timetables). Using timetables and electronic
technologies, students plan and compare the time taken
to travel a specific distance with various modes of
transport to determine the most time-efficient routes.
Contexts could include planning the journey to a job
location some distance from home, or planning a family
holiday travelling interstate or overseas, which would
allow for the investigation and use of time zones in time
calculations.
In other contexts, for example, in schools with a fishing
or horticultural focus, students could interpret more
complex timetables, such as tide charts, sunrise charts,
and moon phases for fishing or planting crops.
How can we use rates to make
comparisons?
 identify common usage of rates; for
example, km/h as a rate to describe speed,
beats/minute as a rate to describe pulse
 convert units of rates occurring in practical
situations to solve problems, including
km/h to m/s, mL/min to L/h
 use rates to make comparisons and to
solve problems
 interpret rate graphs
The extension of time into combining time with other
measurements will introduce the concept of rates. Class
discussion could include questions such as:
 ‘How fast can you walk or run?’
 ‘How could this be measured in comparison with the
speed of a car?’
 Which is faster — 60 kilometres an hour or 15 metres a
second?
Identify appropriate units and conversions for different
activities, such as walking, running, driving, water flow
rates, and petrol consumption.
Determine distances and speed, or other types of
combined measurements using rates (e.g. water loss
from a dripping tap over a given time, heart rate, petrol
usage).
Make comparisons using rates, for example, using unit
prices to compare best buys or comparing heart rates
before and after exercise.
Interpret rate graphs such as distance-versus-time, costversus-time, or car stopping distances at varying speeds.
An extension to this sub-topic can be made through use
of formulas such as speed = distance/time to determine
speed, distance, or time. Alternatively rates can be used
to determine costs of a tradesperson to complete a job;
for example, calculating the cost of a plumber using rates
per hour with a call-out fee included.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
13 of 71
Subtopic 1.3: Ratio and Scale
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
What is a ratio and how do we use them to
solve problems?
 the relationship between fractions and ratio
 express a ratio in simplest form
 find the ratio of two quantities
 divide a quantity in a given ratio
Consideration of ratios and their expression in their
simplest form will lead into problems including: finding
the ratio of two or more quantities; expressing a ratio in
its simplest form; or dividing a quantity in a given ratio.
Use everyday examples such as:
 preparing cordial where water and syrup are combined
in a ratio
 investigating healthy eating guidelines such as 2 fruit
and 5 vegetables every day
 preparing 2-stroke fuel by mixing petrol and oil in ratio
 calculating dosages of medicines (e.g. a vial of
3
3
medicine states 5 mg per cm , how many cm will be
needed for a patient requiring 7.5 mg of medicine?).
How does a scale factor work?
 Scale factor
 Scale diagrams
 Calculation of actual and scale distances
Discus how large distances or spaces can be
represented on a diagram. This leads to a discussion of
the use of scale factor on maps and plans. Investigate
the use of different scales (e.g. a scale on a house plan
in comparison to the scale used on a road map covering
a large area) and use the given scale to calculate actual
lengths or distances.
Construct simple maps or plans where a scale is chosen
by the student (e.g. a scale plan of the school for
students new to the school). Electronic technology, such
as a drawing software package, is used where
appropriate. Emphasis is placed on the conventions of
construction of scale diagrams, such as:
 scales in ratio
 indications of dimensions
 labelling.
Extension to learning about scales include the calculation
of time and costs for a journey from distances estimated
from maps.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
14 of 71
Topic 2: Earning and Spending
This topic examines basic financial calculations in the context of the students’ personal
experiences and intended pathways (e.g. living independently, working, pursuing a
hobby). Students understand the different ways of being paid for work and the impact of
taxation on their income. They learn to manage the spending of their earnings through
budgeting. Much of the learning in this topic is undertaken through investigations in which
individuals or groups of students budget for buying a major item (such as a car) or plan
for a major event (such as a holiday).
Subtopic 2.1: Earning
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
In which different ways can income be
earned and received?
 Work: full-time, part-time, casual, selfemployed, work on commission, contract
 Salary
 Wages
 Hourly paid
 Commission
 Piecework
 Additional bonuses, entitlements,
allowances, overtime, or rewards
Discussion of the various ways of earning an income
(e.g. full-time, part-time, casual work, self-employment,
work on commission, contract). These could be grouped
in remuneration categories (e.g. salary, hourly paid,
contract, commission) and as full-time, part-time, or
casual work, shift work, commission, or piecework.
How can you calculate a weekly, fortnightly,
monthly, or annual income?
 Gross income (salaries, wages)
 Commissions
 Contracts
Students carry out a variety of pay calculations including
maintaining a timesheet, calculating various weekly,
fortnightly, monthly, and annual gross incomes for
salaries, wages, commissions, contracts, and so on.
Allowances, loadings, and bonuses are included.
Technology can be used where appropriate, and the use
of spreadsheets is encouraged.
How much personal taxation will you pay?
 Personal taxation
 Medicare levy
 Other deductions
Once gross incomes have been calculated, personal
income tax, the Medicare levy, and other deductions
(e.g. superannuation, union fees, and private health
cover) are calculated and the amount deducted to find
net income. Students use the taxation tables on the
Australian Taxation Office website.
Explore examples to investigate the impact of changes in
the amount of time worked and how that affects the tax
paid, and hence the net income.
Discussion of leave loadings, overtime, allowances for
uniforms, training, and tools, sick or carer’s leave,
payments in shares or bonuses, and areas of
employment in which these might apply.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
15 of 71
Sub-topic 2.2 Spending
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How, and on what, do you spend your
money?
Explore the different ways in which students spend their
money, and what their expenses might be in the near future.
Questions are posed, such as: ‘What would you like to buy?’,
‘How much money will you need?’, ‘How will you know if you
have a bargain and have spent your money wisely?’
What is a percentage?
Discussion of the term percentage to elicit students’
understanding of the term and its uses in everyday life. Use
of visual diagrams may assist.
How can a percentage be expressed
as a decimal or fraction?
Converting a percentage to both a fraction in its simplest form
and to a decimal. This should include the use of commonly
used percentages such as 5%, 10%, 25%, etc.
How can the calculation of a
percentage provide us with
information?
 finding a percentage of a given
amount
 determining one amount expressed
as a percentage of another
Discussion of where percentages are relevant in practical
situations will lead to calculations that require finding a
percentage of an amount, or describing one amount as a
percentage of another. Examples include distribution of
money (for a job completed), working on commission, trade
discounts, or investigation of what percentage of the
household money is spent on various expenses.
What impact do percentage increases
and decreases have on the price of
goods and services?
 percentage increases and decreases
 calculations involving mark-ups,
discounts, GST.
Advertisements in the media, catalogues, and stores provide
examples of discounts, specials, or sales. Students collect
examples of such advertisement to calculate and check the
accuracy of the advertised savings including investigation of
sale prices, original prices, or percentage discounts.
Estimates of discounts and sale prices from advertised
percentage discounts for commonly used percentages such
as 5%, 10%, and 50% should be practised without the use of
technology.
Discussion of where GST is applied to the cost of individual
goods and services. Students calculate the impact of GST on
the total cost of goods purchased and the services required
where GST is applied in everyday life (e.g. investigate
grocery shopping dockets, painting bill).
How do you determine the overall
change in a quantity following
repeated percentage changes?
What are the real costs and actual
charges when purchasing goods?
Students may investigate the effect of a number of
percentage changes to a given figure:
 volume discount followed by a trade discount based on
payment terms, e.g. 7/10, 5.5/21, n/30
 series discount of two or more percentage reductions e.g.
trade, cash, end of season, mark-downs, end-of-line
clearance.
Investigate the costs associated with different ways in which
money is spent (e.g. debit or credit card or app, cash, lay-by,
direct debit). Assess the advantages and disadvantages of
one or more payment methods.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
16 of 71
Subtopic 2.3: Budgeting
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How do you plan to spend your
money?
Explore the kinds of items that students plan to spend
their money on, and what their expenses might be in
the near future. Questions could be posed such as:
‘What do you want to save up for?’, ‘How much money
will you need?’
How do you balance your personal
budget?
Using a spreadsheet, students list the income that they
receive or earn over a period of time, and their likely
expenses, and determine if they will be in credit or
debt. They extrapolate this information to determine
proposed savings over an extended period of time
such as a month or year. This could lead to
investigation of possible changes to their spending
practices that allow them to save for an event or item.
How can budgeting be applied to a
business?
Discuss the importance to a business entity knowing
what the income and expenses need to be in order to
stay profitable. This could lead to discussions of
various expenses that may be applicable to a business
(e.g. rent, electricity, wages, insurance). Students use
these ideas in designing a fundraising activity to raise
money for a charity.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
17 of 71
Topic 3: Geometry
The earliest geometry involved observing the properties of plane shapes such as
triangles and quadrilaterals and their use in construction. Students name a variety of
common two and three-dimensional figures and classify them according to their
geometric properties. They learn to measure and classify angles, and use instruments,
such as a pair of compasses and a straight edge, to construct geometrical figures. They
identify the geometry involved in structures in the built environment and landscapes.
Subtopic 3.1: Shapes
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How do we identify and classify different 2D
shapes?
 naming of 2D shapes
 classification of shape as regular or
irregular
 classification and naming of different
triangles (e.g. equilateral, isosceles,
scalene, right-angled)
 naming of different representations of
circles (e.g. semicircle and sector)
Consider the names of 2D shapes such as square,
rectangle, rhombus, parallelogram, trapezium, circle, and
triangle. These include naming polygons up to 12-sided
figures. Shapes are further classified as regular or
irregular shapes. Students classify triangles by the length
of their sides, and know the appropriate name. Students
identify right-angled triangles and different
representations of circles such as a semi-circle and
sector.
How do we identify and classify different 3D
shapes?
 naming of 3D shapes
 classification of 3D shapes (e.g. prisms,
pyramids, sphere, cone)
 properties of prisms
 properties of pyramids
 3D nets
Consider the names of 3D shapes such as cube, sphere,
cylinder, and cone. Discuss the properties of a prism,
leading to students naming a variety of prisms such as
cube, cone, rectangular prism, and triangular prism.
Discuss the properties of a pyramid, leading to the
students being able to distinguish between a squarebased pyramid and a triangular-based pyramid.
Students identify these shapes in a range of everyday or
trade contexts (e.g. commercial packaging, cylinders in
car engines, retaining walls). Students investigate a
range of nets used to construct 3D solids such as cubes,
spheres, prisms, pyramids, cylinders, and cones.
Students recognise the 2D shapes that form the faces of
the 3D solids. Students name the 3D shape from a net
as well as draw a net for a given 3D solid.
Students identify 2D shapes in a range of everyday or
trade contexts (e.g. different shapes in tiles, paving,
buildings, and landscape designs).
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
18 of 71
Subtopic 3.2: Angle Geometry
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How do we classify and measure angles?
 classification of angles as acute, right,
obtuse, straight, reflex, revolution
Consider the properties of the various angles such as acute, right,
obtuse, straight, reflex, and a revolution, and recognise these in
everyday shapes and structures. When given a range of diagrams
depicting various angles, students estimate the size of the angles
and then measure the angles using appropriate equipment such
as a geoliner or protractor. The application of this knowledge to
everyday situations such as taking measurements in the building
industry or constructing scale diagrams would provide useful
contexts for problems (e.g. investigation of the angles required in
the different types of roof trusses or laying out a formal garden
design).
