Post on 03-May-2021
NATIONAL SENIOR CERTIFICATE FOR ADULTS
NQF LEVEL 4
DRAFT SUBJECT STATEMENT
MATHEMATICAL LITERACY
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MATHEMATICAL LITERACY
Contents
MATHEMATICAL LITERACY .............................................................................................. 1
MATHEMATICAL LITERACY .............................................................................................. 2
INTRODUCTION ..................................................................................................................... 2
MATHEMATICAL LITERACY .............................................................................................. 2
Some key requirements for Mathematical Literacy ................................................................... 4
1. Mathematical Literacy requires the use of some basic mathematical skills, concepts and
content. ................................................................................................................................... 4
2. Mathematical Literacy requires non-contrived real-life contexts. ..................................... 4
3. Mathematical Literacy requires communication and decision making skills. ................... 4
4. Mathematical Literacy problem solving requires the integration of content ..................... 4
and/or skills. ........................................................................................................................... 4
AIMS.......................................................................................................................................... 5
EXIT LEVEL OUTCOMES ...................................................................................................... 6
ELO 1 – Use numbers and their relationships to estimate, calculate and investigate the
financial aspects of personal, business and social life in order to solve problems in real
contexts. ................................................................................................................................. 6
ELO 2 – Recognise, analyse, interpret, describe and represent various functional
relationships in order to solve problems in real contexts. ...................................................... 6
ELO 3 – Use measurement to estimate and calculate physical quantities in order to
describe and represent properties of and relationships between 2D and 3D objects. .......... 7
ELO 4 – Collect, summarise, represent and analyse data and apply knowledge of statistics
and probability in order to communicate, justify, predict and critically analyse results and
draw conclusions. ................................................................................................................... 7
ASSESSMENT CRITERIA (AC) FOR EXIT LEVEL OUTCOMES ...................................... 8
Mathematical Literacy assessment taxonomy ......................................................................... 15
DESCRIPTION OF THE LEVELS IN THE MATHEMATICAL LITERACY
ASSESSMENT TAXONOMY ................................................................................................ 15
SCHEME OF ASSESSMENTS .............................................................................................. 17
Weightings for Content Areas ................................................................................................. 18
Weightings for Taxonomy Levels ........................................................................................... 18
CONTENT STRUCTURE....................................................................................................... 19
Contexts ................................................................................................................................... 20
CONTENT AREAS ................................................................................................................. 21
FORMULA SHEET................................................................................................................. 35
GLOSSARY ............................................................................................................................ 36
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MATHEMATICAL LITERACY
INTRODUCTION
The Department of Higher Education and Training provides qualifications for adults and out
of school youth as part of contributing to the National Skills Development Strategy 111. This
qualification is known as the National Senior Certificate for Adults (NASCA) and is offered
at Level 4 on the General and Further Education and Training Qualifications sub-framework.
This qualification is aimed primarily at candidates in the age group 18 to 55 years old.
MATHEMATICAL LITERACY
Mathematical Literacy provides candidates with an awareness and understanding of the role
that mathematics plays in the modern world. Mathematical Literacy is a subject driven by
life-related applications of mathematics. It enables candidates to develop the ability and
confidence to think numerically and spatially in order to interpret and critically analyse
everyday situations and to solve problems.
The inclusion of Mathematical Literacy as a fundamental subject in the NASCA curriculum
will ensure that our citizens of the future are highly numerate consumers of mathematics. In
the teaching and learning of Mathematical Literacy, candidates will be provided with
opportunities to engage with reallife problems in different contexts, and so to consolidate and
extend basic mathematical skills. Thus, Mathematical Literacy will result in the ability to
understand mathematical terminology and to make sense of numerical and spatial information
communicated in tables, graphs, diagrams and texts. Furthermore, Mathematical Literacy will
develop the use of basic mathematical skills in critically analysing situations and creatively
solving everyday problems.
In everyday life a person is continually faced with mathematical demands which the
adolescent and adult should be in a position to handle with confidence. These demands
frequently relate to financial issues such as hire-purchase, mortgage bonds, and investments.
There are others, however, such as the ability to read a map, follow timetables, estimate and
calculate areas and volumes, and understand house plans and sewing patterns. Situations,
such as in cooking and the use of medicine, requiring the efficient use of ratio and proportion,
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are encountered on a daily basis. Hence, Mathematical Literacy is required by a self-
managing person.
The workplace requires the use of fundamental numerical and spatial skills in order to
efficiently meet the demands of the job. To benefit from specialised training for the
workplace, a flexible understanding of mathematical principles is often necessary. This
numeracy must enable the person to, for example, deal with work-related formulae, read
statistical charts, deal with schedules and understand instructions involving numerical
components. Such numeracy will enable the person to be a contributing worker.
To be a participating citizen in a developing democracy, it is essential that the adolescent and
adult have acquired a critical stance with regard to mathematical arguments presented in the
media and other platforms. The concerned citizen needs to be aware that statistics can often
be used to support opposing arguments, for example, for or against the use of an ecologically
sensitive stretch of land for mining purposes. In the information age, the power of numbers
and mathematical ways of thinking often shape policy. Unless citizens appreciate this, they
will not be in a position to use their vote appropriately.
Therefore, the NASCA Mathematical Literacy should enable the candidate to become a self-
managing person, a contributing worker and a participating citizen in a developing
democracy.
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Some key requirements for Mathematical Literacy
1. Mathematical Literacy requires the use of some basic mathematical skills, concepts
and content.
The mathematical content required is limited to some basic mathematical concepts and skills
that are in essential in making sense of real life scenarios that require an understanding of
numerical and statistical concepts in the daily lives of individuals and the workplace.
2. Mathematical Literacy requires non-contrived real-life contexts.
The contexts that candidates should be exposed to in Mathematical Literacy should be non-
contrived (i.e. taken from actual and real situations) and relevant, and relate to everyday life,
the working, social and political environments. Candidates are also expected to apply non-
mathematical skills to make sense of real life problems.
