NATIONAL CERTIFICATES (VOCATIONAL) Certificates NQF Level 4/NC(Vocational... · for the National...

NATIONAL CERTIFICATES (VOCATIONAL)

ASSESSMENT GUIDELINES

MATHEMATICS

NQF Level 4

IMPLEMENTATION: JANUARY 2013

Mathematics Level 4 (January 2013) National Certificates (Vocational)

Department of Higher Education and Training 2

MATHEMATICS – LEVEL 4

CONTENTS

SECTION A: PURPOSE OF THE SUBJECT ASSESSMENT GUIDELINES

SECTION B: ASSESSMENT IN THE NATIONAL CERTIFICATES (VOCATIONAL)

1 Assessment in the National Certificates (Vocational) 2 Assessment framework for vocational qualifications

2.1 Internal continuous assessment (ICASS) 2.2 External summative assessment (ESASS)

3 Moderation of assessment 3.1 Internal moderation 3.2 External moderation

4 Period of validity of internal continuous assessment (ICASS) 5 Assessor requirements 6 Types of assessment

6.1 Baseline assessment 6.2 Diagnostic assessment 6.3 Formative assessment 6.4 Summative assessment

7 Planning assessment 7.1 Collecting evidence 7.2 Recording 7.3 Reporting

8 Methods of assessment 9 Instruments and tools for collecting evidence 10 Tools for assessing student performance 11 Selecting and/or designing recording and reporting systems 12 Competence descriptions 13 Strategies for collecting evidence

13.1 Record sheets 13.2 Checklists

SECTION C: ASSESSMENT IN MATHEMATICS

1 Assessment schedule and requirements 2 Recording and reporting 3 Internal assessment of outcomes in Mathematics - Level 4 4 External assessment in Mathematics - Level 4


3 Department of Higher Education and Training

SECTION A: PURPOSE OF THE SUBJECT ASSESSMENT GUIDELINES

This document provides the lecturer with guidelines to develop and implement a coherent, integrated assessment system for Mathematics in the National Certificates (Vocational). It must be read with the National Policy Regarding Further Education and Training Programmes: Approval of the Documents, Policy for the National Certificates (Vocational) Qualifications at Levels 2 to 4 on the National Qualifications Framework (NQF). This assessment guideline will be used for National Qualifications Framework Levels 2-4.

This document explains the requirements for the internal and external subject assessment. The lecturer must use this document with the Subject Guidelines: Mathematics Level 4 to prepare for and deliver Mathematics. Lecturers should use a variety of resources and apply a range of assessment skills in the setting, marking and recording of assessment tasks.

SECTION B: ASSESSMENT IN THE NATIONAL CERTIFICATES (VOCATIONAL)

1 ASSESSMENT IN THE NATIONAL CERTIFICATES (VOCATIONAL)

Assessment in the National Certificates (Vocational) is underpinned by the objectives of the National Qualifications Framework (NQF). These objectives are to:

Create an integrated national framework for learning achievements. Facilitate access to and progression within education, training and career paths. Enhance the quality of education and training. Redress unfair discrimination and past imbalances and thereby accelerate employment opportunities. Contribute to the holistic development of the student by addressing: social adjustment and responsibility; moral accountability and ethical work orientation; economic participation; and nation-building.

The principles that drive these objectives are:

Integration

To adopt a unified approach to education and training that will strengthen the human resources development capacity of the nation.

Relevance

To be dynamic and responsive to national development needs.

Credibility

To demonstrate national and international value and recognition of qualification and acquired competencies and skills.

Coherence

To work within a consistent framework of principles and certification.

Flexibility

To allow for creativity and resourcefulness when achieving Learning Outcomes, to cater for different learning styles and use a range of assessment methods, instruments and techniques.

Participation

To enable stakeholders to participate in setting standards and co-ordinating the achievement of the qualification.

Access

To address barriers to learning at each level to facilitate students’ progress.



Progression

To ensure that the qualification framework permits individuals to move through the levels of the national qualification via different, appropriate combinations of the components of the delivery system.

Portability

To enable students to transfer credits of qualifications from one learning institution and/or employer to another.

Articulation

To allow for vertical and horizontal mobility in the education system when accredited pre-requisites have been successfully completed.

Recognition of Prior Learning

To grant credits for a unit of learning following an assessment or if a student possesses the capabilities specified in the outcomes statement.

Validity of assessments

To ensure assessment covers a broad range of knowledge, skills, values and attitudes (SKVAs) needed to demonstrate applied competency. This is achieved through:

clearly stating the outcome to be assessed; selecting the appropriate or suitable evidence; matching the evidence with a compatible or appropriate method of assessment; and selecting and constructing an instrument(s) of assessment.

Topics should be assessed individually and then cumulatively with other topics. There should be a final summative internal assessment prior to the external assessment.