How can we determine unknown angles?
 complementary angles
 supplementary angles
Students learn the properties of complementary and
supplementary angles, and use these to find unknown angles.
How can we determine angles when lines
are parallel?
 corresponding angles
 alternate angles
 vertically opposite angles
 co-interior angles
The understanding of parallel lines and their representation is
followed by the introduction of the range of rules that can be used
to determine the size of angles created when a transversal line
cuts two or more parallel lines. These rules are used to determine
missing angles in simple questions involving parallel lines.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
19 of 71
Subtopic 3.3: Geometry and Construction
Key Questions and Key Concepts
How do we use mathematical equipment
to construct geometrical figures?
 construction techniques for lines
 construction techniques for angles
Considerations for Developing Teaching and
Learning Strategies
This section reinforces skills associated with the use of
mathematical equipment for making accurate constructions of
angles and lengths using a compass and ruler. Familiarity with
other equipment, such as a protractor, used in the measurement
of angles assist in checking the accuracy of constructions.
Students practise using the equipment by carrying out a variety of
construction exercises such as: bisecting a line segment or angle,
copying a line segment or angle, constructing a range of angle
sizes using a compass, and finding the perpendicular bisector of a
line segment. There is a range of resources available on the web,
such as www.mathopenref.com/constructions.html.
Students use the mathematical equipment with familiarity to
construct accurate geometrical figures.
How do we use mathematical equipment
to construct geometrical shapes?
 construction of triangles (e.g.
equilateral, isosceles, scalene, rightangled)
 constructions involving circles
 construction of a variety of polygons
Students extend their skills in geometrical construction to a range
of geometrical shapes. These include the accurate construction of
a range of triangular shapes given specific side length and angle
properties, constructions involving circles (e.g. constructing a
circle given three points, finding the centre of a given circle and
finding a tangent to a point on a circle), and construction of a
variety of polygons (e.g. squares and hexagons given one side, a
parallelogram given the two side lengths and an included angle).
Students investigate practical applications of these techniques, for
instance in pegging out a building site or landscape design.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
20 of 71
Topic 4: Data in Context
Information, in the form of data, is prevalent and influential in daily life. Data can be
presented in a variety of ways and for many purposes. In this topic students learn to
read and critically interpret data presented to them in various forms. They collect,
organise, analyse, and interpret data to make informed decisions and predictions, or
support a logical argument. Students learn to use various statistical tools and
techniques for working with data. They manipulate and represent data on which to
base sound statistical arguments.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
21 of 71
Subtopic 4.1: Classifying data
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
Why do we collect data?
Discuss the reasons for collecting data, e.g. to answer a
question that has been posed or to solve a problem.
Students develop an understanding of the strengths and
limitations of the use of statistical data to explain
everyday and other phenomena.
What type of data is categorical?
 Nominal
 Ordinal
Discuss the classification of data that is categorical (i.e.
sorted by categories). There are two types of categorical
data: nominal, which has no associated order (items or
subjects that do not have a specific order such as colour
or gender), and ordinal, in which the data collected has a
qualitative scale associated with it (e.g. strongly
disagree, disagree, no opinion, agree, strongly agree).
Students classify types of data. Examples of categorical
data may be collected from the class – both nominal (e.g.
mobile phone brands, favourite sport) and ordinal data
(e.g. students’ opinions of healthy food where a
qualitative scale has been used in the collection of the
data).
What type of data is numerical?
 Discrete
 Continuous
Discuss the classification of data that is numerical and
that can be further classified into discrete numerical data
(i.e. data that has a specific value, such as the number of
mobile phones in a home, number of children in a family)
or continuous numerical data (i.e. data that may take any
value within a range, such as the temperature recorded
hourly for a particular location or other forms of
measurement). Students classify types of numerical
data.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
22 of 71
Subtopic 4.2: Reading and Interpreting Graphs
Key Questions and Key Concepts
How is information presented in tables and
graphs?
 Line graph, step graph, column graph,
picture graph, pie graph
 Two-way tables.
How do the media and different texts display
information in graphs?
Why are some methods of presenting data
more suitable than others?
Considerations for Developing Teaching and
Learning Strategies
Consider the ways in which information can be
represented graphically and what qualities are important
in presenting information (e.g. clear labelling, appropriate
scales). Investigate information presented in two-way
tables as a further method of presenting information.
Students are exposed to a range of graphs or tables from
which they can extract information. Examples include
BMI for age, stopping distances for vehicles, height and
weight charts for babies.
Investigate the variety of places in which graphs are
used to display information. Discuss and interpret how
the way in which information is presented in these
graphs supports understanding. Examples of charts in
magazines, newspapers, and on television displaying a
variety of information such as political polls, indexes, and
share market information could be investigated or
referred to.
Students look for data presented in the media or other
publications in everyday contexts and discuss how
appropriately they have been presented.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
23 of 71
Subtopic 4.3: Drawing Graphs
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
What type of graph is best used to display a
dataset?
Discuss the type of information that is best displayed on
different types of graphs (e.g. pie graphs used to show
how something is divided such as the money in a weekly
budget, line graphs used to show the rise or fall of one
variable, histograms can be used to compare data).
How do we display categorical data?
tables, column graphs, picture graphs, pie
chart
Students recognise categorical data and select an
appropriate method to present it. They select categorical
data that has been presented in the media and display it
in a different format, or collect their own categorical data,
and present the data in a range of appropriate ways.
Students use appropriate labelling, scales, and titles for
graphs.
How do we display numerical data?
as frequency distributions, dot plots, stem
and leaf plots, and histograms
Students recognise numerical data and select an
appropriate method to present it. They select numerical
data that has been presented in the media and display it
in a different format, or collect their own numerical data,
and present the data in a range of appropriate ways.
Students use appropriate labelling, scales, and titles for
all graphs.
How do you use a line graph to represent
data that demonstrates a continuous
change?
Students use a variety of data that demonstrate a
continuous change to produce line graphs. This includes
data such as the hourly temperature over the period of a
day, the rainfall each day of a particular month, the price
of particular companies’ shares every month over the
period of a year). Students use appropriate labelling,
scales, and titles for all graphs.
How do you use spreadsheets to tabulate
and graph data?
Data can be provided to or collected by students to
create a range of different types of graphs using the
graphing package on a spreadsheet. Students choose
appropriate types of graphs to display the different types
of data provided and learn to use the graphing package
to include appropriate labelling, scales, and titles for all
graphs.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
24 of 71
Subtopic 4.4: Summarising and Interpreting Data
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
What is the mode?
Understanding of the mode being the statistical measure that
identifies the most common score/s within a data set. Discuss how
useful it may be as statistical measure in describing data. Students
identify the mode of a given data set.
What is meant by average?
 median and mean
Students examine sets of data with this question in mind, including
median and mean, noting that measures of central location are not
valid for categorical data.
How do we calculate the measures of
central tendency?
 arithmetic mean and median
Understanding of calculations of the mean and median. For small
data sets students calculate these measures without technology.
How useful are the mean and median as
measures of central tendency?
What effect do outliers have on measures
of central tendency?
 recognition of an outlier
 the effect on the measures of central
tendency
Students become aware that the centre, on its own, is of limited use
as the descriptor of a distribution, but that it can be used to compare
two sets of data or to compare a single set of data with a standard.
Students investigate the suitability of mean and median in various
everyday and vocational contexts. Investigation of examples of the
media using these measures inappropriately would give students an
understanding of the need to check the validity of information
presented by different sources.
Discussion of individual data that fall well outside of the observed
pattern (an ‘outlier’) is introduced. Examples of outliers may have
been evident in data that has already been used or collected by the
class. Investigation of the identified outlier should be undertaken to
determine if it is a ‘real’ piece of data or is due to an error in the
collection process (e.g. faulty equipment, error in recording the
value). Outliers should only be excluded from the data set if their
existence can be shown to be due to an error.
The effect of outliers on measures of centre should be discussed.
Discussion of how outliers can distort the different measures of
central tendency of a distribution should be supported by carefully
chosen examples. Students choose the measure of centre most
appropriate for a given purpose and a given set of data (e.g. selling
price of houses, typical daily calorie intake).
What further statistical measures can
support the investigation of data?
 range, interquartile range, standard
deviation
How can we describe data without
statistical measures?
Students use the range, interquartile range, and standard deviation
to obtain more information about aspects of the data. In cases
where a graphics calculator or spreadsheet package is not
accessible to students, the use of small data sets is advised to
support the finding the quartiles of the data. Investigation of
examples of the media, or other sources, using these measures
inappropriately may encourage students to check the validity of
information presented. Further investigation into the calculation and
interpretation of deciles and percentiles is considered.
Discuss the informal ways that can be used to describe the spread
of data, including descriptions such as spread out/dispersed, tightly
packed, clusters, gaps, more/less dense regions, and outliers.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
25 of 71
Subtopic 4.5: Comparing Data Sets
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How can we compare two sets of related
data?
 back-to-back stem plots
 calculate a five-number summary
 construct box and whisker plots using a
five-number summary
 compare the characteristics of the shape of
histograms
Compare the shape of two related distributions of
numerical data using a variety of methods. Two related
sets of data are presented as back-to-back stem plots,
the five-number summaries are calculated and box plots
are produced on the same scale.
Students look for differences in the characteristics of the
shapes of the distributions — e.g. symmetry, skewness,
bimodality. The use of technology for these comparisons
is encouraged, whether through use of a graphics
calculator or a spreadsheet program with a statistics
package.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
26 of 71
Topic 5: Measurement
In this topic students extend their skills in estimating, measuring, and calculating in
practical situations. They identify problems involving length, area, mass, volume, and
capacity and apply relevant techniques to solve them. Units of measurement,
appropriate measuring devices, and the degree of accuracy required for finding
answers in a given situation are considered.
The consideration of units of power and energy consumption extends this topic
beyond spatial measurement into another area relevant to everyday life. This is
investigated through the energy ratings of household items with a focus on reducing
our carbon footprint.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
27 of 71
Subtopic 5.1: Linear Measure
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
Why measure?
 The metric system for linear measurements
 Other systems of measurement of linear
measurement
Consider the history and development of the metric system and
other systems of measurement of linear distances. For students
with an interest in a trade such as building or plumbing, learning
about measurements in inches, feet, and miles should be
considered.
Which metric units are appropriate for
measuring linear distance in a given
situation?
 For example (km, m, cm, mm)
 Appropriate choice of units for linear
measurement
Discuss the most appropriate unit for measuring linear distances
and their abbreviations. Include very large and very small
measurements, and the use of powers of 10 and scientific
notation. Students choose appropriate units for one dimension.
Specific trade examples could be covered such as the use of
millimetres as the standard measure for building plans.
How accurate does the measurement have to
be?
 Using appropriate devices for making linear
measurements
 Consideration of errors in linear
measurement
Collect or research a variety of instruments for measuring linear
distances. Note the scale used for each, and discuss the reason
for the fineness of the scale and the degree of accuracy of the
measure. Use a variety of instruments so that students can read
scales from different tools (e.g. ruler, large measuring tapes and
trundle wheels). Students choose appropriate levels of accuracy
for measuring and recording linear distances.
Discussion and calculations of the errors due to the measuring
equipment used when taking linear measurements, with specific
alternative to absolute and relative errors.