3. Mathematical Literacy requires communication and decision making skills.
A mathematical literate person should be able to evaluate all options for a solution to a
problem, decide on the most appropriate option and then communicate the solution using
appropriate terminology (both mathematical and non-mathematical).
4. Mathematical Literacy problem solving requires the integration of content
and /or skills.
The content, skills and contexts in this document are organised and grouped according to
content areas. The solving of real-life problems generally requires the use of content and/or
skills taken from a variety of content areas which requires the skill to identify and use
different techniques integrated from various content areas.
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AIMS
The National Senior Certificate for Adults (NASCA) aims to provide
evidence that the adult candidates are equipped with a sufficiently
substantial basis of discipline-based knowledge, skills and values to
enhance meaningful social, political and economic participation, to
form a basis for further and/or more specialist learning and possibly
to enhance the likelihood of employment. In these respects, the
NASCA promotes the holistic development of adult candidates. The
intention is also that the quality of the learning offered by the NASCA
will reinvigorate an interest in learning for many who have had
negative experiences in school.
NASCA aims to service an identifiable need in the basic adult education
system, not currently met by other qualifications on the NQF and to create
pathways for further learning. It is designed to provide opportunities for
people who have limited or no access to continuing education and training
opportunities.
NASCA aims to produce candidates that are able to:
Use mathematical and non-mathematical knowledge, skills and understanding to
solve problems in real-life contexts, in a range of workplace, personal, further
learning and community settings.
Demonstrate the necessary applied knowledge and skills identified
for competence including the use of necessary technology.
Communicate arguments and strategies when solving problems
using appropriate mathematical and non-mathematical language
language
Use mathematics effectively, efficiently and critically to make informed
decisions in their daily lives.
Reflect on own learning in order to re-establish an interest in
learning and further study.
Gain an opportunity to prepare for post-school options of employment and
further training.
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Improve their quality of life and free the potential of each person.
EXIT LEVEL OUTCOMES
Assessment is a critical element of NASCA. It is a process of collecting and interpreting
evidence in order to determine the candidate’s progress in learning and to make a
judgment about a candidate’s performance.
When assessment is used to record a judgment of the competence or performance of the
candidate, it serves a summative purpose. Summative assessment gives a picture of a
candidate’s competence or progress at any specific moment. It can occur at the end of a
single learning activity, a unit, cycle, term, semester or year of learning. Summative
assessment in NASCA will take the form of a year-end examination to enable candidates
to demonstrate competence.
The following Exit Level Outcomes (ELO’s) are expected of Mathematical Literacy
candidates:
ELO 1 – Use numbers and their relationships to estimate, calculate and investigate the
financial aspects of personal, business and social life in order to solve problems in real
contexts.
The emphasis of this Exit Learning Outcome is on the investigation and solution of
real life problems requiring an understanding of numbers and their use in
calculations, especially in financial situations, ranging from personal to social issues.
Candidates should develop appropriate estimation and mental calculation skills to
perform simple calculations. Appropriate and effective calculator skills are essential
for the solving of problems.
ELO 2 – Recognise, analyse, interpret, describe and represent various functional
relationships in order to solve problems in real contexts.
The emphasis of this Exit Learning Outcome is to give candidates opportunities to
investigate and make sense of functional relationships arising in the context of other
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subjects, work or real life situations. Candidates should be able to reflect on
relationships between variables and use various techniques to determine values of
variables in solving problems.
ELO 3 – Use measurement to estimate and calculate physical quantities in order to
describe and represent properties of and relationships between 2D and 3D objects.
The emphasis of this Exit Learning Outcome is on the development of spatial
understanding and geometric skills relating to real life contexts. A variety of
applications are available in design, art, geography and other fields to develop
measurement skills.
ELO 4 – Collect, summarise, represent and analyse data and apply knowledge of
statistics and probability in order to communicate, justify, predict and critically analyse
results and draw conclusions.
The emphasis of this Exit Learning Outcome is on the ability of candidates as
consumers to interpret and use data. Candidates should be able to reflect on the use,
misuse and meaning of different graphical representations of data in a variety of real
life situations. The critical awareness of data manipulation to prove opposing views
should be developed by candidates.
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ASSESSMENT CRITERIA (AC) FOR EXIT LEVEL OUTCOMES
AC 1.1 Analyse personal and business finances and evaluate the effects of taxation,
inflation and changing interest rates on personal credit and investment growth
options to make decisions.
We know this when the candidate is able to:
1.1.1 Solve problems in different contexts (including financial) by
estimation and accurate calculation skills( including calculator skills)
where appropriate, inclusive of:
(a) Working with formulae, including simple interest, compound
interest, annuities and loans (direct application of formulae only)
(b) Hire-purchase calculations
(c) Using the basic order of operations with numbers
(d) Calculating personal Income tax
(e) VAT calculations
1.1.2 Analyse and critically interpret different financial situations inclusive
of:
(a) Personal and business finances
(b) The effects of taxation and fluctuating interest rates
(c) Income and expenditure
(d) The impact of changing interest (simple and compound) within
personal finance contexts
(e) Estimating and calculating profit/loss.
1.1.3 Interpret calculated answers appropriately to the given problem by:
(a) Interpreting answers in context
(b) Reworking a problem if the calculated answer is not suitable or if
the initial conditions change
(b) Logically interpreting and communicating calculated answers in
relation to the problem.
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AC 1.2 Analyze provincial and national budgets in terms of health and welfare issues,
job opportunities, etc., using mathematical skills and knowledge.
We know this when the candidate is able to:
1.2.1 Critically analyse budgets by performing relevant calculations to verify
budgeted amounts
1.2.2 Compare budgets by performing relevant calculations.
AC 1.3 Compare different currencies for best investment opportunities.
We know this when the candidate is able to:
1.3.1 Calculate currency conversions relating to investments
1.3.2 Select the most appropriate currency for an investment
1.3.3 Determine the effects of currency fluctuations for investments.
AC 1.4 Understand inflation using mathematical investigation.