Reliability

To assure that assessment practices are consistent so that the same result or judgment is arrived at if the assessment is replicated in the same context. This demands consistency in the interpretation of evidence; therefore, careful monitoring of assessment is vital.

Cumulative and summative assessments must be weighted more than single topic tests for the internal mark.

There should be at least one standardised or norm test in each trimester. All standardised or norm tests must be moderated by a subject specialist.

Fairness and transparency

To verify that no assessment process or method(s) hinders or unfairly advantages any student. The following could constitute unfairness in assessment:

Inequality of opportunities, resources or teaching and learning approaches Bias based on ethnicity, race, gender, age, disability or social class Lack of clarity regarding Learning Outcome being assessed Comparison of students’ work with other students, based on learning styles and language

Assessment in Mathematics must take into consideration that the process or method carries more weight than the final answer.

Practicability and cost-effectiveness

To integrate assessment practices within an outcomes-based education and training system and strive for cost and time-effective assessment.

2 ASSESSMENT FRAMEWORK FOR VOCATIONAL QUALIFICATIONS

The assessment structure for the National Certificates (Vocational) qualification is as follows:

2.1 Internal continuous assessment (ICASS)

Knowledge, skills values, and attitudes (SKVAs) are assessed throughout the year using assessment instruments such as projects, tests, assignments, investigations, role-play and case studies. All internal continuous assessment (ICASS) evidence is kept in a Portfolio of Evidence (PoE) and must be readily



available for monitoring, moderation and verification purposes. This component is moderated and quality assured both internally and externally.

2.2 External summative assessment (ESASS)

The external summative assessment comprises TWO papers set to meet the requirements of the Subject and Learning Outcomes. It is administered according to relevant assessment policies and requirements.

External summative assessments will be conducted annually between October and December, with provision made for supplementary sittings.

3 MODERATION OF ASSESSMENT

3.1 Internal moderation

Assessment must be moderated according to the internal moderation policy of the Further Education and Training (FET) College. Internal college moderation is a continuous process. The moderator’s involvement starts with the planning of assessment methods and instruments and follows with continuous collaboration with and support to the assessors. Internal moderation creates common understanding of Assessment Standards and maintains these across vocational programmes.

3.2 External moderation

External moderation is conducted according to relevant quality assurance bodies’ standards, policies, and requirements (currently the South African Qualifications Authority (SAQA) and Umalusi.)

The external moderator:

monitors and evaluates the standard of all summative assessments; maintains standards by exercising appropriate influence and control over assessors; ensures proper procedures are followed; ensures summative integrated assessments are correctly administered; observes a minimum sample of ten (10) to twenty-five (25) percent of summative assessments; gives written feedback to the relevant quality assurer; and moderates in case of a dispute between an assessor and a student.

Policy on inclusive education requires that assessment procedures be customised for students who experience barriers to learning, and supported to enable these students to achieve their maximum potential.

4 PERIOD OF VALIDITY OF INTERNAL CONTINUOUS ASSESSMENT (ICASS)

The period of validity of the internal continuous assessment mark is determined by the National Policy on the Conduct, Administration and Management of the Assessment of the National Certificates (Vocational).

The internal continuous assessment (ICASS) must be re-submitted with each examination enrolment for which it constitutes a component.

5 ASSESSOR REQUIREMENTS

Assessors must be subject specialists and a competent assessor.

6 TYPES OF ASSESSMENT

Assessment benefits the student and the lecturer. It informs students about their progress and helps lecturers make informed decisions at different stages of the learning process. Depending on the intended purpose, different types of assessment can be used.



6.1 Baseline assessment

At the beginning of a level or learning experience, baseline assessment establishes the knowledge, skills, values and attitudes (SKVAs) that students bring to the classroom. This knowledge assists lecturers to plan learning programmes and learning activities.

6.2 Diagnostic assessment

This assessment diagnoses the nature and causes of learning barriers experienced by specific students. It is followed by guidance, appropriate support and intervention strategies. This type of assessment is useful to make referrals for students requiring specialist help.

6.3 Formative assessment

This assessment monitors and supports teaching and learning. It determines student strengths and weaknesses and provides feedback on progress. It determines if a student is ready for summative assessment.

6.4 Summative assessment

This type of assessment gives an overall picture of student progress at a given time. It determines whether the student is sufficiently competent to progress to the next level.

7 PLANNING ASSESSMENT

An assessment plan should cover three main processes:

7.1 Collecting evidence

The assessment plan indicates which Subject Outcomes and Assessment Standards will be assessed, what assessment method or activity will be used and when this assessment will be conducted.

7.2 Recording

Recording refers to the assessment instruments or tools with which the assessment will be captured or recorded. Therefore, appropriate assessment instruments must be developed or adapted.

7.3 Reporting

All the evidence is put together in a report to deliver a decision for the subject.