What do estimation and approximation
mean?
Students estimate a variety of lengths and check their accuracy,
using an appropriate measuring device, and justify their choice of
particular units. For example, trade mathematics students pacing
out distances for quote estimates.
Discuss percentage error; students consider if the accuracy of
their estimates increases or decreases with practice.
When is it appropriate to use estimation?
Discuss the different purposes of measurement and the level of
accuracy required (e.g. building an object, determining the
distance you need to walk to a café).
Why convert units of measurement?
 Converting metric units of length
 Convert metric units of length to other
length units (e.g. metric to imperial
measurements)
A discussion of the need to convert units (e.g. building plans have
measurements indicated in millimetres, however wood is often
purchased per lineal metre). This will lead students into practising
the conversion of units.
How do you calculate the perimeter of regular
and non-regular shapes?
 Circumference and perimeters of regular
shapes (triangles, squares, rectangles and
circles)
 Perimeters of non-regular shapes,
including composite shapes
Some ‘rule of thumb’ conversions between other length units are
discussed and used (e.g. 1 metre is approximately equal to 3
feet), and the accuracy of the rules of thumb to the accurate
conversion discussed. A conversion graph could be constructed
manually or via a spreadsheet to assist with the conversion of
metric units to non-metric units of measure.
Calculation of the circumference of circles and perimeters of
regular, non-regular and composite shapes. The problems are
presented within a context where possible in both written and
visual format. For circumference of circles students are introduced
to rearranging the formula to find the radius or diameter. Notation
on diagrams indicating equal lengths are covered, and emphasis
is placed on the appropriate conventions of constructing labelled
diagrams.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
28 of 71
Subtopic 5.2: Area Measure
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How can the area of standard and nonstandard shapes be found?
 The use of grids to estimate and calculate
areas
Students develop an understanding of the concept of
area and use grids to estimate standard and nonstandard areas. For example, students could use the
grid method on graph paper to estimate the area of a
hand, foot, or body length. Maples’ method could also
be used.
Which metric units are used for area?
2
2
2
2
 For example (km , m , cm , mm )
Students discuss the most appropriate unit for
measuring areas and their abbreviations. Students
choose appropriate units for the area being measured.
How do we convert units of measurement
used for area?
 Converting metric units of area
 Convert metric units of area to other area
units (e.g. metric to imperial)
Calculating:
 Calculate areas of regular shapes
 Calculate areas of composite shapes
Students carry out a range of conversions between
metric units of area. Use correct units of area and
convert between different measurements of area such
as square metres, square kilometres, hectares, and
acres.
Students calculate areas such as squares, rectangles,
triangles, and circles both with and without electronic
technology. For the calculation of the area of circles,
students should use 22/7 as an approximation for 𝜋 in
non-calculator computations. This is extended to
Heron’s formula and calculation of composite shapes.
The calculations of areas are presented as contextual
problems. When given the area of a regular shape,
students rearrange the formula to find a missing
variable. Students choose appropriate units.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
29 of 71
Subtopic 5.3: Mass
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
What is the difference between mass and
weight?
Students develop an understanding of the concept of
mass, and the difference between mass and weight.
Which metric units are used for mass?
 For example (tonnes, kg, gm, mg)
Students discuss the most appropriate unit for measuring
mass and their abbreviations. Students choose
appropriate units for the mass being measured.
How do we convert units of measurement
used for mass?
 Converting metric units of mass
Students carry out conversions between tonnes,
kilograms, grams, and milligrams and select the correct
units of mass for the problem posed. For students with
an interest in a trade vocation such as hospitality or
nursing, imperial measurements such as pounds and
ounces should be considered.
Estimate the mass of different objects.
Students estimate the mass of a variety of items and
check their accuracy, using an appropriate measuring
device (where appropriate), and justify their choice of
particular units. For instance, students estimate weights
of food for dietary requirements or cooking.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
30 of 71
Subtopic 5.4: Volume and Capacity
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
What is the relationship between volume and
capacity?
Students understand the relationship between volume
and capacity.
Which metric units are used for volume?
 For example (kL, l, ml)
Students discuss the most appropriate unit for measuring
volume and their abbreviations. Students choose
appropriate units for the volume being measured.
How do we convert units of measurement
used for volume?
 Converting metric units of volume
When is it appropriate to estimate?
Students carry out a range of conversions between
metric units of volume. Students convert cubic metres to
litres and cubic centimetres to millilitres. These
conversions will be particularly relevant for students
interested in an automotive trade pathway.
Discussion of the appropriateness of estimation,
choosing everyday situations such as cooking, paint
tinting, and taking medicine.
Students estimate the volume and capacity of a variety of
containers (e.g. milk or fruit juice containers or tins of
soup), check their accuracy using an appropriate
measuring device where possible, and select appropriate
units for the measurement being made.
How do we calculate the volume of objects?
 Volume of cubes, rectangular and
triangular prisms and cylinders
Develop the link between the volume and the area of the
base of a prism multiplied by the height and apply these
formulas to a range of contextual problems such as the
amount of water in a swimming pool or rain water tank,
the mulch required to cover a garden bed, or the volume
of a cylinder in a car engine.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
31 of 71
Subtopic 5.5: Units of Energy and Power
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
What is the difference between energy and
power?
Students understand the difference between energy and
power, and also the connection between the two
measures.
Which units are appropriate for measuring
power?
 Energy in the form of electricity (such as
watts, kilowatts, megawatts and gigawatts)
Students investigate the amount of power used by
different types of TVs (e.g. plasma v LCD v LED). Using
an assumed usage time per day they determine a
monthly and yearly amount of power used by the item.
They gain an understanding of the conversions between
the units of power, and which units are appropriate given
the quantity being described.
Which units are appropriate for measuring
the energy used by electrical items?
 Energy in the form of electricity (kilowatts
hours)
 How do we convert from watts to kilowatt
hours?
Define the link between power usage and kilowatt hours.
This includes investigating energy bills, and the energy
rating of different household items. The formula allowing
the conversion between power and energy could be
introduced to calculate the number of kilowatt hours used
per day for a variety of electrical items:
𝐸(𝑘𝑊ℎ) = [𝑃(𝑤) 𝑥 𝑡(ℎ𝑟) ]/1000
where:
𝐸 =
𝑃 =
𝑡 =
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
energy in kilowatt hours
power in watts
time in hours
32 of 71
Topic 6: Investing
Students investigate interest, term deposits, and the costs of credit, using current and
relevant examples. To explore the concepts and uses of simple and compound interest,
students collect and analyse materials of various financial institutions outlining their
financial products. They examine the effects of changing interest rates, terms, and
investment balances on interest earned, and make comparisons. Emphasis is placed on
the use of technology, particularly spreadsheets and graphical packages, to enhance
students’ opportunities to investigate interest generated on investments.
Subtopic 6.1: Simple Interest
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
Why do people save?
Discussion of the reasons that people save money.
Discussion of the role of banks and other financial
institutions in building savings.
How is simple interest calculated, and in
which situations is it used?
 Using the simple interest formula to find
the:
- simple interest
- principal
- interest rate
- time invested in years.
Investigation into everyday situations in which simple
interest calculations are used to build the students
understanding of its applications (e.g. term deposits).
The formula for simple interest is used to explore a range
of examples of simple interest calculations. These
calculations should allow students to be able to solve
problems with and without electronic technology.
Students rearrange the simple interest formula to find all
of the variables.
Investigations draw the students to recognise the effects
of changing the principal, interest rate, and time invested.
Use of spreadsheets assist students to further
investigate simple interest problems through the ability to
make changes quickly and see the immediate impact of
the change.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
33 of 71
Subtopic 6.2: Compound Interest
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How is compound interest different from
simple interest?
 The way in which interest is accrued?
Discussion of the differences between the way in which
interest is accrued in a simple interest account in
comparison to a compound interest account.
How is the compound interest formula
derived?
 Find the
- amount accrued
- principal
 Find the
- amount accrued
- principal
- interest rate
- time
Exposure to the derivation of the formula for compound
interest leads to the students carrying out basic
calculations for problems in everyday contexts to
determine the amount accrued and the principal, using
the compound interest formula. The effect of changing
the number of compounding periods per year is
investigated to develop the understanding of the effect
that the compounding period has on the interest earned.
Using examples collected from financial institutions,
students use technology to calculate the amount
accrued, principal, interest rate, and time. This leads to
students investigating and recognising the effects of
changing the principal, interest rate, and time. Use of
spreadsheets assist students to further investigate
compound interest problems through the ability to make
changes quickly and see the immediate impact of the
change.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
34 of 71
Subtopic 6.3: Investing for Interest
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
Which is the better option: simple interest or
compound interest
 graphical comparisons
 effective rate calculations (both simple and
compound interest investment rates)
Students investigate both simple and compound interest
alternatives for a range of everyday problems to
determine which option presents them with the best
outcome in saving scenarios. The presentation of the
interest earned for both a simple and compound interest
scenario, is presented graphically to assist students
understanding of the different way in which the interest
accrues. Using a larger simple interest rate and a smaller
(annual) compound interest rate, students determine
when compound interest becomes the better option.
Calculate the simple interest rate that, over a number of
years, will earn the same amount as a given compound
interest rate.
Study the use of effective rates to compare both simple
and compound interest rate investment options.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
35 of 71
ASSESSMENT SCOPE AND REQUIREMENTS
Assessment at Stage 1 is school based.
EVIDENCE OF LEARNING
The following assessment types enable students to demonstrate their learning in
Stage 1 Essential Mathematics:
Assessment Type 1: Skills and Applications Tasks
Assessment Type 2: Practical Report
For a 10-credit subject, students provide evidence of their learning through four
assessments. Each assessment type should have a weighting of at least 20%.
Students undertake:
 at least two skills and applications tasks
 at least one practical report.
For a 20-credit subject, students provide evidence of their learning through eight
assessments. Each assessment type should have a weighting of at least 20%.
Students undertake:
 at least four skills and applications tasks
 at least two practical reports.
It is anticipated that from 2018 all assessments will be submitted electronically.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
36 of 71
ASSESSMENT DESIGN CRITERIA
The assessment design criteria are based on the learning requirements and are used
by teachers to:
 clarify for the student what he or she needs to learn
 design opportunities for the student to provide evidence of his or her learning
at the highest level of achievement.
The assessment design criteria consist of specific features that:
 students need to demonstrate in their evidence of learning
 teachers look for as evidence that students have met the learning
requirements.
For this subject, the assessment design criteria are:
 concepts and techniques
 reasoning and communication.
The specific features of these criteria are described below.
The set of assessments, as a whole give students opportunities to demonstrate each
of the specific features by the completion of study of the subject.
Concepts and Techniques
The specific features are as follows:
CT1 Knowledge and understanding of concepts and relationships
CT2 Selection and application of techniques and algorithms to solve problems in a
variety of contexts
CT3 Application of mathematical models
CT4 Use of electronic technology to find solutions to mathematical problems.
Reasoning and Communication
The specific features are as follows:
RC1 Interpretation and evaluation of mathematical results, with an understanding
of their reasonableness and limitations
RC2 Knowledge and use of appropriate mathematical notation, representations,
and terminology
RC3 Communication of mathematical ideas, and reasoning, to develop logical
arguments
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
37 of 71
SCHOOL ASSESSMENT
Assessment Type 1: Skills and Applications Tasks
For a 10-credit subject, students complete at least two skills and applications tasks.