We know this when the candidate is able to:
1.4.1 Calculate annual inflation for given products over multiple periods
1.4.2 Calculating and comparing annual inflation rates
1.4.3 Analyse and critically interpret the effects of inflation
AC 1.5 Use ratio, rate, proportions and percentages to solve problems.
We know this when the candidate is able to:
1.5.1 Perform calculations using given percentages, ratios, rates
1.5.2 Perform calculations using proportions(direct or inverse)
1.5.3 Solve problems by calculating percentage, ratio, rate or proportion.
1.5.4 Increase or decrease values in a given ratio or percentage.
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AC 2.1 Translate between different representations of functions to solve
problems and analyze situations.
We know this when the candidate is able to:
2.1.1 Work with tabulated data and formulae in different real life situations to
determine input and output values
2.1.2 Draw graphs from tabulated data and formulae by point by point
plotting
2.1.3 Critically interpret tables and graphs relating to real life situations by:
(a) Finding values of variables at certain points
(b) Describing overall trends
(c) Identify maximum and minimum values
(d) Estimate input and output values
2.1.4 Critically interpret graphs in real life situations inclusive of more than
one graph on a system of axes.
2.1.5 Find break-even points involving linear functions from tabulated data or
from graphs.
AC 2.2 Analyze and interpret graphs of sine, cosine and tangent functions to
solve problems.
We know this when the candidate is able to:
2.2.1 Read input and output values for sine, cosine and tangent functions
from given graphs
2.2.2 Identify type of trigonometric graphs from given graphs.
2.2.3 Solve simple trigonometrical equations using given graphs.
AC 2.3 Solve design and planning problems through optimizing a function in two
discreet variables.
We know this when the candidate is able to:
2.3.1 Identify the feasible region by plotting discrete values.
2.3.2 Determine if given quantities will be feasible.
2.3.3 Identify optimum solution for a given situation.
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AC 3.1 Represent and identify views of scale drawings of plans, maps and models
and calculate values from plans, maps and models.
We know this when the candidate is able to:
3.1.1 Convert units of measurement within the metric system.
3.1.2 Convert units of measurement between different scales and systems.
3.1.3 Convert units of measurement between different scales and systems in
context.
3.1.4 Use and interpret scale drawings of plans, maps and models to identify
views, estimate and calculate values according to scale.
3.1.5 Determine scale of maps and plans
3.1.6 Represent different perspective views of plans and models
AC 3.2 Use maps and layout plans to interpret and analyze spatial relations.
We know this when the candidate is able to use map grids, including compass
directions to:
3.2.1 Determine locations and grid reference
3.2.2 Plan trips
3.2.3 Describe routes between two different locations
3.2.4 Describe relative positions.
AC 3.3 Solve 2D and 3D problems using measurement and calculations.
We know this when the candidate is able to:
3.3.1 Solve problems in 2-dimensional and 3-dimensional contexts
by estimating, measuring and calculating values which involve:
(a) Lengths, distances and perimeters(circumference)
(b) Areas of common polygons and circles (including combinations of
these shapes)
(c) Surface areas of right prisms and right cylinders
(d) Volumes of right prisms, right cylinders and spheres.
3.3.2 Verifying values of the solutions against the contexts in terms of
suitability and degree of accuracy.
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AC 3.4 Use basic trigonometric ratios to interpret situations and solve problems.
We know this when the candidate is able to:
3.4.1 Name the parts of a right angled triangle
3.4.2 Find the missing side of a right angled triangle using the
Pythagorean Theorem
3.4.3 Find the measure of an unknown side or angle of a right
angled triangle using sine, cosine or tangent ratios using a scientific
calculator.
3.4.4 Solve problems using right angled triangle trigonometry
3.4.5 Finding angles of elevation and depression
3.4.6 Find heights and horizontal distances
AC 4.1 Use a representative sample from a population to solve a problem with
due sensitivity to issues relating to bias.
We know this when the candidate is able to:
Investigate a problem on issues relating to social, environmental and political
factors and people’s opinions by:
4.1.1 Using appropriate statistical methods
4.1.2 Selecting a representative sample
4.1.3 Comparing data from different sources and samples
AC 4.2 Calculate measures of central tendency and spread of data.
We know this when the candidate is able to:
4.2.1 Understand that data can be summarized in different ways by
calculating:
(a) Mean
(b) Median
(c) Mode
(d) Range
4.2.2 Make comparisons and draw conclusions using :
(a) Mean
(b) Median
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(c) Mode
(d) Range
(e) Quartiles (Interpretations only)
(f) Percentiles (Interpretations only)
AC 4.3 Represent and analyze data and statistics
We know this when the candidate is able to:
4.3.1 Select, justify and use a variety of methods to summarise and display
data, inclusive of:
(a) Tallies
(b) Tables, including frequency tables
(c) Pie charts
(d) Single and compound bar graphs
(e) Line and broken line graphs
(f) Histograms
(g) Frequency Polygons
(h) Stem and Leaf diagram
4.3.2 Select, justify and use a variety of methods to summarise and display
data to describe trends
AC 4.4 Working with simple notions of probability in order to make sense of
probability statements.
We know this when the candidate is able to:
4.4.1 Express probability values in terms of fractions, ratios and percentages.
4.4.2 Use tree diagrams to determine the probability of non-independent
events.
4.4.3 Effectively communicate conclusions and predictions that can be made
from the analysis and representation of data, using appropriate terminology
such as, trends, increase, decrease, constant, impossible, likely and even
chance.
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AC 4.5 Critically evaluate and make recommendations to statistically based
arguments.
We know this when the candidate is able to:
4.5.1 Critically interpret data and representations thereof (with the awareness
of sources of error and bias) in order to draw conclusions, make
predictions, predict trends and critique other interpretations.