8 METHODS OF ASSESSMENT

Methods of assessment refer to who carries out the assessment and includes lecturer assessment, self-assessment, peer assessment and group assessment.

LECTURER ASSESSMENT The lecturer assesses students’ performance against given criteria in different contexts, such as individual work, group work, etc.

SELF-ASSESSMENT Students assess their own performance against given criteria in different contexts, such as individual work, group work, etc.

PEER ASSESSMENT Students assess another student’s or group of students’ performance against given criteria in different contexts, such as individual work, group work, etc.

GROUP ASSESSMENT Students assess the individual performance of other students within a group or the overall performance of a group of students against given criteria.

9 INSTRUMENTS AND TOOLS FOR COLLECTING EVIDENCE

All evidence collected for assessment purposes is kept or recorded in the student’s PoE.

The following table summarises a variety of methods and instruments for collecting evidence. A method and instrument is chosen to give students ample opportunity to demonstrate that the Subject Outcome has been attained. This will only be possible if the chosen methods and instruments are appropriate for the target group and the Specific Outcome being assessed.



METHODS FOR COLLECTING EVIDENCE

Observation-based (Less structured)

Task-based (Structured)

Test-based (More structured)

Assessment instruments

Observation Class questions Lecturer, student,

parent discussions

Assignments or tasks Projects Investigations or

research Case studies Practical exercises Demonstrations Role-play Interviews

Examinations Class tests Practical examinations Oral tests Open-book tests

Assessment tools Observation sheets Lecturer’s notes Comments

Checklists Rating scales Rubrics

Marks (e.g. %) Rating scales (1-7)

Evidence

Focus on individual students

Subjective evidence based on lecturer observations and impressions

Open middle: Students produce the same evidence but in different ways. Open end: Students use same process to achieve different results.

Students answer the same questions in the same way, within the same time.

10 TOOLS FOR ASSESSING STUDENT PERFORMANCE

Rating scales are marking systems where a symbol (such as 1 to 7) or a mark (such as 5/10 or 50%) is defined in detail. The detail is as important as the coded score. Traditional marking, assessment and evaluation mostly used rating scales without details such as what was right or wrong, weak or strong, etc.

Task lists and checklists show the student what needs to be done. These consist of short statements describing the expected performance in a particular task. The statements on the checklist can be ticked off when the student has adequately achieved the criterion. Checklists and task lists are useful in peer or group assessment activities.

Rubrics provide a hierarchy (graded levels) of criteria with benchmarks that describe the minimum level of acceptable performance or achievement for each criterion. Use of rubrics provides a different way of assessing that cannot be compared to tests. Each criterion described in the rubric must be assessed separately. Mainly two types of rubrics, namely holistic and analytical, are used.

11 SELECTING AND/OR DESIGNING RECORDING AND REPORTING SYSTEMS

The selection or design of recording and reporting systems depends on the purpose of recording and reporting student achievement. Why particular information is recorded and how it is recorded determine which instrument will be used.

Computer-based systems, for example spreadsheets, are cost and time effective. The recording system should be user-friendly and information should be easily accessed and retrieved.

12 COMPETENCE DESCRIPTIONS

All assessment should award marks to evaluate specific assessment tasks. However, marks should be awarded against rubrics and not be simply a total of ticks for right answers. Rubrics should explain the competence level descriptors for the skills, knowledge, values and attitudes (SKVAs) that a student must demonstrate to achieve each level of the rating scale.

When lecturers or assessors prepare an assessment task or question, they must ensure that the task or question addresses an aspect of a Subject Outcome. The relevant Assessment Standard must be used to create the rubric to assess the task or question. The descriptions must clearly indicate the minimum level of attainment for each category on the rating scale.



13 STRATEGIES FOR COLLECTING EVIDENCE

A number of different assessment instruments may be used to collect and record evidence. Examples of instruments that can be (adapted and) used in the classroom include:

13.1 Record sheets

The lecturer observes students working in a group. These observations are recorded in a summary table at the end of each project. The lecturer can design a record sheet to observe students’ interactive and problem-solving skills, attitudes towards group work and involvement in a group activity.

13.2 Checklists

Checklists should have clear categories to ensure that the objectives are effectively met. The categories should describe how the activities are evaluated and against what criteria they are evaluated. Space for comments is essential.



ASSESSMENT IN MATHEMATICS

LEVEL 4



SECTION C: ASSESSMENT IN MATHEMATICS

1 ASSESSMENT SCHEDULE AND REQUIREMENTS

Internal and external assessments are conducted and the results of both are contributing to the final mark of a student in the subject The internal continuous assessment (ICASS) mark accounts for 25 percent and the external examination mark for 75 percent of the final mark. A student needs a minimum final mark of 30 percent to enable a pass in the subject. 1.1 Internal assessment Lecturers must compile a detailed assessment plan/schedule of formal assessments to be undertaken during the year in the subject. (e.g. date, assessment task/or activity, rating code/marks allocated, assessor, moderator.) Formal assessments are then conducted according to the plan/schedule using appropriate assessment instruments and tools for each assessment task (e.g. tests, assignments, practical tasks/projects and memorandum, rubric, checklist)

The marks allocated to both the practical and written formal assessment tasks conducted during the internal continuous assessment (ICASS) are kept and recorded in the Portfolio of Evidence (PoE) which is subjected to internal and external moderation. A year mark out of 100 is calculated from the ICASS marks contained in the PoE and submitted to the Department on the due date towards the end of the year.