For a 20-credit subject, students complete at least four skills and applications tasks.
Students find solutions to mathematical problems that may:
 be routine, analytical, and/or interpretative
 be posed in a variety of familiar and new contexts
 require discerning use of electronic technology.
In setting skills and applications tasks, teachers may provide students with
information in written form or in the form of numerical data, diagrams, tables, or
graphs. A task should require students to demonstrate an understanding of relevant
mathematical concepts and relationships.
Students select appropriate techniques or algorithms and relevant mathematical
information to find solutions to routine and some analytical and/or interpretative
problems.
Students provide explanations and use correct mathematical notation, terminology,
and representation throughout the task. They may be required to use electronic
technology appropriately to aid and enhance the solution of some problems.
Skills and applications tasks can be presented in different formats.
For this assessment type, students provide evidence of their learning in relation to
the following assessment design criteria:
 concepts and techniques
 reasoning and communication.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
38 of 71
Assessment Type 2: Practical Report
For a 10-credit subject, students complete at least one practical report.
For a 20-credit subject, students complete at least two practical reports.
Students, either individually or in a group, undertake planning, apply their numeracy
skills to gather, represent, analyse, and interpret data, and propose or develop a
solution to a mathematical problem based in an everyday or workplace context. The
subject of the mathematical problem may be derived from one or more subtopics,
although it can also relate to a whole topic or across topics.
A mathematical problem may be initiated by a student, a group of students, or the
teacher. Teachers may give students clear and detailed advice and instructions for
setting and solving the mathematical problem, or may provide guidelines for students
to develop contexts, themes, or aspects of their own choice. Teachers should give
some direction about the appropriateness of each student’s choice and guide and
support students’ progress in a mathematical solution.
If a mathematical problem is undertaken by a group, students explore the problem
and gather data together to develop a model or solution individually. Each student
must submit an individual practical report.
Students demonstrate their knowledge. They are encouraged to use a variety of
mathematical and other software (e.g. statistical packages, spreadsheets, CAD,
accounting packages) to solve their mathematical problem.
The practical report may take a variety of forms, but would usually include the
following:
 an outline of the problem to be explored
 the method used to find a solution
 the application of the mathematics, including, for example:
– generation or collection of relevant data and/or information, with a
summary of the process of collection
– mathematical calculations and results, using appropriate representations
– discussion and interpretation of results, including consideration of the
reasonableness and limitations of the results
 the results and conclusions in the context of the problem
 a bibliography and appendices, as appropriate.
The format of a practical report may be written or multimodal.
Each practical report should be a maximum of 8 pages if written, or the equivalent in
multimodal form.
For this assessment type, students provide evidence of their learning in relation to
the following assessment design criteria:
 concepts and techniques
 reasoning and communication.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
39 of 71
PERFORMANCE STANDARDS
The performance standards describe five levels of achievement, A to E. Each level of
achievement describes the knowledge, skills and understanding that teachers refer to
in deciding how well a student has demonstrated his or her learning on the basis of
the evidence provided.
During the teaching and learning program the teacher gives students feedback on
their learning, with reference to the performance standards.
At the student’s completion of study of a subject, the teacher makes a decision about
the quality of the student’s learning by:
 referring to the performance standards
 taking into account the weighting given to each assessment type
 assigning a subject grade between A and E.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
40 of 71
Performance Standards Stage 1 Essential Mathematics
A
Concepts and Techniques
Reasoning and Communication
Comprehensive knowledge and understanding of concepts
and relationships.
Comprehensive interpretation and critical evaluation of
mathematical results, with an in-depth understanding of
their reasonableness and possible limitations.
Highly effective selection and application of techniques and
algorithms to find efficient and accurate solutions to routine
and complex problems in a variety of contexts.
Successful development and application of mathematical
models to find concise and accurate solutions.
Proficient and accurate use of appropriate mathematical
notation, representations, and terminology.
Highly effective communication of mathematical ideas
and reasoning to develop logical and concise arguments.
Appropriate and effective use of electronic technology to find
accurate solutions to routine and complex problems.
B
Some depth of knowledge and understanding of concepts and
relationships.
Mostly effective selection and use of mathematical techniques
and algorithms to find accurate solutions to routine and some
complex problems in a variety of contexts.
Attempted development and successful application of
mathematical models to find accurate solutions.
Mostly appropriate and effective use of electronic technology
to find accurate solutions to routine and some complex
problems.
C
Generally competent knowledge and understanding of
concepts and relationships.
Generally effective selection and use of mathematical
techniques and algorithms to find mostly accurate solutions to
routine problems in a variety of contexts.
Application of mathematical models to find generally accurate
solutions.
Mostly appropriate interpretation and some critical
evaluation of mathematical results, with some depth of
understanding of their reasonableness and possible
limitations.
Mostly accurate use of appropriate mathematical
notation, representations, and terminology.
Mostly effective communication of mathematical ideas
and reasoning to develop logical arguments.
Generally appropriate interpretation and evaluation of
mathematical results, with some understanding of their
reasonableness and possible limitations.
Generally appropriate use of mathematical notation,
representations, and terminology with some inaccuracies.
Generally effective communication of mathematical ideas
and reasoning to develop some logical arguments.
Generally appropriate and effective use of electronic
technology to find mostly accurate solutions to routine
problems.
D
Basic knowledge and some understanding of concepts and
relationships.
Some effective selection and use of mathematical techniques
and algorithms to find some accurate solutions to routine
problems in some contexts.
Some application of mathematical models to find some
accurate or partially accurate solutions.
Some interpretation and attempted evaluation of
mathematical results, with some awareness of their
reasonableness and possible limitations.
Some appropriate use of mathematical notation,
representations, and terminology, with some accuracy.
Some communication of mathematical ideas with
attempted reasoning and/or arguments.
Some appropriate use of electronic technology to find some
accurate solutions to routine problems.
E
Limited knowledge or understanding of concepts and
relationships.
Attempted selection and limited use of mathematical
techniques or algorithms, with limited accuracy in solving
routine problems.
Attempted application of mathematical models, with limited
accuracy.
Limited interpretation or evaluation of mathematical
results, with limited awareness of their reasonableness
and possible limitations.
Limited use of appropriate mathematical notation,
representations, or terminology, with limited accuracy.
Attempted communication of mathematical ideas, with
limited reasoning.
Attempted use of electronic technology, with limited accuracy
in solving routine problems.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
41 of 71
ASSESSMENT INTEGRITY
The SACE Assuring Assessment Integrity Policy outlines the principles and processes
that teachers and assessors follow to assure the integrity of student assessments. This
policy is available on the SACE website (wwww.sace.sa.edu.au) as part of the SACE
Policy Framework.
The SACE Board uses a range of quality assurance processes so that the grades
awarded for student achievement in the school assessment are applied consistently
and fairly against the performance standards for a subject, and are comparable across
all schools.
Information and guidelines on quality assurance in assessment at Stage 1 are available
on the SACE website (www.sace.sa.edu.au).
SUPPORT MATERIALS
SUBJECT-SPECIFIC ADVICE
Online support materials are provided for each subject and updated regularly on the
SACE website (www.sace.sa.gov.au). Examples of support materials are sample
learning and assessment plans, annotated assessment tasks, annotated student
responses, and recommended resource materials.
ADVICE ON ETHICAL STUDY AND RESEARCH
Advice for students and teachers on ethical study and research practices is available in
the guidelines on the ethical conduct of research in the SACE on the SACE website.
(www.sace.sa.edu.au).
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
42 of 71
Stage 2
Essential Mathematics
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
43 of 71
LEARNING SCOPE AND REQUIREMENTS
LEARNING REQUIREMENTS
The learning requirements summarise the knowledge, skills, and understanding that
students are expected to develop and demonstrate through learning in Stage 2
Essential Mathematics.
In this subject, students are expected to:
1. understand mathematical concepts, demonstrate mathematical skills, and apply
mathematical techniques
2. investigate and analyse mathematical information in a variety of contexts
3. recognise and apply the mathematical techniques needed when analysing and
finding a solution to a problem, including the forming and testing of conjectures
4. interpret results, draw conclusions, and reflect on the reasonableness of solutions
in context
5. make discerning use of electronic technology
6. communicate mathematically and present mathematical information in a variety of
ways.
CONTENT
Stage 2 Essential Mathematics is a 20-credit subject.
In this subject students extend their mathematical skills in ways that apply to practical
problem solving in everyday and workplace contexts. A problem-based approach is
integral to the development of mathematical skills and associated key ideas in this
subject.
Stage 2 Essential Mathematics consists of the following list of five topics:
Topic 1: Scales, Plans, and Models
Topic 2: Measurement
Topic 3: Business Applications
Topic 4: Statistics
Topic 5: Investment and Loans
The suggested order of the topics is a guide only; however, students study all five
topics.
Each topic consists of a number of subtopics. These are presented in the subject
outline in two columns as a series of ‘key questions and key ideas’ side by side with
‘considerations for developing teaching and learning strategies’.
The ‘key questions and key ideas’ cover the prescribed content for teaching, learning,
and assessment in this subject. The considerations for developing teaching and
learning strategies are provided as a guide only.
A problem-based approach is integral to the development of the computational models
and associated key ideas in each topic. Through key questions teachers can develop
the concepts and processes that relate to the mathematical models required to address
the problems posed.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
44 of 71
The ‘considerations for developing teaching and learning strategies’ present suitable
problems and guidelines for sequencing the development of ideas. They also give an
indication of the depth of treatment and emphases required.
Although the external examination (see Assessment Type 3) will be based on the ‘key
questions and key ideas’ outlined in the three topics specified for examination, the
‘considerations for developing teaching and learning strategies’ may provide useful
contexts for examination questions.
Stage 2 Essential Mathematics prepares students with the mathematical knowledge,
skills, and understanding needed for entry to a range of practical trades and vocations.
In the ‘considerations for developing teaching and learning strategies’ the term trade is
used to suggest a context in a generic sense to cover a range of industry areas and
occupations such as automotive, building and construction, electrical, hairdressing,
hospitality, nursing and community services, plumbing, and retail.
It is recommended that students carry out calculations by hand to develop an
understanding of the processes involved, before using electronic technology to enable
more complex problems to be solved when it is more efficient to do so.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
45 of 71
Topic 1: Scales, Plans, and Models
Students extend their understanding of the properties of plane shapes and solids, and
construct the nets of a range of three dimensional shapes. They use scaled representations
such as maps, plans, and three dimensional models to determine full scale measurements
in practical contexts. Students develop practical skills in measuring and scaling down to
create their own maps, scaled plans or models. Comparison of lengths determined using a
scale with those found by direct measurement allows students to consider the effect of
errors.
Subtopic 1.1: Geometry
Key Questions and Key Concepts
What are the properties of 2D shapes?
 Number of vertices and edges
Considerations for Developing Teaching and
Learning Strategies
Consideration of properties of 2D shapes such as
square, rectangle, rhombus, parallelogram, trapezium,
circle, and triangle. These include the naming of
polygons with a variety of number of sides up to 12
sided figures.
Students identify these shapes in a range of everyday
or trade contexts (e.g. different shapes in building
structures such as windows, doors, and roofs).
What are the properties of 3D shapes?
 Number of faces, vertices and edges
How do we recognise a 3 dimensional solid
from a 2 dimensional representation?
