NATIONAL SENIOR CERTIFICATE FOR ADULTS NQF … Mathematical Literacy.pdf · DRAFT SUBJECT STATEMENT...

NATIONAL SENIOR CERTIFICATE FOR ADULTS

NQF LEVEL 4

DRAFT SUBJECT STATEMENT

MATHEMATICAL LITERACY

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Contents

MATHEMATICAL LITERACY .............................................................................................. 1


INTRODUCTION ..................................................................................................................... 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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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 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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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


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.


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.


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.


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.


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.


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.


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.


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.

24

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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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.


AC 2.1 Translate between different representations of functions to solve problems and analyses situation


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.

26

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.


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.


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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.


AC 3.1 Represent and identify views of scale drawings of plans; calculate values and build models


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.


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.


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

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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.


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.


AC 4.1 Use a representative sample from a population to solve a problem with due sensitivity to issues relating to bias.


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.


4.2.1 Understand that data can be summarized in different ways by calculating:

(a) Mean

(b) Median

(c) Mode

(d) Range

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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


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

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AC 4.4 Working with simple notions of probability in order to make sense of probability statements.


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.


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

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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)

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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.

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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.

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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.

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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.

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