4.5.2 Identify and interpret misuse of statistics.
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Mathematical Literacy Assessment Taxonomy
Assessment can encompass different levels of cognitive demands ranging from the simple
reproduction of facts to the detailed analysis and the use of varied and complex methods and
approaches. Assessment in Mathematical Literacy follows the same principles and to
determine the level of cognitive demand it is useful to use a hierarchy or taxonomy.
The PISA (Programme for International Candidate Assessment) Assessment Framework
(OECD,
2003) provides a possible taxonomy for assessment of Mathematical Literacy based on what
it
calls competency clusters. The TIMSS (Trends in Mathematics and Sciences Study)
Assessment
Framework (IEA, 2001) provides another framework for cognitive domains. Using these two
frameworks the following taxonomy for Mathematical Literacy is proposed:
Level 1: Knowing
Level 2: Applying routine procedures in familiar contexts
Level 3: Applying multistep procedures in a variety of contexts
Level 4: Reasoning and reflecting
DESCRIPTION OF THE LEVELS IN THE MATHEMATICAL LITERACY
ASSESSMENT TAXONOMY
Level 1: Knowing
Tasks at the knowing level of the Mathematical Literacy taxonomy require learners to:
(a) Calculate using the basic operations including:
algorithms for +, -, ×, and ÷;
appropriate rounding of numbers;
estimation;
calculating a percentage of a given amount; and
measurement.
(b) Know and use appropriate vocabulary such as equation, formula, bar graph, pie chart,
table of values, mean, median and mode.
(c) Know and use formulae such as the area of a rectangle, a triangle and a circle where
each of the required dimensions is readily available.
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(d) Read information directly from a table (e.g. the time that bus number 1234 departs
from the terminal).
Level 2: Applying routine procedures in familiar contexts
Tasks at the applying routine procedures in familiar contexts level of the Mathematical
Literacy taxonomy require learners to:
(a) Perform well-known procedures in familiar contexts. Learners know what procedure
is
required from the way the problem is posed. All of the information required to solve
the problem is immediately available to the candidate.
(b) Solve equations by means of trial and improvement or algebraic processes.
(c) Draw data graphs for provided data.
(d) Draw algebraic graphs for given equations.
(e) Measure dimensions such as length, weight and time using appropriate measuring
instruments sensitive to levels of accuracy.
Level 3: Applying multistep procedures in a variety of contexts
Tasks at the applying multistep procedures in a variety of contexts level of the Mathematical
Literacy taxonomy require learners to:
(a) Solve problems using well-known procedures. The required procedure is, however, not
immediately obvious from the way the problem is posed. Learners will have to decide
on the most appropriate procedure to solve the solution to the question and may have
to perform one or more preliminary calculations before determining a solution.
(b) Select the most appropriate data from options in a table of values to solve a problem.
(c) Decide on the best way to represent data to create a particular impression.
Level 4: Reasoning and reflecting
Tasks at the reasoning and reflecting level of the Mathematical Literacy taxonomy require
learners to:
(a) Pose and answer questions about what mathematics they require to solve a problem
and
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then to select and use that mathematical content.
(b) Interpret the solution they determine to a problem in the context of the problem and
where necessary to adjust the mathematical solution to make sense in the context.
(c) Critique solutions to problems and statements about situations made by others.
(d) Generalise patterns observed in situations, make predictions based on these patterns
and/or other evidence and determine conditions that will lead to desired outcomes.
SCHEME OF ASSESSMENTS
The nationally set, marked and moderated examination will consist of two 3 hour papers:
Paper 1
A ‘basic knowing and routine applications paper’.
Will consist of between five to eight questions.
Each content area will have equal weighting in this paper.
All, but the first question will focus on a single context.
All questions should integrate Assessment criteria from more than content area.
All questions will include sub-questions from the different cognitive levels of the
Mathematical Literacy taxonomy levels appropriate for this paper.
Paper 2
An ‘applications, reasoning and reflecting’ paper.
Will consist of between four to six questions.
Questions will require more interpretation and application of the information
provided.
Each content area will have equal weighting in this paper.
All of the questions will focus on a single context.
All questions will integrate Assessment Criteria from more than one content
area.
All questions will include sub-questions from the different cognitive levels of the
Mathematical Literacy taxonomy levels appropriate to this paper.
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Weightings for Content Areas
Content
Area (CA)
Numbers &
Operations
Functions &
Relationships
Space, Shape
Measurement
Data
Handling
TOTAL
Paper 1 38 (±5) 38 (±5) 37 (±5) 37 (±5) 150
Paper 2 38 (±5) 38 (±5) 37 (±5) 37 (±5) 150
Weightings for Taxonomy Levels
Taxonomy
Levels
LEVEL 1
Knowing
LEVEL 2
Applying
routine
procedures in
familiar
contexts
LEVEL 3
Apply multi-
step
procedures in a
variety of
contexts
LEVEL 4
Reasoning
and
reflecting
Paper 1 60% 40% 0% 0%
Paper 2 0% 20% 40% 40%
Average Total 30% 30% 20% 20%
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CONTENT STRUCTURE
The following Content Areas (CA’s) will constitute the Mathematical Literacy
curriculum:
CA 1 Numbers and Operations – estimate and calculate, investigate and
monitor the financial aspects of personal, business and national life and
to investigate and solve problems in other contexts.
CA 2 Functions and Relationships – recognize, analyze, interpret,
describe and represent various functional relationships in order to
solve problems in real and simulated contexts.
CA 3 Space, Shape and Measurement using appropriate measuring
instruments to estimate, calculate physical quantities and to describe
and represent properties of and relationships, between 2- and 3-
dimensional objects in a variety of orientations and positions.
CA 4 Data Handling - collect, summarize, display and analyze data and apply
knowledge of statistics and basic probability to communicate, justify, predict
and critically interrogate findings and draw conclusions.
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Contexts
Contexts are essential for the development of Mathematical Literacy and require that the
subject be embedded in the everyday lives of the candidates. Local community practices, the
home environment and local industry provide a good source of relevant contexts to explore.