The following internal assessment units GUIDE the assessment of Mathematics Level 4.

TASKS Time-frame

Type of assessment activity

(the time and proposed mark allocation can be increased but not reduced)

Scope of assessment

% contribution to the year

mark Do not confuse the weightings of topics in

the Subject Guidelines with the % contribution to the year mark

1 Term 1 Test 1 Hour (50 marks)

Topics completed in term 1 10%

2 Term 1 **Assignment Assignment on one or more topics completed to date

10%

3 Term 2 Test 1 Hour (50 marks)

Topics completed in term 2 10%

4 Term 2 **Assignment Topics completed in term 2 10%

5 Term 2 *Test 2 Hours (70 marks)

Topics completed in term 1 and 2

20%

6 Term 3

Practical Assessment /**Assignment

Topics completed, any related Subject Outcomes, for example:

1. Work with practical

problems involving the

construction of scatter

plots, lines of best fit by

10%



regression analysis and

predictions based on

those results.


problems involving tax

tables.

3. Construct at least three

circle geometrical riders

and prove (by using a

protractor) some of the

major circle theorems.

4. Sketch the graph of a

function e.g. xy for a

specified domain,

example [-3 ; 3] and

calculate the area by first

using area of the triangles

and then by using

integration. Compare

results of at least three

different equations.


activities involving

probability models. Use

dices, coins and cards.

7 Term 3

*Internal Examination External examination papers serves as guidelines for content duration and mark allocation Paper 1 Paper 2

All topics completed to date Paper 1=15% Paper 2=15%

30%

TOTAL 100% *The internal examination (Term 3) and the test (Term 2) can be swapped around to allow the examination to be written either during the second term or the third term. If the examination is written at the end of the second term, at least 60% of the curriculum must have been covered. If the examination is written in the third term at least 80%-90% of the curriculum must have been covered. **The assignment must be completed within 5 days. A clear instruction sheet outlining the task and the resources required to complete the task must be given to students.



2 RECORDING AND REPORTING

Mathematics is assessed according to seven levels of competence. The level descriptions are explained in the following table.

Scale of achievement for the Fundamental component

The planned/scheduled assessment should be recorded in the Lecturer’s Portfolio of Assessment (PoA) for each subject. The minimum requirements for the Lecturer’s Portfolio of Assessment should be as follows:

o Lecturer information

o A contents page

o Subject and Assessment Guidelines

o Year plans /Work schemes/Pace Setters

o A subject assessment plan

o Instrument(s) (tests, assignments, practical) and tools (memorandum, rubric, checklist) for each assessment task

o A mark/result sheet for assessment tasks

The college could standardise these documents. The minimum requirements for the student’s Portfolio of Evidence (PoE) should be as follows:

o Student information/identification

o A contents page/list of content (for accessibility)

o A subject assessment schedule

o A record/summary/ of results showing all the marks achieved per assessment for the subject

o The evidence of marked assessment tasks and feedback according to the assessment schedule

o Where tasks cannot be contained as evidence in the Portfolio of Evidence (PoE), its exact

location must be recorded and it must be readily available for moderation purposes.

RATING CODE RATING MARKS (%)

7 Outstanding 80 – 100

6 Meritorious 70 – 79

5 Substantial 60 – 69

4 Adequate 50 – 59

3 Moderate 40 – 49

2 Elementary 30 – 39

1 Not achieved 0 – 29



3 INTERNAL ASSESSMENT OF SUBJECT OUTCOMES IN MATHEMATICS - LEVEL 4

Topic 1: Complex Numbers (Approximately 10 hours)

SUBJECT OUTCOME

1.1 Work with complex numbers.

ASSESSMENT STANDARD LEARNING OUTCOME

Operations are performed on complex numbers in standard and polar form.

De Moivre’s theorem is used to raise complex

numbers to powers (excluding fractional powers)

Conversion of the form of complex numbers is

done if needed to perform advanced operations on complex numbers.

Perform addition, subtraction, multiplication and division on complex numbers in standard form. (Includes i-notation) Note: Leave answers with positive argument

Perform multiplication and division on complex

numbers in polar form. Use De Moivre’s theorem to raise complex

numbers to powers (excluding fractional powers)

Convert the form of complex numbers where

needed to enable performance of advanced operations on complex numbers (a combination of standard and polar form may be assessed in one expression)

ASSESSMENT TASKS OR ACTIVITIES

Assignments Test Internal Examination

SUBJECT OUTCOME

1.2 Solve problems using complex numbers.


Variables in identical complex numbers in rectangular/standard form are solved using simultaneous equations.