 Nets
Review of the properties of 3D shapes such as cube,
sphere, prisms, pyramids, cylinder and cone. Students
identify these shapes in a range of everyday or trade
contexts (e.g. prisms in food containers such as cereal
boxes or powdered drink).
Students investigate the range of nets used to
construct 3D solids such as cubes, spheres, prisms,
pyramids, cylinders, and cones. Students recognise
the 2D shapes that form the faces of the 3D solids.
Students name the 3D shape from a net as well as
draw a net for a given 3D solid. Perspective diagrams
allow students to extend their capacity to identify and
represent 3D solids.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
46 of 71
Subtopic 1.2: Scale Diagrams
Key Questions and Key Concepts
How can scaled representations be
constructed from data?
 Determining an appropriate scale
 Commonly used symbols and
abbreviations
 Representing large scale information in
2D
Considerations for Developing Teaching and
Learning Strategies
Students take measurements to construct a scaled
representation. These may take the form of a map of an area
surveyed such as the local park, constructing an orienteering
course, a scaled plan of a site, or a diagram of an object to be
constructed. Students use a clinometer or compass to measure
angles, and a trundle wheel or large measuring tape to measure
distances.
Discussion leads students to understand that the measurements
they collected need to be displayed using a scale. Students
present the collected measurements as a scale diagram using
appropriate representations (e.g. scale in ratio, clear labelling and
indication of dimensions). Students construct scale diagrams by
hand, however where software packages are available these may
be used to enhance students understanding.
What information can be gained from
scaled representations?
 Finding lengths, perimeters and areas
from scaled representations
 Solve problems using bearings
How accurate are measurements gained
from scaled diagrams?
 Discussion of accuracy of
measurements
 Effect of errors on the calculations
Students determine information from scaled representations such
as lengths, perimeters, and areas. Students investigate a variety
of situations in which problems could be posed such as:
finding the length of fencing needed for a farm paddock
designing and costing a garden
finding the distance between two locations on a road map
finding a ship that is in trouble near the coast, when the same
distress flare has been sighted from two different places along the
coast.
Students consider the reliability of their answers in the context of
the problem.
Discussion of the accuracy of measurements from a scaled
representation assists in the understanding of limitations of some
scale representations (e.g. a scale diagram will not allow for hills
and valleys in the calculation of a distance between two towns on
a map). Measurements of angles and their effect on the accuracy
of measurements gained from scale diagrams should also be
investigated (e.g. the effect of a small error in angle measurement
is magnified by distance).
Scaled three dimensional models may also be considered in this
topic. Students determine information from plans and elevations
for house constructions or other 3D objects.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
47 of 71
Topic 2: Measurement
Students extend the concepts from Topic 5 in Stage 1 to include practical problems in
two dimensions involving circles, polygons, and composite shapes, and in three
dimensions involving cones, cylinders, pyramids and spheres. Methods for estimating
the size of irregular areas and volumes are investigated.
Through the study of Pythagoras’ theorem and the trigonometry of right and non-right
triangles students solve for unknown sides and angles in triangle problems posed in
everyday and workplace contexts. Students solve problems involving the calculation of
volume, mass, and density posed in practical contexts.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
48 of 71
Topic 2: Measurement
Subtopic 2.1: Linear Measure
Key Questions and Key Concepts
Which metric units are appropriate for
measuring linear distance in a given
situation?
 For example (km, m, cm, mm)
 Appropriate choice of units for linear
measurement
Why convert units of measurement?
 Converting metric units of length
 Convert metric units of length to other
length units (e.g. metric to imperial
measurements)
When is it appropriate to use estimation?
Considerations for Developing Teaching and Learning
Strategies
Consideration of the units for measuring linear distances and their
abbreviations. Include very large and very small measurements,
and review the use of powers of 10 and scientific notation.
Students choose appropriate units for one dimension. Specific
trade examples could be covered such as the use of millimetres
as the standard measure for building plans.
Discussion of the rules for converting between metric units.
Familiarity of other length measurement systems such as the
Imperial system using inches, feet, and yards may be necessary
for students in certain trades. Some ‘rule of thumb’ conversions
between other length units can be discussed and used (e.g. 1
metre is approximately equal to 3 feet).
Students estimate a variety of lengths and check their accuracy
using an appropriate measuring device, and justify their choice of
particular units. Where possible the contexts should be based on
everyday and workplace examples, with a particular focus on
trade scenarios (e.g. students pacing out distances for quote
estimates and building an awareness of their level of accuracy).
Discuss the different purposes of measurement and the level of
accuracy required (e.g. determining the length of a fence,
constructing a frame of wood for a chicken coop).
How do you calculate the perimeter of
familiar shapes?
 Circumference and perimeters of familiar
shapes (triangles, squares, rectangles,
polygons, circles and arc lengths)
 Composite shapes
Calculation of the circumference of circles and arc lengths and
perimeters of regular, non-regular, and composite shapes.
Students rearrange the formula to find the radius or diameter for
circle and arc lengths. Consider notation on diagrams indicating
equal lengths, and emphasise the appropriate conventions of
constructing labelled diagrams. Calculations are carried out
without electronic technology where possible.
How can we find the lengths of missing sides
when working with right-angled triangles?
 Pythagoras’ theorem
 Sine, cosine, and tangent ratios
 Angle of elevation and angle of depression
Right-angled triangles are commonly encountered in con
construction problems such as roof designs and trusses. Consider
the use of Pythagoras’ theorem, sine, cosine, and tangent ratios to
provide students with tools to find missing sides of right-angled
triangles. Students use the concepts angle of elevation and angle
of depression. They construct diagrams displaying information
provided about angles of elevation and/or depression, and use
these in solving problems.
How can we find the lengths of missing sides
when working with non-right-angled
triangles?
 Sine rule
 Cosine rule
How can unknown angles be found for nonright-angled triangles using the sine and
cosine rules?
The sine rule and cosine rule are introduced to allow for angles of
non-right-angled triangles. Students use them to find missing
sides of non-right-triangles. These calculations require the use of
electronic technology. Students are encouraged, when
appropriate, to confirm their results from a scaled representation.
Students see how the sine and cosine rule formulae can be
rearranged to find unknown angles in non-right-angled triangles.
Problems posed in practical contexts need not involve the
ambiguous case of the sine rule.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
49 of 71
Subtopic 2.2: Area Measure
Key Questions and Key Concepts
Which metric units are used for area and
how do we convert between them?
2
2
2
2
 For example (km , m , cm , mm )
 Converting metric units of area
How do we calculate:
 areas of regular shapes
 areas of composite shapes
 areas of irregular shapes using:
- Simpson’s rule
𝐴=
-
1
𝑤 (𝑑1 + 4𝑑2 + 𝑑3𝑛 )
3
an approximation using simple
mathematical shapes
Considerations for Developing Teaching and
Learning Strategies
Students discuss the most appropriate unit for
measuring areas and their abbreviations. Students
choose appropriate units for the area being measured
and carry out conversions between metric units of area
where necessary for problems being solved. Students
should be familiar with different measurements of area
such as square metres, square kilometres, hectares,
acres, including conversion of metric units of area to
imperial measures of area.
Contextual problems involving everyday and workplace
scenarios will provide opportunities for students to
apply formulae to solve various area calculations. The
range of problems covered include calculations of area
for triangles, squares, rectangles, parallelograms,
trapeziums, circles and sectors, and composites of
these shapes. Contexts such as finding, the area of a
floor to be covered in tiles, the layout of a car park or
playground, require students to find areas of regular
and composite shapes.
The area of irregular shapes like kidney-shaped garden
beds, fish ponds, golfing greens or swimming pools can
be calculated by using Simpson’s rule.
Simpson’s rule:
𝐴=
1
𝑤 (𝐿0 + 4𝐿1 + 2𝐿2 + . . . . . +4𝐿𝑛−1 + 𝐿𝑛 )
3
where:
𝑤 = distance between offsets
𝐿𝑘 = is the length of the 𝑘 𝑡ℎ offset
To use this method accurately 𝐿0 and 𝐿𝑛 must be the
measurement of the strips at the ends of the irregular
area, even if these lengths are of size zero. An
alternative method that should be used, where
appropriate, is the deconstruction of an irregular shape
into regular shapes to estimate the area. Discussion of
the accuracy of these methods in determining the areas
of irregular shapes encourage students to consider the
reliability of their answers, and the importance of
accuracy to the original scenario.
How do we calculate the surface area of:
 cubes, rectangular and triangular based
prisms, cylinders, pyramids, cones and
spheres
 simple composites of these shapes
Contextual problems involving everyday and workplace
scenarios will provide opportunities for students to solve
surface area problems. The range of contexts covered,
include calculations of the glass to cover a glass house,
the paint to cover a water tank. Students consider
which parts of the shapes are important in the
calculation (e.g. can you paint the base of a rain water
tank?).
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
50 of 71
Subtopic 2.3: Mass, Volume and Capacity
Key Questions and Key Concepts
Considerations for Developing Teaching and
Learning Strategies
How do we convert units of measurement
used for mass?
 Converting metric units of mass
Students carry out conversions between tonnes,
kilograms, grams, and milligrams and select the correct
units of mass for the problem posed. They discuss these
measurements in various contexts, e.g. hospitality,
nursing, and nutrition. Imperial measurements such as
pounds and ounces could be considered.
What is the relationship between volume and
capacity and the relevant units?
Students review the relationship between volume and
capacity in contexts and the appropriate metric units (kL,
L, mL).
How do we convert units of measurement
used for volume?
 Converting metric units of volume
 Converting between volume and capacity
3
3
(e.g. 1cm = 1mL and 1m =1kL)
Students carry out a range of conversions between
metric units of volume. Students convert cubic metres to
litres and cubic centimetres to millilitres.
How do we calculate the volume of objects?
 Volume of cubes, rectangular and
triangular based prisms, cylinders,
pyramids, cones and spheres
Students use formula to find the volume and capacity of
regular objects in a range of contextual problems such as
the volume of water in a cylindrical drink container,
volume of water in an aquarium. Students choose
appropriate units for the volume being measured.
What is the connection between volume and
mass?
 Calculating density
3
 Units of measurement for density (Kg/m
3
and g/cm )
A practical approach is useful in the development of an
understanding of the idea of density. Compare solid
objects with similar volume but different masses.
Through experiment students investigate the different
densities of liquids.
Problem involving density should be posed in familiar
contexts, e.g. finding the volume of a given mass of
building materials (e.g. wood, garden soils, or cement)
from its density or finding how much a given volume of a
substance will weigh.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
51 of 71
Topic 3: Business Applications
In setting up and running a business there are several considerations: the location and
spatial requirements of the premises; pricing policies and a mathematical analysis of their
impact on the profitability of the business; the impact of taxation depending on the
business structure. Students investigate these physical and financial planning aspects of a
small business.
Linear programming provides the opportunity to investigate optimal solutions to problems
involving two variables. Students investigate the effects of changing the initial parameters
and consider the reasonableness of solutions.
In breakeven and linear programming problems emphasis is placed on the use of
graphing technology to facilitate solutions and provide students with opportunities to
investigate ‘What if …’ questions.
Subtopic 3.1: Planning a Business Premises
Key Questions and Key Ideas
What factors influence a business in
choosing a location for their business
premises?