Resources obtained from media can be used effectively to expose candidates to current
happenings (locally, nationally and internationally).
The approach to developing Mathematical Literacy is to engage with contexts rather than to
apply Mathematics already learnt to a context. Being mathematically literate is an essential
for the development of the responsible citizen, the contributing worker and the self-managing
individual.
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CONTENT AREAS
CONTENT AREA 1 (CA 1): NUMBERS AND OPERATIONS
Estimate , calculate, investigate and monitor the financial aspects of personal, business and
national life and to investigate and solve problems in other contexts.
EXIT LEVEL OUTCOME 1 (ELO 1)
Use numbers and their relationships to estimate, calculate and investigate the financial aspects of personal, business and social life in
order to solve problems in real contexts.
The emphasis of this Exit Learning Outcome is on the investigation and solution of real life problems requiring an understanding of
numbers and their use in calculations, especially in financial situations, ranging from personal to social issues. Candidates should
develop appropriate estimation and mental calculation skills to perform simple calculations. Appropriate and effective calculator skills
are essential for the solving of problems.
ASSESSMENT CRITERIA (AC)
AC 1.1 Analyze personal and business finances and evaluate the effects of taxation, inflation and changing
interest rates on personal credit and investment growth options to make decisions.
We know this when the candidate is able to:
1.1.1 Solve problems in different contexts (including financial) by
estimation and accurate calculation skills( including calculator skills)
where appropriate, inclusive of:
(a) Working with formulae, including simple interest, compound
interest, annuities and loans (direct application of formulae only)
22
(b) Hire-purchase calculations
(c) Using the basic order of operations with numbers
(d) Calculating personal Income tax
(e) VAT calculations
1.1.2 Analyse and critically interpret different financial situations inclusive
of:
(a) Personal and business finances
(b) The effects of taxation and fluctuating interest rates
(c) Income and expenditure
(d) The impact of changing interest (simple and compound) within
personal finance contexts
(e) Estimating and calculating profit/loss.
1.1.3 Interpret calculated answers appropriately to the given problem by:
(a) Interpreting answers in context
(b) Reworking a problem if the calculated answer is not suitable or if
the initial conditions change
(c) Logically interpreting and communicating calculated answers in
relation to the problem.
23
AC 1.2 Analyze provincial and national budgets in terms of health and welfare issues, job opportunities, etc.,
using mathematical skills and knowledge.
We know this when the candidate is able to:
1.2.1 Critically analyse budgets by performing relevant calculations to verify budgeted amounts
1.2.2 Compare budgets by performing relevant calculations.
AC 1.3 Compare different currencies for best investment opportunities.
We know this when the candidate is able to:
1.3.1 Calculate currency conversions relating to investments
1.3.2 Select the most appropriate currency for an investment
1.3.3 Determine the effects of currency fluctuations for investments.
AC 1.4 Understand inflation using mathematical investigation.
We know this when the candidate is able to:
1.4.1 Calculate annual inflation for given products over multiple periods
1.4.2 Calculating and comparing annual inflation rates
1.4.3 Analyse and critically interpret the effects of inflation
AC 1.5 Use ratio, rate, proportions and percentages to solve problems.
We know this when the candidate is able to:
1.5.1 Perform calculations using given percentages, ratios, rates
1.5.2 Perform calculations using proportions(direct or inverse)
1.5.3 Solve problems by calculating percentage, ratio, rate or proportion.
1.5.4 Increase or decrease values in a given ratio or percentage.
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CONTENT
Fractions, decimals, percentages
Rounding and estimation
Positive and negative numbers
The associative, commutative and distributive properties ( Candidates are not expected to know
these properties by name).
Positive exponents , square and cube roots
Rate , Ratio
Direct proportion.
Inverse (indirect) proportion.
Direct substitution in formulae.
Simple and compound growth and decay (Formulae to be provided on Formula Sheet)
Inflation
Annuities and Loans (Formulae to be provided on Formula Sheet)
Hire Purchase Agreements .
Cost price and selling price.
Profit and Loss
Taxation – VAT
Taxation – Personal income tax
Currency conversions and fluctuations.
CONTENT AREA 2 (CA 2) : FUNCTIONS AND RELATIONSHIPS
Recognize, analyse, interpret, describe and represent various functional relationships in order to solve
problems in real and simulated contexts.
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EXIT LEVEL OUTCOME 2 (ELO 2)
Recognise, analyse, interpret, describe and represent various functional relationships in order to solve problems in real contexts.
The emphasis of this Exit Learning Outcome is to give candidates opportunities to investigate and make sense of functional
relationships arising in the context of other subjects, work or real life situations. Candidates should be able to reflect on relationships
between variables and use various techniques to determine values of variables in solving problems.
ASSESSMENT CRITERIA (AC)
AC 2.1 Translate between different representations of functions to solve problems and analyses situation
We know this when the candidate is able to:
2.1.1 Work with tabulated data and formulae in different real life situations to determine input and output values
2.1.2 Draw graphs from tabulated data and formulae by point by point plotting
2.1.3 Critically interpret tables and graphs relating to real life situations by:
(a) Finding values of variables at certain points
(b) Describing overall trends
(c) Identify maximum and minimum values
(d) Estimate input and output values
2.1.4 Critically interpret graphs in real life situations inclusive of more than one graph on a system of axes.
2.1.5 Find break-even points involving linear functions from tabulated data or from graphs.
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AC 2.2 Analyze and interpret graphs of sine, cosine and tangent functions to solve problems.
2.2.1 Read input and output values for sine, cosine and tangent functions from given graphs
2.2.2 Identify type of trigonometric graphs from given graphs.
2.2.3 Solve simple trigonometrical equations using given graphs.
AC 2.3 Solve design and planning problems through optimizing a function in two discreet variables.