Complex numbers are used to solve problems

which cannot be solved by using the real number system by applying:

o Factorisation o Quadratic formula

Solve identical complex numbers in rectangular/standard form using the concept of simultaneous equations.

Use complex numbers to solve equations that

cannot be solved using the real number system by applying:

o Factorisation o Quadratic formula





Topic 2: Functions and Algebra. (Approximately 40 hours)

SUBJECT OUTCOME

2.1 Work with algebraic expressions using the remainder and the factor theorems.


Know and apply the remainder and the factor theorem.

Third degree polynomials are factorised by using the factor theorem.

Use and apply the remainder and the factor theorem.

o Find the remainder o Prove that an expression is a factor o Find an unknown variable in order to

make an expression, a factor or to leave a remainder.

Factorise third degree polynomials including

examples that require the factor theorem. (Long division or any other method may be used)



SUBJECT OUTCOME

2.2 Use a variety of techniques to sketch and interpret information for graphs of the inverse of a function.


An ability to sketch and work with various types of functions, relations and their inverses:

0;

2

aay

axy

qaxy

x

Range: A maximum of two graphs per system of axis will be assessed.

Determine the equations of the inverses of the functions:

0;

2

aay

axy

qaxy

x

( xay may be left with x as the subject of the

formula. Note: No logarithms required) Sketch the graphs of the inverse of the

functions:

0;

2

aay

axy

qaxy

x

Note: Sketching the graphs using point by point plotting is an option

Obtain the equation of any of the following inverse graphs given as a sketch.

0;

2

aay

axy

qaxy

x



The following characteristics are identified with respect to the above-mentioned functions:

o Domain and range. o Intercepts with axes. o Turning points, minima and maxima. o Asymptotes o Shape and symmetry. o Functions or non functions. o Continuous or discontinuous. o Intervals at which a function

increases/decreases.

Identify characteristics as listed below with respect to the following functions and their inverses.

0;

2

aay

axy

qaxy

x

o Domain and range. o Intercepts with axes. o Turning points, minima and maxima. o Asymptotes o Shape and symmetry. o Functions or non-functions. o Continuous or discontinuous. o Intervals at which a function

increases/decreases.



SUBJECT OUTCOME

2.3 Use mathematical models to investigate linear programming problems.


Linear constraints are found and formulated from

a given problem.

Linear programming problems are solved by

optimizing a function in two variables, subject to

one or more linear constraints, using the search

line method

Method: o Sketch the functions/constraints.

o Determine and shade the feasible region.

o Work out the gradient of the search line.

o Use the search line to optimise the maximum or minimum from the objective function.

Find and formulate the linear constraints from a

given problem.

Solve linear programming problems by

optimizing a function in two variables, subject to

one or more linear constraints, using the search

line method.





SUBJECT OUTCOME

2.4 Investigate and use instantaneous rate of change of a variable when interpreting models both in mathematical and real life situations


First principles are used to establish the derivatives of the following functions.

x

axf

xxf

axxf

xxf

baxxf

baxxf

bxf

)(

1)(

)(

)(

)(

)(

)(

3

3

2

Range: a and b will be integers only

The derivatives of the following functions are found

kxaxf

kxaxf

kxaxf

aexf

kxaxf

axxf

kx

n

tan)(

cos)(

sin)(

)(

ln)(

)(

Note: Use the following notations,

dx

dyxf );( , for the first derivative.

Derivatives are found using the constant, sum and/or

difference, product, quotient and chain rules for differentiation after simplifying the expression.

Establish the derivatives of the following functions from first principles:

x

axf

xxf

axxf

xxf

baxxf

baxxf

bxf

)(

1)(

)(

)(

)(

)(

)(

3

3

2

Note: The binomial theorem does not form part of the curriculum.

Find the derivatives of the functions in the form:

kxaxf

kxaxf

kxaxf

aexf

kxaxf

axxf

kx

n

tan)(

cos)(

sin)(

)(

ln)(

)(

Where

kxkaxfkxaxf

kxkaxfkxaxf

kexfexf

x

kxfkxxf

xanxfaxxf

kxkx

nn

sin)('cos)(

cos)('sin)(

)(')(

)('ln)(

)(')( 1

Examples to include are

.;tan4;2

cos3

1;3sin2

;2

1;3ln2;

2;

3;3 2

3 232

etcxx

x

exxx

x x

Use the constant, sum and/or

difference, product, quotient and chain rules for differentiation.