 Comparison of location and facilities
 Comparison of costs of premises
Considerations for Developing Teaching and
Learning Strategies
A variety of retail businesses (e.g. hairdressing salon,
delicatessen, real-estate firm, mechanic business,
mobile dog wash) provide suitable discussion points
about choice of location and facilities for a business.
The productivity of a business can depend on the
location of its premises. Discussion focuses on the size
of premises needed for a variety of business activities.
Consideration is given to the cost of the premises and
its impact on the businesses ability to make a profit.
The premises cost calculations are carried out without
the use of electronic technology. Costs are provided in
a variety of periods of time (e.g. weekly, fortnightly,
monthly, quarterly and annually).
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
52 of 71
Subtopic 3.2: Costing Calculations
Key Questions and Key Ideas
Considerations for Developing Teaching and
Learning Strategies
What terms may be used in describing the
costing of goods to be sold?
 manufacturer’s cost, wholesaler’s cost,
retail cost, profit margin, discount, GST,
input tax credits
Students are introduced to the relevant terms associated with the
selling of goods. Goods are produced by a manufacturer, sold to a
wholesaler, and distributed to a retail business. The wholesaler then
deals with a retail business. At each stage of this process costs are
added. Discussion could surround when a manufacturer may sell direct
to a retailer. Terms such as GST, input tax credits, profit margin, and
discount are covered in relation to the retail sector.
How are pricing structures used to decide
the price of a product?
 trade discount based on payment terms
e.g. 7/10, 5.5/21, n/30
 series discount of two or more
percentage reductions e.g. trade, cash,
end of season, mark-downs, end-of-line
clearance.
 GST
 profit margin (mark up)
Students calculate the pricing structure of buying and selling goods.
These calculations include examples of trade discounts and series
discounts. Students investigate the cost of the goods and services tax
(GST) and how it affects the price of items between the manufacturer
and the retailer. Examples are drawn from larger manufacturers and
supply stores, which are used by small business owners servicing the
electronic, building, automotive, and plumbing sectors. Students
determine the selling price of an item from a given profit margin.
What other factors may affect the viability
of the business?
 calculation of depreciation (straight line
and reducing balance depreciation)
 construction of deprecation graphs
 discussion of other insurance costs
including WorkCover and public liability
 input tax credits
Discussion of how businesses can use depreciation to minimise their
taxation leads into consideration of what items can be depreciated, and
the two methods that can be used. Calculation of the depreciation of an
item using both methods allows for decisions to be made about which
is the better option for the business to minimise their taxation liability.
Students make comparisons between the two methods using graphical
methods. These graphs are constructed initially without the aid of
electronic technology.
Insurance is another cost factor for most small businesses and the
types of insurance needed are discussed. In particular, WorkCover and
public liability are explored for a variety of businesses.
Input tax credits/rebates and the manner in which a business can gain
a rebate for previous GST payments on the items that they sell are
investigated. Calculations covering the GST costs through the
manufacturer, wholesaler, retailer/tradesperson, and customer chain
allow for students to investigate the return of GST through input tax
credits.
What determines the break-even point?
 calculation of fixed costs and variable
costs
 determination of break-even point
graphically
 calculation break-even point using the
marginal income
How viable is the business?
 construction of profit-and-loss
statements (including COGS)
 profit projections.
Students investigate the costs involved in the production of an item for
sale. They identify these costs as fixed or variable. To investigate at
what point a business will break-even, the sale price of the item is
determined, and a break-even investigation is carried out graphically
and using the marginal income method. Discussion of the
reasonableness of the break-even point can lead to investigation of
increasing or decreasing the sale price of the item or ways in which the
fixed or variable costs may be reduced. Students investigate business
ideas creating a product that can be sold to small retailers and
investigate through a break-even analysis.
Students understand that the profit-and-loss statement for a business
provides an overview of the revenue and expenses incurred each
financial year. Discussion covers: how the figures in the statement are
calculated from details of the income and costs for a business. The
place of depreciation in a profit and loss statement are covered, and
also why depreciation is an important factor to be included in these
calculations. Students construct a profit and loss statement from a list
of expenses and revenue, including businesses that earn their income
from selling goods. This leads to questions such as: What constitutes a
profit or loss?
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
53 of 71
Subtopic 3.3: Business Structures and Taxation
Key Questions and Key Ideas
What are common business
structures?
 Sole Trader
 Partnership
 Company
How does the business structure
affect the taxation liability of a
business?
 Calculations comparing the
taxation payable under sole trader
or partnership structures, or
different proportioning of ownership
within the partnership
Considerations for Developing Teaching and
Learning Strategies
Students understand that the legal structure of a
business determines the taxation payable by the
business.
Students understand that the business structure
chosen can minimise the taxation liability for a
business, depending on their level of taxable income.
A variety of income situations are considered for the
same business with both sole trader and partnership
structures discussed. Students recognise the
importance of projecting future income levels and the
way in which these projections can inform the settingup of the best business structure when the business is
established as well as throughout the life of the
business. For example, there may be a need to
change a business structure from a sole trader to a
partnership when the business grows too large to
manage. Discussion of other factors affecting taxation
could cover concessions for businesses that have a
turnover less than $2 million other than income tax
(e.g. FBT, PAYG and Primary Producer etc.).
Note: The calculation of the taxation payable by a
business with a company structure involves
consideration of the personal taxation liability of
directors as well as the company taxation liability on
actual net profit, so this would be best covered in a
discussion, and no calculations are required.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
54 of 71
Subtopic 3.4: Linear Programming
Key Questions and Key Ideas
How do we determine the best use of a
resource?
Considerations for Developing Teaching and Learning
Strategies
Students investigate a problem scenario in which two items
are produced by a business from a limited number of parts.
Being a business the aim is to make the greatest profit from
selling the items made, or to minimise the cost of producing
the items.
The textiles class make two types of soft toy to sell at the
school open day. The soft toys will be made from two types
of material, and will vary in selling price. Students investigate
the combinations of each soft toy that can be made from the
available material using a trial and error approach. In
suggesting possible combinations and then testing for their
feasibility, students gain an appreciation of the dynamic
nature of the problem. The number of one type of toy made
will affect the materials available to make the other. Students
calculate the profit made from each scenario to determine
which combination will produce the greatest profit overall.
The mathematical investigations are carried out without
electronic technology.
How can we more efficiently solve these
problems?
 Identifying variables
 Constructing constraints (two or more
active constraints)
 What is the objective function to be
minimised or maximised?
 Graphing constraints Identifying the
feasible region
 Identifying vertices of the feasible
region (integral solutions only)
 Finding the optimal solution
 Interpreting the solution and its
ramifications in the context of the
problem
 Discussing the appropriateness and
limitations of the model.
What is the effect on the optimal solution
if there are changes to the original
problem?
 Change of the objective function
 Change of one or more of the
constraints
At this point a better way of organising the information will be
introduced. Students are introduced to the terminology of
linear programming problems. Students investigate a range
of problems with two or more constraints to determine the
optimal solution. Interpretation of the solution is provided in
the context of the initial problem posed (e.g. the maximum
profit occurs when we make two small soft toys and three
large soft toys). Discussion about the limitations of the
solutions to these problems could include:
 the assumption that all items made would be sold
 a bakery making two items where the optimal solution is to
only produce one type of item, hence limiting customer
choice
 that no parts are broken in the production of the items.
Changes to the original constraints or the objective function
will lead to investigation of the effect on the optimal solution.
Extensions to these problems deal with wastage of materials
in the optimal solution and what should be done if integral
solutions are required at a non-integral corner.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
55 of 71
Topic 4: Statistics
Students consider the collection of data through various methods of sampling. Emphasis
is placed on the importance of eliminating bias and ensuring the validity and reliability of
results used from the sample. Two or more sets of data examining a single variable, from
the same or similar populations, are compared using calculated statistics and graphical
representations.
Students extend their skills introduced in Topic 4 in Stage 1 to analyse their data critically
and use this analysis to form and support reasonable conjectures.
Linear regression techniques are used to investigate the relationship between two variable
characteristics. Students analyse data graphically and algebraically to determine the
strength and nature of the relationship and use it, where appropriate, to make predictions.
Throughout this topic there is a focus on the use of electronic technology to facilitate the
calculations of statistics and construction of graphs.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
56 of 71
Subtopic 4.1: Sampling from Populations
Key Questions and Key Ideas
Considerations for Developing Teaching and Learning
Strategies
What is a sample and what is the purpose
of sampling?
 Census
 Population
 Sample
 Survey
Students investigate the Census through the ABS website.
They develop an understanding of the purpose of the census in
collecting data about the entire Australian population.
Describing the 2011 Census as the “largest logistical
peacetime operation ever undertaken in Australia” will form the
basis of discussion of the costs and time involved in carrying
out a Census. Discussion of how data can be collected without
the cost and time of surveying the entire population leads a
discussion of sampling. Students develop the understanding
that sampling provides an estimate of the population values
when a census is not used. Discussion of the size of a sample
includes the reasonableness of the sample to reflect the views
of the population.
What are some methods of sampling?
 Simple random
 Stratified, including appropriate
calculation
 Systematic
 Self-selected
A range of contexts are explored where data is to be collected
from an identified population via sampling, and the need for
different methods of sampling are required to ensure that the
data collected best represents the population. The problems
posed relate to the students’ interests and/or the community
context. Students use the various methods of sampling to
select a sample group from a variety of population scenarios.
What are the advantages and
disadvantages of each sampling method?
Students discuss the advantages and disadvantages of each
method of sampling. For example in a self-selected sample,
people or organizations choose to participate in a sample: they
are not directly approached by the researcher. Using this
method of sampling may lead to a sample consisting of people
with a particular point of view being involved, leading to
possible bias.
Students examine the four methods of determining samples:
simple random, stratified (including appropriate calculation),
systematic, and self-selected. The validity of the results from
each method of sampling for the described population is
discussed. For example, a simple random sample of all
employees of a business may not be appropriate if the question
under consideration only affects one part of the business.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
57 of 71
Key Questions and Key Ideas
How can bias occur in sampling?
 Faults in the process of collecting data
 Errors in surveys:
- sampling errors
- measurement errors
- coverage errors
- non-response error
What influence does sample size have on
the reliability of findings?
 Sample mean compared with population
mean
Considerations for Developing Teaching and Learning
Strategies
Discussion of the factors impacting on the collection of
data for analysis and errors in the survey include:
 questions in survey that don’t elicit the required
information
 incorrect instructions given to those collecting data
 inaccuracies associated with using a sample to
represent the population (sampling error)
 inaccuracies in measurement processes to collect data
(measurement error)
 sample selected using an inappropriate method
(coverage error)
 sample selected not large enough (coverage error)
 not getting all survey responses returned (nonresponse error)
Students combine two samples and see what difference
increasing the sample size makes to the sample mean.
The addition of increasingly more data from the
population when determining the sample mean leads to
the understanding that the larger the sample size is, the
more closely it will reflect the population characteristics.
The balance required between confidence in the results
and the expense in cost and/or time is discussed.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
58 of 71
Subtopic 4.2: Analysing and Representation of Sets of Data
Key Questions and Key Ideas
Considerations for Developing Teaching and
Learning Strategies
What do we do with the data now that we
have collected it?
Students discuss how the data collected can be
analysed. For each problem the focus is on
having two sets of data from the same population,
or from similar populations. Students analyse two
sets of data for the testing of two products for a
particular characteristic, or the size of a product
being produced by two different machines.