We know this when the candidate is able to:
2.3.1 Identify the feasible region by plotting discrete values.
2.3.2 Determine if given quantities will be feasible.
2.3.3 Identify optimum solution for a given situation.
CONTENT
Tables of values.
Formulae depicting relationships between variables.
Cartesian co-ordinate system restricted to first quadrant only
Linear functions.
Inverse proportion.
Compound growth.
Graphs depicting the relationship between two variables.
Maximum and minimum points.
Rates of change (speed, distance, time).
Solution to simple linear equations using non-algebraic methods.
Graphical solution to two simultaneous linear equations.
Breakeven –point
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Simple linear programming (design and planning problems) – Discrete values only.
(Candidates not required to represent feasible regions for a given optimization problem by
plotting constraints. Interpretations from graphical representations. Optimization of restricted
to two variables . At most three constraints)
Interpreting trigonometric graphs of the following types:
siny a x cosy a x siny b x where 0 and 0 360o oa x
(Candidates are Not required to draw trigonometry graphs. Read input and output
values from trigonometric graphs drawn on scaled paper).
Examples:
Use the graph to calculate the value of 25sin30o ; 1 tan45o
Use the graph to solve the following 2cosx 0,8
CONTENT AREA 3 (CA 3): SPACE, SHAPE and MEASUREMENT
Using appropriate measuring instruments to estimate, calculate physical quantities and to describe and
represent properties of and relationships, between 2- and 3-dimensional objects in a variety of orientations
and positions.
EXIT LEVEL OUTCOME 3 (ELO 3)
28
Use measurement to estimate and calculate physical quantities in order to describe and represent properties of and relationships
between 2D and 3D objects.
The emphasis of this Exit Learning Outcome is on the development of spatial understanding and geometric skills relating to real life contexts.
A variety of applications are available in design, art, geography and other fields to develop measurement skills.
ASSESSMENT CRITERIA (AC)
AC 3.1 Represent and identify views of scale drawings of plans; calculate values and build models
We know this when the candidate is able to:
3.1.1 Convert units of measurement within the metric system.
3.1.2 Convert units of measurement between different scales and systems.
3.1.3 Convert units of measurement between different scales and systems in context.
3.1.4 Use and interpret scale drawings of plans, maps and models to identify views, estimate and calculate values
according to scale.
3.1.5 Determine scale of maps and plans
3.1.6 Represent different perspective views of plans and models
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AC 3.2 Use sketches, models and technology to represent and analyses spatial relation.
We know this when the candidate is able to use map grids, including compass directions to:
3.2.1 determine locations and grid reference
3.2.2 plan trips and describe routes between two different locations
3.2.4 describe relative positions.
AC 3.3 Solve 2D and 3D problems using measurement and calculations.
We know this when the candidate is able to:
3.3.1 Solve problems in 2-dimensional and 3-dimensional contexts by estimating, measuring and calculating
values which involve:
(a). lengths, distances and perimeters(circumference)
(b). areas of common polygons and circles (including combinations of these shapes)
(c). surface areas and volumes of right prisms, right cylinders
3.3.2 Verifying values of the solutions against the contexts in terms of suitability and degree of accuracy.
AC 3.4 Use basic trigonometric ratios to interpret situations and solve problems.
We know this when the candidate is able to:
3.4.1 Name the parts of a right angled triangle
3.4.2 Find the missing side of a right angled triangle using the Pythagorean Theorem
3.4.3 Find the measure of an unknown side or angle of a right angled triangle using sine, cosine or tangent
ratios using a scientific calculator.
3.4.4 Solve problems using right angled triangle trigonometry
3.4.5 Finding angles of elevation and depression, heights and horizontal distances
CONTENT
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Conversion of units within the metric system
Conversion of measurements between different scales and systems. (Conversion tables will be provided)
Temperature conversions (Formulae to be provided on Formula Sheet)
Scale drawings.
Floor plans, layout plans, seating plans and elevation views
Maps, compass directions, location and position on grids.
Travel timetables
Measurement of time (including international time zones).
Measurement of length, distance, volume, area, perimeter.
Polygons commonly encountered (triangles, squares, rectangles) and combinations of these shapes and circles
Calculation of perimeter and area of common polygons and circles (Formulae to be provided on Formula Sheet)
Calculation of surface area and volumes of right prisms, right circular cylinders and spheres (Formulae to be provided on
Formula Sheet)
Right angled triangle terminology: Right Angle ; Hypotenuse ; Adjacent Side ; Opposite Side.
Theorem of Pythagoras (Formulae to be provided on Formula Sheet)
Application of trigonometric ratios: sin x, cos x, tan x (Formulae to be provided on Formula Sheet)
[Application of the trigonometric ratios can also be integrated with questions on the graphical representation of these
ratios ( Refer AC 2.2)]
Solution of right angled triangles
Problem solving using trigonometry using right angled triangles only.
[ Diagrams relating to problems must be provided to candidates and should involve at most two right angled triangles]
CONTENT AREA 4 (CA 4) : DATA HANDLING
31
Collect, summarize, display and analyze data and apply knowledge of statistics and basic probability to communicate,
justify, predict and critically interrogate findings and draw conclusions.
EXIT LEVEL OUTCOME 4 (ELO 4)
Collect, summarise, represent and analyse data and apply knowledge of statistics and probability in order to communicate, justify,
predict and critically analyse results and draw conclusions.
The emphasis of this Exit Learning Outcome is on the ability of candidates as consumers to interpret and use data. Candidates should be
able to reflect on the use, misuse and meaning of different graphical representations of data in a variety of real life situations. The critical
awareness of data manipulation to prove opposing views should be developed by candidates.
ASSESSMENT CRITERIA (AC)
AC 4.1 Use a representative sample from a population to solve a problem with due sensitivity to issues relating to bias.
We know this when the candidate is able to:
Investigate a problem on issues relating to social, environmental and political factors and people’s opinions by:
4.1.1 Using appropriate statistical methods
4.1.2 Selecting a representative sample
4.1.3 Comparing data from different sources and samples
AC 4.2 Calculate measures of central tendency and spread of data.