Range: The following rules of differentiation are used: o If axfy )( and a is a constant function, then

0)( xfdx

dy

o If

)()(

)()()()(

)()(

xfdx

dkxkf

dx

d

xgdx

dxf

dx

dxgxf

dx

d

thenxgxfy

o If

thenxgxfy )().(

)().()().( xgxfxgxfdx

dy

o If thenxg

xfy

)(

)(

2)(

)().()().(

xg

xgxfxfxg

dx

dy

o dx

du

du

dy

dx

dy

(u-substitution)

Note: The chain rule may also be used without u-substitution. Note: Combinations of rules in the same problem are excluded.

The equations of tangents to graphs at specific points are

found.

Practical problems regarding rates of change are solved

Cubic functions are sketched by using differentiation.

Work with maximum, minimum turning points and point of

inflection.

Note: Use the following notations,

2

2

);(dx

ydxf ,for the second order derivative.

Note: Combinations of rules in the same problem are excluded. Find the equation of the tangent to a

graph at a specific point.

Solve practical problems involving rates of change. Note: velocity and acceleration may be included

Draw graphs of cubic functions by determining: o y-intercept o roots (x-intercepts) o turning points using derivatives

Determine/prove maximum and

minimum turning points by making use of second order derivatives (Only: quadratic and cubic functions)

Determine the point of inflection of cubic graphs by using second order derivatives.





SUBJECT OUTCOME

2.5 Analyse and represent mathematical and contextual situations using integrals and find areas under curves by using integration rules.


Integrals are found by using appropriate rules and simplifications correctly.

Range:

dxkxa

dxkxa

dxkxa

dxae

dxx

a

dxax

kx

n

2sec

cos

sin

Definite integrals are calculated.

Find the integrals of the following:

dxkxa

dxkxa

dxkxa

dxae

dxx

a

dxax

kx

n

2sec

cos

sin

Where:

ck

kxadxkxa

ck

kxadxkxa

ck

eadxae

cxadxx

a

cn

xadxax

kxkx

nn

sincos

cossin

ln

1

1

Note:

o Simplifications may be required where necessary

o Integrals of polynomials may be assessed o Integration by parts is excluded

Use the upper and lower limits to calculate

definite integrals.



The area under a curve is found by splitting the area into two intervals when the graph crosses the x -axis.

Determine the area under a curve by: o Working from a given graph or by sketching a

graph o Working with an area bounded by a curve,

the x -axis, an upper and a lower limit. o Splitting the area into two intervals when the

graph crosses the x -axis Note: o Integrals with respect to the x -axis only. o Areas between two curves are excluded. o The y -axis ( 0x ) may be used as an

upper or lower limit.


Assignments Test Internal Examination Practical exercise.

Topic 3: Space, Shape and Measurement. (Approximately 35 hours)

SUBJECT OUTCOME

3.1 Use the Cartesian co-ordinate system to derive and apply equations.


The Cartesian co-ordinate system is used to derive and apply the equation of a circle with any centre.

The Cartesian co-ordinate system is used to derive and apply the equation of a tangent to a circle given a point on the circle.

Use the Cartesian co-ordinate system to derive and apply the equation of a circle (any centre).

Use the Cartesian co-ordinate system to derive and apply the equation of a tangent to a circle given a point on the circle.

Note: o Straight lines to be written in the following

forms only:

)(0/

)(; 11

formgeneralcbyaxorand

xxmyycmxy

o o Learners are expected to know and be able

to use as an axiom “the tangent to a circle is perpendicular to the radius drawn to the point of contact.”





SUBJECT OUTCOME

3.2 Explore, interpret and justify geometric relationships.


Geometry involving straight lines and triangles are used to solve problems in geometrical figures.

The major theorems of circles are stated and applied.

Range:

o If a line is drawn from the centre of a circle to the midpoint of a chord, then that line is perpendicular to the chord.

o If a line is drawn from the centre of the circle perpendicular to the chord, then it bisects the chord.

o If an arc subtends an angle at the centre of the circle and at any point on the circumference, then the angle at the centre is twice the measure of the angle at the circumference.

o If the diameter of a circle subtends an angle at the circumference, then the angle subtended is a right angle triangle.

o If an angle subtended by a chord at a point on the circumference is a right angle, then the chord is a diameter.

o Angles in the same segment of a circle are equal.

o The opposite angles of a cyclic quadrilateral are supplementary.

o An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

o If the exterior angle of a quadrilateral is equal to the interior opposite angle the quadrilateral will be a cyclic quadrilateral.

o The four vertices of a quadrilateral in which the opposite angles are supplementary will be a cyclic quadrilateral.

o If a tangent to a circle is drawn, then it is perpendicular to the radius at the point of contact.

o If a line is drawn perpendicular to a radius at the point where the radius meets the circle, then it is a tangent to the circle.

o If two tangents are drawn from the same

Use geometry of straight lines and triangles to solve problems and to justify relationships in geometric figures. Concepts to include are: o angles of a triangle; o exterior angles, o straight lines, o vertically opposite angles; o corresponding angles, o co-interior angles and o alternate angles.