How do we compare two or more
distributions?
 Calculation of measures of central
tendency and spread
The measures considered include mean, median,
range, interquartile range, and standard deviation
for comparing distributions. For smaller data sets
the mean, median, range and interquartile range
can be calculated without the use of electronic
technology. The use of technology for
calculations of the standard deviation and for
large data sets is recommended.
The appropriateness of each measure as a
representative of the particular set of data is
emphasised.
What is the effect of outliers on the
distributions?
Students investigate the difference between
resistant and non-resistant measures of central
tendency and spread. Discussion of how outliers
can distort the different measures of central
tendency of a distribution is supported by
carefully chosen examples. Students choose the
measure of centre most appropriate for a given
purpose and a given set of data. Investigation of
the identified outlier is undertaken to determine if
it is a ‘real’ piece of data or due to an error in
collection (e.g. faulty equipment, error in
recording the value). Outliers should only be
excluded from the data set if they can be shown
to be invalid due to a data acquisition error or the
data is invalid in the context of the question.
Which types of representation are most
suitable for comparing two or more
distributions?
 Stem-and-leaf plots
 Box-and-whisker diagrams
Stem plots are useful for comparing smaller sets
of data. Students select the size of stem and the
treatment of data (truncation, rounding) that best
shows the distribution. Discussion of the general
shape and specific features of the plots is
essential. Box plots are useful for larger sets of
data and for comparing several distributions.
Students are able to transfer box plots created by
the calculator. Discussion of the general
distribution and specific features of the plots is
essential. The emphasis is on the interpretation of
these representations and comparisons of two or
more sets of data.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
59 of 71
Subtopic 4.3: Linear Correlation
Key Questions and Key Ideas
Is there a linear relationship between two
variables?
 Dependent and independent variable
 Patterns and features of scatter plots
 Description of association – direction,
form and strength
 Causality between variables
Considerations for Developing Teaching and Learning
Strategies
Students consider the relationship between two variables
(bivariate data). At least 10 data pairs are considered when
constructing scatter plots. Initially students produce the
scatter plots without electronic technology displaying
approximate scales, axis labels, and approximate positioning
of points. Where a strong relationship is evident students
indicate an approximate line of best fit by visual inspection
and describe the direction, form, and strength of the
relationship. Students understand that bivariate analysis
allows a causal relationship between two variables, to be
identified (e.g. that one variable is a consequence of the
other variable).
The scenarios chosen relate to workplace and everyday
contexts such as the amount of fertilizer on a crop and its
production of fruit or the stopping distance for a vehicle at
various speeds.
How can the degree of linear relationship
between two variables be established?
 Pearson’s correlation coefficient
When is it appropriate to draw a least
squares regression line?
2
 Coefficient of determination (r )
 Least squares regression line (‘line of
best fit’)
Students investigate the strength of the linear association by
calculating Pearson’s correlation coefficient (as indicated by
r), but also realise that even a strong correlation does not
imply causality.
Students understand that the square of r indicates the
fraction, or percentage, of the variation in the values that is
explained by least squares regression.
Students use electronic technology to calculate Pearson’s
correlation coefficient and the coefficient of determination to
decide if the relationship is strong enough to warrant drawing
a least squares regression line. If the strength of the
relationship is sufficient, students determine the location of,
and draw, the line of best fit using electronic technology. As a
2
guide an r ≥ 0.70 is sufficiently large to proceed with using a
least squares regression line.
How may the least squares regression
line be used?
Students interpolate values and, where appropriate,
extrapolate values, with an understanding of the validity of
the results.
How do outliers affect the degree of
linear relationship and the least squares
regression line?
Students that note neither r nor the least squares regression
line is resistant to outliers. Outliers are identified visually from
the scatterplot. Removing them from the data can strengthen
the correlation and improve the way the line of best fit
predicts. However, whether it is appropriate to do this, needs
to be considered carefully. Any removal of outliers should
only be made with reasonable justification.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
60 of 71
Topic 5: Investments and Loans
Students investigate a range of ways of investing and borrowing money. The simple
and compound interest calculations from Topic 6 in Stage 1 are extended by seeking
the best return on a lump sum investment. Students consider the effects of taxation
and inflation on the investment return.
Annuity calculations are developed by considering that investing generally involves
making regular deposits into an account, and borrowing requires regular repayments of
a loan.
Emphasis is placed on the use of technology, particularly spreadsheets and financial
graphical package in a graphics calculator, to enhance students’ opportunities to
investigate investments and loans. The use of technology is useful in encouraging
students to ask and investigate ‘What if …’.
Subtopic 5.1: Lump Sum Investments
Key Questions and Key Ideas
How can we invest a lump sum of money?
 Simple interest investments
 Compound interest investments
Considerations for Developing Teaching and
Learning Strategies
Consideration to earning money through investing
a lump sum into an account should include both
simple and compound interest calculations. As
these calculations are covered in Stage 1, the
focus is on rearranging the simple interest
formula to find all of the variables. However
electronic technology is utilised to allow students
to easily investigate problems requiring the range
of variables to be found for compound interest
problems. Graphical comparisons may assist
students in seeing the manner in which the
compounding of interest increases the growth of
the accumulated savings.
Students learn interest rate terminology including
variable rates and fixed rates, and develop an
understanding of the implications of the
terminology to the investment.
What may impact on the earnings of the
investment?
 Taxation of interest earned
 Inflation
Discussion and calculations of the impact of
taxation on investment earnings and calculations
of inflation provide students with an
understanding of the factors that diminish the
value of investment earning.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
61 of 71
Subtopic 5.2: Annuity Investments
Key Questions and Key Ideas
Considerations for Developing Teaching and Learning
Strategies
What is an investment called when a
regular payment is made into an account?
 Future-value annuity
It is likely that students will need to save money to make a
significant purchase in the future (e.g. a car or house). The
idea of making a regular deposit into an account rather than
investing a lump sum and waiting for it to build is introduced.
Discussion of the likelihood that students will have access to
a lump sum amount for investing leads them to an
understanding that it is common that a small regular
investment is used as a method of building up an amount of
savings towards a significant purchase.
What mathematics is used in calculating
future-value annuities?
 Future value
 The regular deposit
 The number of periods
 The interest rate
 The interest earned by the annuity
investment
 Assumptions made in long term annuity
calculations
Students understand that the calculations to determine how
much money a regular small investment could build up to
over time requires multiple applications of the simple or
compound interest formula. The class construct a
spreadsheet that allows either of these formulas to be used to
investigate the future value of a regular investment into a
saving account.
The students are introduced to the variables that are used in
the future-value annuity formula. Students use graphics
calculators, spreadsheets, or other appropriate technology for
their calculations.
Students investigate ‘What if …’ questions with varying future
values, payments, rates, and times and consider the interest
accrued after a given period.
The assumptions and limitations of these calculations are
discussed.
What applications of future-value annuity
savings plans are available? Discussion
and calculations should cover:
 long term investments
 superannuation
Personal or business saving is an essential part of any
investment strategy. The context of problems focus on
scenarios such as the regular deposit of money to save
towards a future savings goal, or the investment in
superannuation as a means of planning for retirement.
Calculations are related to ‘What if …’ questions such as:
 What if the interest rate changes?
 What if the regular deposit amount changes?
 What if the term of the investment is extended or reduced?
What may impact on the earnings of the
future value annuity?
 taxation of interest earned
 inflation
Students investigate the impact of taxation on the investment
earnings by calculating the tax liability. The effect of inflation
on the value of investments is investigated through
calculations, particularly when the investment is over a longer
term.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
62 of 71
Subtopic 5.3: Loan Annuities
Key Questions and Key Ideas
What is a present-value annuity?
What mathematics is used in calculating
the cost of a loan?
 Present value
 The regular payment
 The number of periods
 The interest rate
 The interest paid
 Assumptions made in long term annuity
calculations
How can we determine the most
appropriate loan option?
 Charges on loan accounts
 Comparison rates (no calculations
required)
Considerations for Developing Teaching and Learning
Strategies
It is likely that students will borrow to purchase a car,
property, or start a business. In this sub-topic students
investigate the cost of a loan, and consider if it is better
to take out a loan, or to save the money required to
purchase an item instead.
The students are introduced to variables that are used
in a present-value annuity formula. Students use
graphics calculators, spreadsheets, or other
appropriate technology for their calculations Students
investigate ‘What if …’ questions including varying
present values, payments, rates, and times.
Students use a loan simulator calculator from the
Internet to obtain results that they can compare with
their calculated result. Examination of a loan using
graphical methods highlights the high cost of interest in
a loan, particularly in its early stages.
Discussion of different loan terminology including split
loans and offset accounts familiarize students with the
variety of options that people taking out a home loan
can consider.
Students investigate strategies that can minimise the
interest paid on a loan. The methods of interest
minimisation that could be investigated include:
 Reducing the term
 Lump-sum payments
 Making larger repayments per period
 Making more frequent payments
Students investigate the various charges that can be
levied on loan accounts. They gain an understanding
that comparison rates take into account the fees and
charges connected with the loan. They use comparison
rates to determine the best loan option from two or
more loans being investigated. Discussion focuses on
the importance of factors other than the interest rate
when choosing a loan. Students are not required to
carry out comparison rate calculations.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
63 of 71
ASSESSMENT SCOPE AND REQUIREMENTS
All Stage 2 subjects have a school assessment component and an external
assessment component.
EVIDENCE OF LEARNING
The following assessment types enable students to demonstrate their learning in
Stage 2 Essential Mathematics.
School Assessment (70%)
 Assessment Type 1: Skills and Applications Tasks (30%)
 Assessment Type 2: Practical Report (40%)
External Assessment (30%)
 Assessment Type 3: Examination (30%)
Students provide evidence of their learning through eight assessments, including the
external assessment component. Students undertake:
 four skills and applications tasks
 three practical reports
 one examination.
It is anticipated that from 2018 all school assessments will be submitted
electronically.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
64 of 71
ASSESSMENT DESIGN CRITERIA
The assessment design criteria are based on the learning requirements and are used
by:
 teachers to clarify for the student what he or she needs to learn
 teachers and assessors to design opportunities for the student to provide
evidence of his or her learning at the highest possible level of achievement.
 The assessment design criteria consist of specific features that:
 students should demonstrate in their learning
 teachers and assessors look for as evidence that students have met the
learning requirements.
For this subject, the assessment design criteria are:
 concepts and techniques
 reasoning and communication.
The specific features of these criteria are described below.
The set of assessments, as a whole give students opportunities to demonstrate each of
the specific features by the completion of study of the subject.
Concepts and Techniques
The specific features are as follows:
CT1 Knowledge and understanding of concepts and relationships
CT2 Selection and application of techniques and algorithms to solve problems in a
variety of contexts
CT3 Development and application of mathematical models
CT4 Use of electronic technology to find solutions to mathematical problems
Reasoning and Communication
The specific features are as follows:
RC1 Interpretation and evaluation of mathematical results with an understanding of
their reasonableness and limitations
RC2 Knowledge and use of appropriate mathematical notation, representations, and
terminology.
RC3 Communication of mathematical ideas and reasoning, to develop logical
arguments
RC4 Forming and testing of valid conjectures *
* In this subject the development and testing of conjectures (RC4) is not intended to
include formal mathematical proof.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
65 of 71
SCHOOL ASSESSMENT
Assessment Type 1: Skills and Applications Tasks (30%)
Students complete four skills and applications tasks.