We know this when the candidate is able to:
4.2.1 Understand that data can be summarized in different ways by calculating:
(a) Mean
(b) Median
(c) Mode
(d) Range
32
4.2.2 Make comparisons and draw conclusions using :
(a) Mean
(b) Median
(c) Mode
(d) Range
(e) Quartiles (Interpretations only)
(f) Percentiles (Interpretations only)
AC 4.3 Represent and analyze data and statistics
We know this when the candidate is able to:
4.3.1 Select, justify and use a variety of methods to summarise and display data, inclusive of:
(a) Tallies
(b) Tables, including frequency tables
(c) Pie charts
(d) Single and compound bar graphs
(e) Line and broken line graphs
(f) Histograms
(g) Frequency Polygons
(h) Stem and Leaf diagram
4.3.2 Select, justify and use a variety of methods to summarise and display data to describe trends
33
AC 4.4 Working with simple notions of probability in order to make sense of probability statements.
We know this when the candidate is able to:
4.4.1 Express probability values in terms of fractions, ratios and percentages.
4.4.2 Use tree diagrams to determine the probability of non-independent events.
4.4.3 Effectively communicate conclusions and predictions that can be made from the analysis and representation of
data, using appropriate terminology such as, trends, increase, decrease, constant, impossible, likely and even chance.
AC 4.5 Critically evaluate and make recommendations to statistically based arguments.
We know this when the candidate is able to:
4.5.1 Critically interpret data and representations thereof (with the awareness
of sources of error and bias) in order to draw conclusions, make
predictions, predict trends and critique other interpretations.
4.5.2 Identify and interpret misuse of statistics.
CONTENT
Construction of questionnaires (Not examinable)
Selection of populations and samples
Classification of variables: Qualitative, quantitative, discreet, continuous, dependent and independent.
Methods of tabulating data.
Absolute and relative frequencies
Tally and frequency tables (Grouped and ungrouped)
Graphical representations:
Single and compound bar graphs
Pie charts
34
Histograms
Line and broken-line graphs.
Stem and leaf diagram
Calculation of measures of central tendencies and spread:
Mean, median, mode
Range
Quartiles and Percentiles ( interpretations only)
Critically interpret misuse of statistics
Calculating simple Probability including from simple contingency tables
Representing and calculating probability using tree diagrams
Critically interpret probability statements
(Terminology such as, trends, increase, decrease, constant, impossible, likely and even chance)
35
FORMULA SHEET
9 5
F C C 32 325 9
o o o oo oF
1 1
1 1
1 1 1 1
n n
n n
V V
A P i n A P i n
A P i A P i
x i x i
i iF P
distanceAverage speed
time
2 212
3
Perimeter ( Circumference) :
4 2 2
Area :
A A A
Volume:
A
P s P b C r
A s b b h r
V s b h
2 343
V r h V r
2 2 2
Theorem of Pythagoras :
hypotenues opposite side adjacent side
opposite sidesin
adjacent sidecos
opposite sidetan
hypotenues
hypotenues
adjacentside
36
GLOSSARY
association – a general term to describe the relationship between two variables. Two
variables in bivariate data are associated or dependent if the pattern of frequencies of their
bivariate values cannot be explained only by the frequencies of the univariate values. In
contrast, two variables are not associated or independent, if the frequencies of bivariate
values can be determined simply from the frequencies of the values of each variable.
associative law/property – the property of an operation which allows for the operation to
be carried out by grouping the terms differently (e.g. for addition of real numbers: (a + b )
+ c = a + (b + c) and for multiplication (a × b) × c = a × (b × c).
bar graph/diagram – a diagram that uses horizontal or vertical bars to represent the
frequency of classes (or groups or labels) in data consisting of observations of a
categorical variable. The height or length of each bar is proportional to the frequency of
the corresponding class, but the thickness of a bar has no meaning. The bars are not
required to touch each other, and may be separated. A bar graph is not a histogram.
bias – a distortion of the data in a set due to irregularities in the collection of the data; an
unjustified tendency to favour a particular point of view.
bivariate data – two dimensions (of each object under observation) that are recorded as a
pair of variables (usually to investigate or describe an association or correlation or
relationship between the variables).
break-even point – the value of the independent variable at which the costs associated
with various (two) pricing structures for a commodity become equal; the point at which
expenditure and income are equal.
Cartesian plane – the system whereby position in a plane is determined with reference to
two axes (number lines) which are at right angles to each other and intersecting at the
origin (the 0s on the number lines); any point on the plane is fixed by an ordered pair of
real numbers (co-ordinates) which are the numbers the point refers to on the axes.
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circumference – the (measure of) the perimeter of a circle.
commutative law/property – the property of an operation which allows for the order of
the values operated with to be interchanged (e.g. for the addition of real numbers a + b = b
+ a and for multiplication a × b = b ×a.
compass direction – the direction indicated with reference to the globe of the earth as
north, south, east or west; the direction in degrees from the northerly direction in an
anticlockwise sense.
compound growth – the accelerated effect in the manner in which a quantity increases (or
decreases) due to the factor causing the increase (or decrease) also acting on the increase
(or decrease) in the amount itself (e.g. the growth in the amount invested when interest is
calculated on interest, as in compound interest).
compound interest – the calculation of the new amount A when the original amount (the
principal), P, of money is subjected to interest being calculated on interest at the end of a
period according to the following formula A = P(1+i)n.
conjecture – a tentative solution or generalisation inferred from collected data.
constraint – limiting condition (usually translated into a linear inequality) in a linear
programming problem.
contingency table – classification of a population or group in a two-way table of
qualitative (categorical) elements. The rows of the table denote the categories of the first
variable and the columns denote the categories of the second variable.
continuous variable – a variable which ranges through all the real numbers on the interval
applicable to it.
cumulative frequency – for data that has been ordered (from minimum to maximum
values) the successive values can be assigned frequencies. The cumulative frequency for a
value x is the total count of all the data values that are less than or equal in value to x.
data – items of information that have been observed and recorded; can be categorical (e.g.
gender), numerical (e.g. age), univariate, bivariate or multivariate, and are often
arranged in a list or table.
dependent variable – the element of the range of a function which depends on the
corresponding value(s) of the domain (e.g. in y = f(x) = πx2 the area of a circle (y) depends
of the radius, x. x is the independent variable and y the dependent variable).