State and apply the following theorems of

circles:

o If a line is drawn from the centre of a circle to the midpoint of a chord, then that line is perpendicular to the chord.

o If a line is drawn from the centre of the circle perpendicular to the chord, then it bisects the chord.

o If an arc subtends an angle at the centre of the circle and at any point on the circumference, then the angle at the centre is twice the measure of the angle at the circumference.

o If the diameter of a circle subtends an angle at the circumference, then the angle subtended is a right angle triangle.

o If an angle subtended by a chord at a point on the circumference is a right angle, then the chord is a diameter.

o Angles in the same segment of a circle are equal.

o The opposite angles of a cyclic quadrilateral are supplementary.

o An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

o If the exterior angle of a quadrilateral is equal to the interior opposite angle the quadrilateral will be a cyclic quadrilateral.

o The four vertices of a quadrilateral in which the opposite angles are supplementary will be a cyclic quadrilateral.

o If a tangent to a circle is drawn, then it is perpendicular to the radius at the point of contact.

o If a line is drawn perpendicular to a radius at the point where the radius meets the circle, then it is a tangent to the circle.

o If two tangents are drawn from the same point outside a circle then they are equal in length.



point outside a circle then they are equal in length.

o The angle between a tangent to a circle and a chord drawn from the point of contact is equal to an angle in the alternate segment. (tan-chord theorem)

Note: Proofs of the above theorems are excluded

o The angle between a tangent to a circle and a chord drawn from the point of contact is equal to an angle in the alternate segment. (tan-chord theorem)

Note: Proofs of the above theorems are excluded



SUBJECT OUTCOME

3.3 Solve problems by constructing and interpreting trigonometric models.


The following compound angle identities are derived and used.

sinsincoscos)cos(

sincoscossin)(sin

to derive and apply the following double angle identities:

2

2

22

sin21

1cos2

sincos

2cos

cossin22sin

Trigonometric equations are solved using

compound and double angle identities without a calculator.

Compound angle identities are used to simplify trigonometric expressions and to prove trigonometric equations.

Specific solutions of trigonometric equations are

solved. Range:

o Solutions: ]360;0[

o Identities limited to:

Use the following compound angle identities:

sinsincoscos)cos(

sincoscossin)(sin

to derive and apply the following double angle identities:

2

2

22

sin21

1cos2

sincos

2cos

cossin22sin

Determine the specific solutions of trigonometric expressions using compound and double angle identities without a calculator. (e.g. sin 1200, cos 750 etc.)

Use compound angle identities to simplify trigonometric expressions and to prove trigonometric equations.

Determine the specific solutions of trigonometric

equations by using knowledge of compound angles and identities.

Note:

o Solutions: ]360;0[

o Identities limited to:



1cossin

cos

sintan

22

o Double and compound angle identities

Note: radians are excluded

Problems in two and three dimensions are

solved from a given diagram by using the sine and cosine rules.

1cossin

cos

sintan

22

o Double and compound angle identities are included.

Note: radians are excluded

Solve problems from a given diagram in two and three dimensions by applying the sine and cosine rule. Note: Area formula and compound angle identities are excluded.



Topic 4: Data Handling and Probability Models. (Approximately 18 hours)

SUBJECT OUTCOME

4.1 Represent, analyse and interpret data using various techniques.


Situations or issues that can be dealt with through statistical methods are identified.

Note: Not for examination purposes.

Resolutions are made to maximize efficiency from given data which has been organised and graphically represented.

Note: For discussion only on the use of knowledge acquired in Level 2 and 3

Identify situations or issues that can be dealt with through statistical methods.

Range: Data given should include problems relating to health, social, economic, cultural, political and environmental issues.

Note: Not for examination purposes but for class activities only.

Discuss the use of appropriate and efficient

methods to record, organise and interpret given data by making use of: - Manageable data sample sizes

o (less than or equal to 10) and which are representative of the population.

- Graphical representations and numerical

summaries which are consistent with the data, and clear and appropriate to the situation and target audience. Note: Discussion only, not expected to draw again.

- Compare different representations of given data.

Justify and apply statistics to answer questions

about problems.



Questions that arise from the modelling of data are discussed

Statistics is applied to answer questions about problems.

Discuss new questions that arise from the modelling of data.

Take a position on an issue by comparing different representations of given data.


Discussions Assignments Group activities

SUBJECT OUTCOME

4.2 Use variance and regression analysis to interpolate and extrapolate bivariate data.


Measures of deviation and dispersion of bivariate

data are used, calculated and interpreted

Scatter plots are used to represent bivariate

numerical data

Range:

o Data given should include problems related

to health, social, economic, cultural, political

and environmental issues.

o For small sets of data only (limited to 8)

Determine the equation of the line of best fit by

using least squares regression.