Skills and applications tasks are completed under the direct supervision of the teacher.
Students find solutions to mathematical problems that may:
 be routine, analytical, and/or interpretative
 be posed in a variety of familiar and new contexts
 require discerning use of electronic technology.
In setting skills and applications tasks, teachers may provide students with information
in written form or in the form of numerical data, diagrams, tables, or graphs. A task
should require the student to demonstrate an understanding of relevant mathematical
concepts and relationships.
Students select appropriate techniques or algorithms and relevant mathematical
information and apply them to find solutions to routine, analytical, and/or interpretative
problems. Some of these problems should be set in everyday and workplace contexts.
Students provide explanations arguments, and use correct notation, terminology, and
representation throughout the task. They may be required to use electronic technology
appropriately to aid and enhance the solution of some problems.
For this assessment type, students provide evidence of their learning in relation to the
following assessment design criteria:
 concepts and techniques
 reasoning and communication.
Assessment Type 2: Practical Report (40%)
Students complete three practical reports.
Students, either individually or in a group, undertake planning, apply their numeracy
skills to gather, represent, analyse, and interpret data, and propose or develop a
solution to a mathematical problem based in an everyday or workplace context. The
subject of the mathematical problem may be derived from one or more subtopics,
although it can also relate to a whole topic or across topics.
A mathematical problem may be initiated by a student, a group of students, or the
teacher. Teachers may give students clear and detailed advice and instructions for
setting and solving the mathematical problem, or may provide guidelines for students to
develop contexts, themes, or aspects of their own choice. Teachers should give some
direction about the appropriateness of each student’s choice and guide and support
students’ progress in a mathematical solution.
If a mathematical problem is undertaken by a group, students explore the problem and
gather data together to develop a model or solution individually. Each student must
submit an individual practical report.
The practical report may take a variety of forms, but would usually include the
following:
 an outline of the problem to be explored
 the method used to find a solution
 the application of the mathematics, including, for example:
– generation or collection of relevant data and/or information, with a summary
of the process of collection
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
66 of 71


– mathematical calculations and results, using appropriate representations
– discussion and interpretation of results, including consideration of the
reasonableness and limitations of the results
the results and conclusions in the context of the problem
a bibliography and appendices, as appropriate.
The format of a practical report may be written or multimodal.
Each practical report should be a maximum of 8 pages if written, or the equivalent in
multimodal form.
For this assessment type, students provide evidence of their learning in relation to the
following assessment design criteria:
 concepts and techniques
 reasoning and communication.
EXTERNAL ASSESSMENT
Assessment Type 3: Examination (30%)
Students undertake a 2-hour external examination in which they answer questions on
the following three topics:
 Topic 2: Measurement
 Topic 4: Statistics
 Topic 5: Investment and Loans.
The examination is based on the ‘key questions and key ideas’ in the topics 2, 4, and 5.
The ‘considerations for developing teaching and learning strategies’ are provided as a
guide only, although applications described under this heading may provide contexts
for examination questions.
The examination consists of a range of problems, some focusing on knowledge, routine
skills, and applications, and others focusing on analysis and interpretation. Students
provide explanations and arguments, and use correct mathematical notation,
terminology, and representation throughout the examination.
The examination is divided into two parts:
 Part 1 (40%, 50 minutes): Calculations without electronic technology
(graphics/scientific calculators). Part 1 covers aspects of:
– Topic 2: Measurement
– Topic 4: Statistics.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
67 of 71

Part 2 (60%, 70 minutes): Calculations with access to approved electronic
technology (graphics/scientific calculators). Part 2 covers aspects of:
– Topic 2: Measurement
– Topic 4: Statistics
– Topic 5: Investment and Loans.
Students have 10 minutes in which to read both Part 1 and Part 2 of the examination
before commencing to write their answers. At the end of the reading time, students
begin their answers to Part 1. For this part, students do not have access to electronic
technology (graphics or scientific calculators).
At the end of the specified time for Part 1, students stop writing. Students submit Part 1
to the invigilator.
Students have access to Board-approved calculators for Part 2 of the examination. The
invigilator coordinates the distribution of calculators (graphics and scientific). Once all
students have received their calculators, the time allocated for Part 2 begins, and
students resume writing their answers.
The SACE Board will provide a list of approved graphics calculators for use in
Assessment Type 3: Examination that meet the following criteria:








have flash memory that does not exceed 5.0 MB (this is the memory that can be
used to store add-in programs and other data)
can calculate derivative and integral values numerically
can calculate probabilities
can calculate with matrices
can draw a graph of a function and calculate the coordinates of critical points
using numerical methods
solve equations using numerical methods
do not have a CAS (Computer Algebra System)
do not have SD card facility (or similar external memory facility).
Graphics calculators that currently meet these criteria and are approved for 2017 are
as follows:
Casio fx-9860G AU
Casio fx-9860G AU Plus
Hewlett Packard HP 39GS
Sharp EL-9900
Texas Instruments TI-83 Plus
Texas Instruments TI-84 Plus
Texas Instruments – TI 84 Plus C –silver edition
Texas Instruments – TI 84 Plus CE
Other graphic calculators will be added to the approved calculator list as they become
available.
Students may bring two graphics calculators or one scientific calculator and one
graphics calculator into the examination room.
There is no list of Board-approved scientific calculators. Any scientific calculator,
except those with an external memory source, may be used.
For this assessment type, students provide evidence of their learning in relation to the
following assessment design criteria:
 concepts and techniques
 reasoning and communication.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
68 of 71
PERFORMANCE STANDARDS
The performance standards describe five levels of achievement, A to E.
Each level of achievement describes the knowledge, skills, and understanding that
teachers and assessors refer to in deciding how well a student has demonstrated his or
her learning, on the basis of the evidence provided.
During the teaching and learning program the teacher gives students feedback on their
learning, with reference to the performance standards.
At the student’s completion of study of each school assessment type, the teacher
makes a decision about the quality of the student’s learning by:
 referring to the performance standards
 assigning a grade between A+ and E- for the assessment type.
The student’s school assessment and external assessment are combined for a final
result, which is reported as a grade between A+ and E-.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
69 of 71
Performance Standards for Stage 2 Essential Mathematics
A
Concepts and Techniques
Reasoning and Communication
CT1 Comprehensive knowledge and understanding of
concepts and relationships.
RC1 Comprehensive interpretation and critical evaluation of
mathematical results, with an in-depth understanding of their
reasonableness and possible limitations.
CT2 Highly effective selection and application of techniques
and algorithms to find efficient and accurate solutions to
routine and complex problems in a variety of contexts.
CT3 Successful development and application of
mathematical models to find concise and accurate
solutions.
CT4 Appropriate and effective use of electronic technology
to find accurate solutions to routine and complex problems.
B
CT1 Some depth of knowledge and understanding of
concepts and relationships.
CT2 Mostly effective selection and use of mathematical
techniques and algorithms to find accurate solutions to
routine and some complex problems in a variety of
contexts.
C
RC4 Formation and testing of appropriate conjectures, using
sound mathematical evidence.
RC1 Mostly appropriate interpretation and some critical
evaluation of mathematical results, with some depth of
understanding of their reasonableness and possible
limitations.
RC2 Mostly accurate use of appropriate mathematical
notation, representations, and terminology.
RC3 Mostly effective communication of mathematical ideas
and reasoning to develop logical arguments.
CT4 Mostly appropriate and effective use of electronic
technology to find accurate solutions to routine and some
complex problems.
RC4 Formation and testing of mostly appropriate
conjectures, using some mathematical evidence.
CT1 Generally competent knowledge and understanding of
concepts and relationships.
RC1 Generally appropriate interpretation and evaluation of
mathematical results, with some understanding of their
reasonableness and possible limitations.
RC2 Generally appropriate use of mathematical notation,
representations, and terminology with some inaccuracies.
CT3 Application of mathematical models to find generally
accurate solutions.
RC3 Generally effective communication of mathematical
ideas and reasoning to develop some logical arguments.
CT4 Generally appropriate and effective use of electronic
technology to find mostly accurate solutions to routine
problems.
RC4 Formation of an appropriate conjecture and some
attempt to test it using mathematical evidence.
CT1 Basic knowledge and some understanding of concepts
and relationships.
RC1 Some interpretation and attempted evaluation of
mathematical results, with some awareness of their
reasonableness and possible limitations.
CT2 Some effective selection and use of mathematical
techniques and algorithms to find some accurate solutions
to routine problems in some contexts.
E
RC3 Highly effective communication of mathematical ideas
and reasoning to develop logical and concise arguments.
CT3 Attempted development and successful application of
mathematical models to find accurate solutions.
CT2 Generally effective selection and use of mathematical
techniques and algorithms to find mostly accurate solutions
to routine problems in a variety of contexts.
D
RC2 Proficient and accurate use of appropriate mathematical
notation, representations, and terminology.
RC2 Some appropriate use of mathematical notation,
representations, and terminology, with some accuracy.
CT3 Some application of mathematical models to find some
accurate solutions.
RC3 Some communication of mathematical ideas with
attempted reasoning and/or arguments.
CT4 Some appropriate use of electronic technology to find
some accurate or partially accurate solutions to routine
problems.
RC4 Attempted formation of a conjecture with limited attempt
to test it using mathematical evidence.
CT1 Limited knowledge or understanding of concepts and
relationships.
RC1 Limited interpretation or evaluation of mathematical
results, with limited awareness of their reasonableness and
possible limitations.
CT2 Attempted selection and limited use of mathematical
techniques or algorithms, with limited accuracy in solving
routine problems.
RC2 Limited use of appropriate mathematical notation,
representations, or terminology, with limited accuracy.
CT3 Attempted application of mathematical models, with
limited accuracy.
RC3 Attempted communication of mathematical ideas, with
limited reasoning.
CT4 Attempted use of electronic technology, with limited
accuracy in solving routine problems.
RC4 Limited attempt to form or test a conjecture.
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
70 of 71
ASSESSMENT INTEGRITY
The SACE Assuring Assessment Integrity Policy outlines the principles and
processes that teachers and assessors follow to assure the integrity of student
assessments. This policy is available on the SACE website (www.sace.sa.edu.au)
as part of the SACE Policy Framework.
The SACE Board uses a range of quality assurance processes so that the grades
awarded for student achievement, in both the school assessment and the external
assessment, are applied consistently and fairly against the performance standards
for a subject, and are comparable across all schools.
Information and guidelines on quality assurance in assessment at Stage 2 are
available on the SACE website (www.sace.sa.gov.au)
SUPPORT MATERIALS
SUBJECT-SPECIFIC ADVICE
Online support materials are provided for each subject and updated regularly on the
SACE website (www.sace.sa.edu.au). Examples of support materials are sample
learning and assessment plans, annotated assessment tasks, annotated student
responses, and recommended resource materials.
ADVICE ON ETHICAL STUDY AND RESEARCH
Advice for students and teachers on ethical study and research practices is available
in the guidelines on the ethical conduct of research in the SACE on the SACE
website (www.sace.sa.edu.au).
Stage 1 and Stage 2 Essential Mathematics Subject Outline
Draft for online consultation – 11 March – 17 April 2015
Ref: A374762
71 of 71