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direct proportion – two variables, x and y, which are related by the equation y = kx, are
said to be in direct proportion.
discrete variables – variables for which the values do not take on all the real numbers
within the range over which they vary; a discrete variable is often associated with a count
and so takes the values of the counting numbers (e.g. 0, 1, 2, 3 …).
event – any subset of all the possible outcomes of an experiment. An event occurs at a
particular experimental trial if any one of its constituent outcomes is the outcome
observed for that trial.
exchange rate – the price of a unit of the currency of one country in terms of the currency
of another.
experiment – a repeatable activity or process for which each repetition gives rise to
exactly one outcome drawn from the sample space (statistical experiment); gives rise to
univariate data on the outcome of each trial (e.g. the observed face of a die) (simple
experiment). The number of trials observed is the sample size n.
extrapolate – to estimate an unknown quantity by projecting from the basis of what is
already known, but outside the limits of the known data (in contrast see interpolate).
feasible region – the set of all points that satisfy the constraints of a linear programming
problem.
frequency – a count of the number of times a particular outcome or event was observed in
data with a sample size n.
frequency polygon – a polygon formed by joining the mid-points of the top of the
columns of a histogram.
frequency table – a table reporting the groups into which data values were organised, and
the frequency of each group.
function – a relationship between two sets of variables such that each element of the one
set (the domain) is associated with a unique element of the second set (the range).
functional relationship – a relationship between variables which is a function.
global positioning system – a system using satellite and electronic technology, whereby a
particular location on the earth’s surface is determined in terms of its latitude and
longitude.
grid – a pattern of lines usually drawn at right angles to each other to form rectangles.
39
grouped data – data arising from organising n observed values into a smaller number of
disjoint groups of values, and counting the frequency of each group; often presented as a
frequency table or visually as a histogram.
hire-purchase – the system whereby goods are bought by putting down a deposit and then
periodically paying off the balance of the purchase price plus interest. Simple interest
usually applies while insurance costs are commonly also included.
histogram – a visual representation used for grouped data for a numerical variable which
has been divided into class intervals. The histogram consists of adjacent rectangles, each
standing on a class interval. The area of each rectangle is proportional to the frequency of
observations falling in a class interval. Class intervals are often plotted on the horizontal
axis so that all have the same width, while the rectangles are vertical. Frequency density is
read on the vertical axis. A bar graph is not a histogram.
independent events – the idea that two events do not connect with each other in any
observable pattern, and hence that neither event can give any useful information about the
other event (in contrast see associated). Numerical variables that are said to be correlated
are not independent.
independent variable – as used in dealing with functions, the value that determines the
value of the dependent variable (e.g. in y = f(x) = πx2 the area of a circle (y) depends on
the radius, x; x is the independent variable and y the dependent variable).
inflation rate – a quantitative measure which indicates the rate at which the price of
consumer goods are increasing over time.
interpolate – to estimate an unknown quantity within the limits of what is already known
(in contrast see extrapolate).
inverse proportion – two variables, x and y, which are related to each other by the
equation y = k/x are said to be in inverse proportion.
latitude – the number of degrees that a location is north or south of the equator.
longitude – the number of degrees that a location is east or west of a line passing through
the poles and Greenwich in Britain.
median – a value that splits the sample data of a numerical variable into two parts of equal
size, one part consisting of all values less than the median and one part with all values
greater than the median; most easily established if the data values are arranged in
increasing or decreasing order.
mode – the most frequently occurring observation in a set of data.
mortgage bond – a loan from a bank, usually for the purchase of property. The loan is
subject to the payment of compound interest and is paid off in regular instalments which
include interest and capital.
40
multivariate data – two or more dimensions (of each object under observation) are
recorded as an ordered string of variables (often to investigate or describe any association
or correlation or relationship between some of the variables). The data is often arranged
in a rectangular format of rows and columns, where a specific column is reserved for each
variable, and a row is allocated to each object that is observed (e.g. 5 rows of data, one row
for the 5 variables height, mass, age, eye-colour, and gender observed for each person, and
5 columns each with n entries). Categorical variables are summarised by frequency
counts. Numerical multivariate data are often presented two variables at a time, with a
scatter plot for each pair.
outcome – the result of an experiment (in statistics) (e.g. the outcome of an experiment in
which a dice is rolled can be any one of the natural numbers 1 through 6).
percentiles – values of ranked data separated into one hundred groups of equal size,
especially when sample size n is very large.
polygon – a figure in a plane formed by some number of straight sides.
probability – for equally likely outcomes, the number of favourable outcomes divided by
the total number of possible outcomes of an experiment.
qualitative data – information or data arising from observations which are not numerical;
can be categorical.
quantitative data – data with values that are numerical; can be discrete (counted) or
continuous (measured).
quartiles – three values which split the ordered sample values into four groups of equal
size. The second quartile is the median.
relative frequency – (of a particular outcome in a statistical experiment) the number of
occurrences of a particular outcome divided by the number of trials.
right cylinder – a solid that has one axis of symmetry through the centre of the circular
base and a uniform, circular cross-section.
right prism – a prism whose lateral sides are perpendicular to its base.
sample – in statistics, a group of data chosen from all the possible data.
tree diagrams – a diagram in which the possible outcomes of trials involving one or more
events are indicated by line segments.
trial – each repetition of a statistical experiment.
41
univariate data – one-dimensional data; any quantity or attribute whose value varies from
one observation to another gives rise to univariate data, which may be qualitative or
quantitative.