The regression line is used to predict the outcome

of a given problem.

Calculate: o variance and o standard deviation manually for small sets of

data only.

Interpret the meaning of variance and standard deviation for small sets of data only.

Represent bivariate numerical data as a scatter

plot

Identify intuitively whether a linear, quadratic or

exponential function would best fit the data.

Draw the intuitive line of best fit.

Range:

o Data given should include problems related

to health, social, economic, cultural, political

and environmental issues.

o For small sets of data only (limited to 8)

Use least squares regression method to

determine a function which best fits a given set

of bivariate data.

Use the regression line to predict the outcome of

a given problem





SUBJECT OUTCOME

4.3 Use experiments, simulation and probability distribution to set and explore probability models.


The following terminology are explained:

o Probability

o Dependent events

o Independent events

o Mutually exclusive

o Mutually inclusive

o Complimentary events

Predictions are made based on validated

experimental or theoretical probabilities.

Explain and distinguish between the following

terminology/events:

o Probability

o Dependent events

o Independent events

o Mutually exclusive

o Mutually inclusive

o Complimentary events

Make predictions based on validated

experimental or theoretical probabilities taking

the following into account:

o P(S) = 1

(where S is the sample space);

o Disjoint (mutually exclusive) events,

and is therefore able to calculate the

probability of either of the events

occurring by applying the addition

rule for disjoint events:

P(A or B) = P(A) + P(B);

o Complementary events and is

therefore able to calculate the

probability of an event not occurring;

o P(A or B) = P(A) + P(B) - P(A and B) (where A and B are events within a sample space);

o Correctly identify dependent and

independent events

(e.g. from two-way contingency

tables or Venn diagrams) and

therefore appreciate when it is

appropriate to calculate the

probability of two independent

events occurring by applying the

product rule for independent events:

P (A and B) = P (A).P (B).



Tree diagrams, Venn diagrams and contingency

two-way tables are drawn/completed to solve

probability problems (where events are not

necessarily independent).

Range:

o Venn diagrams to be limited to two

subsets.

o Tree diagrams where the sample

space in manageable. (not more

than 15 possible outcomes)

Results are interpreted correctly and clearly

communicated.

Draw Tree diagrams, Venn diagrams and

complete contingency two-way tables to

solve probability problems (where events are

not necessarily independent).

Range:

o Venn diagrams to be limited to two

subsets.

o Tree diagrams where the sample

space in manageable. (not more

than 15 possible outcomes)

Interpret and clearly communicate results of the

experiments correctly in terms of real context.


Assignments Test Internal Examination Practical exercises.

Topic 5: Financial Mathematics (Approximately 7 hours)

SUBJECT OUTCOME

5.1 Use mathematics to plan and control financial instruments.


Simple and compound growth formulae

)1( inPA A and niPA )1( and mt

m

rPA

1001 are used to solve

problems, including interest, hire-purchase and inflation. Range: Manipulation of only iPA ;;

Tax tables are understood, used and interpreted.

Simple and compound decay formulae growth

formula, )1( niPA and niPA )1( , are

used to solve problems. Range: Unknown values to calculate will only include iPA ;;

Use simple and compound growth formulae

)1( inPA A and niPA )1( and mt

m

rPA

1001 to solve

problems, including interest, hire-purchase and inflation.

Understand, use and interpret tax tables. Use simple and compound decay formulae,

)1( niPA and niPA )1( , to solve

problems (straight line depreciations and depreciation on a reducing balance).


Assignments Test Internal Examination Practical test.



4 SPECIFICATIONS FOR EXTERNAL ASSESSMENT IN MATHEMATICS - LEVEL 4

A national examination is conducted annually in October or November each year by means of two three hour examination papers set externally and marked and moderated externally. One of the papers will consist of algebra, and complex numbers and the other of geometry, trigonometry, statistics and financial mathematics. The content covered in each is described in Subject Guidelines Mathematics Level 4. The following distribution of cognitive application is suggested:

Details in respect of relative weightings of the topics are contained in Subject Guidelines Mathematics

Proposed mark distribution between paper 1 and paper 2 for external examination papers for

Level 4.

Paper 1 (3 hours)

TOPICS WEIGHTED VALUE

1. Complex numbers 20 2. Functions and Algebra 2.1 Functions & Algebra 25 2.2 Linear programming 15 2.3 Differentiation 25 2.4 Integration 15

TOTAL 100

Paper 2 (3 hours)

TOPICS/THEMES WEIGHTED VALUE

3. Space ,shape and measurement 3.1 Geometry 25 3.2 Trigonometry 25 4. Statistics and probability models 4.1 Statistics 15 4.2 Probability 15 5. Financial Mathematics 20

TOTAL 100

LE

VE

L 4

KNOWLEDGE COMPREHENSION AND

APPLICATION ANALYSIS

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