Technical CAPS Technical Mathematics Grades 10-12

8/13/2019 Technical CAPS Technical Mathematics Grades 10-12

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CURRICULUM AND ASSESSMENTPOLICY STATEMENT

TECHNICAL MATHEMATICS

GRADES 10 -12


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T ABLE OF CONTENTS

1.1 Background 4

1.2 Overview 4

1.3 General aims of the South African Curriculum 5

1.4 Time allocation 6

1.4.1 Foundation Phase 6

1.4.2 Intermediate Phase 6

1.4.3 Senior Phase 7

1.4.4 Grades 10 12 7

2.1 What is Technical Mathematics? 82.2 Specific Aims 8

2.3. Specific Skills 9

2.4 Focus of Content Areas 92.5 Weighting of Content Areas 102.6 Technical Mathematics in the FET 10

3.1 Specification of content to show Progression 123.1.1 Overview of topics 12

3.2 Content clarification with teaching guidelines 183.2.1 Allocation of teaching time 183.2.2 Sequencing and pacing of topics 19

3.2.3 Topic allocation per term and clarification 24Grade 10 Term: 1 26

Grade 10 Term: 2 32

Grade 10 Term: 3 36


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Grade 10 Term: 4 39

Grade 11 Term: 1 42

Grade 11 Term: 2 46

Grade 11 Term: 3 49

Grade 11 Term: 4 54

Grade 12 Term: 1 56

Grade 12 Term: 2 65

Grade 12 Term: 3 71

Grade 12 Term: 4 73

Assessment Guidelines4.1.Introduction 74

4.2.Informal or Daily Assessment 75

4.3.Formal Assessment 75

4.4.Programme of Assessment 764.5.Recording and reporting 784.6.Moderation of Assessment 794.7.General 79


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1.1

BackgroundThe National Curriculum Statement Grades R 12 (NCS) stipulates policy on curriculum and assessment inthe schooling sector. To improve its implementation, the National Curriculum Statement was amended, withthe amendments coming into effect in January 2012. A single comprehensive National Curriculum andAssessment Policy Statement was developed for each subject to replace the old Subject Statements,Learning Programme Guidelines and Subject Assessment Guidelines in Grades R - 12.

The amended National Curriculum and Assessment Policy Statements (January 2012) replace the NationalCurriculum Statements Grades R - 9 (2002) and the National Curriculum Statements Grades 10 - 12 (2004) .

1.2 Overview

(a) The National Curriculum Statement Grades R 12 (January 2012) represents a policy statement forlearning and teaching in South African schools and comprises the following: National Curriculum and Assessment Policy s tatemen ts for each approved school subject as listed in the policy document, and the National policy pertaining to the programme and promotion requirements ofthe National Curriculum Statement Grades R 12, which replaces the following policy documents:(i) National Senior Certificate: A qualification at Level 4 on the National Qualifications Framework

(NQF); and2680(ii) An addendum to the policy document, the National Senior Certificate: A qualification at Level 4

on the National Qualifications Framework (NQF), for learners with special needs , published in theGovernment Gazette, No.29466 of 11 December 2006.

(b) The National Curriculum Statement Grades R 12 (January 2012) should be read in conjunction with the National Protocol for Assessment Grade R 12, which replaces the policy document, An addendum to the

policy document, the National Senior Certificate: A qualification at Level 4 on the National Qualifications Framework (NQF), for the National Protocol for Assessment Grade R 12, published in the GovernmentGazette, No. 29467 of 11 December 2006.

(c) The Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines for GradesR - 9 and Grades 10 - 12 have been repealed and replaced by the National Curriculum and Assessment

Policy Statements for Grades R 12 (January 2012) .(d) The sections on the Curriculum and Assessment Policy as discussed in Chapters 2, 3 and 4 of this

document constitute the norms and standards of the National Curriculum Statement Grades R 12.T therefore , in terms of section 6A of the South African Schools Act, 1996 (Act No. 84 of 1996,) thisforms the basis on which the Minister of Basic Education will determine minimum outcomes andstandards, as well as the processes and procedures for the assessment of learner achievement , that willapply in public and independent schools.

CHAPTER 1CURRICULUM AND A SSESSMENT P OLICY S TATEMENT (CAPS)

TECHNICAL FET MATHEMATICS


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1.4 Time Allocation

1.4.1 Foundation Phase

(a) The instructional time for each subject in the Foundation Phase is indicated in the table below:

Subject Time allocation per week (hours)i. Languages (FAL and HL)

ii. Mathematicsiii. Life Skills

Beginning Knowledge Creative Arts Physical Education Personal and Social Well-

being

10 (11)7

6 (7)1 (2)

221

(b) Total instructional time for Grades R, 1 and 2 is 23 hours and for Grade 3 is 25 hours.

(c) To Languages 10 hours are allocated in Grades R-2 and 11 hours in Grade 3. A maximum of 8 hours anda minimum of 7 hours are allocated to Home Language and a minimum of 2 hours and a maximum of 3hours to First Additional Language in Grades R 2. In Grade 3 a maximum of 8 hours and a minimum of7 hours are allocated to Home Language and a minimum of 3 hours and a maximum of 4 hours to FirstAdditional Language.

(d) To Life Skills Beginning Knowledge 1 hour is allocated in Grades R 2 and 2 hours are allocated forgrade 3 as indicated by the hours in brackets.

1.4.2 Intermediate Phase

(a) The table below shows the subjects and instructional times allocated to each in the Intermediate Phase.

Subject Time allocation per week (hours)i. Home Language

ii. First Additional Languageiii. Mathematicsiv. Science and Technologyv. Social Sciences

vi. Life Skillsvii. Creative Arts

viii. Physical Educationix. Religious Studies

656

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1.4.3 Senior Phase

(a) The instructional time for each subject in the Senior Phase is allocated as follows:

Subject Time allocation per week (hours)


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i. Home Language

ii. First Additional Language

iii. Mathematics

iv. Natural Sciences

v. Social Sciences

vi. Technology

vii. Economic Management Sciences

viii. Life Orientation

ix. Arts and Culture

5

4

4.5

3

3

2

2

2

2

1.4.4 Grades 10-12

(a) The instructional time for each subject in Grades 10-12 is allocated as follows:

Subject Time allocation per week(hours)

i. Home Language

ii. First Additional Language

iii. Mathematics

iv. Technical Mathematics

v. Mathematical Literacy

vi. Life Orientation

vii. Three Electives

4.5

4.5

4.5

4.5

4.5

2

12 (3x4h)

The allocated time per week may be utilised only for the minimum number of required NCS subjects as

specified above, and may not be used for any additional subjects added to the list of minimum subjects .

Should a learner wish to offer additional subjects, additional time has to be allocated in order to offer these

subjects .


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Introduction

In Chapter 2, the Further Education and Training (FET) Phase Technical Mathematics CAPS provides teachers

with a definition of technical mathematics, specific aims, specific skills, focus of content areas, and the

weighting of content areas.

2.1 What is Technical Mathematics?

Mathematics is a universal science language that makes use of symbols and notations for describing numerical,

geometric and graphical relationships. It is a human activity that involves observing, representing and

investigating patterns and qualitative relationships in physical and social phenomena and between mathematical

objects themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy and

problem solving that will contribute in decision-making.

Mathematical problem solving enables us to understand the world (physical, social and economic) around us,

and, most of all, to teach us to think creatively. The aim of Technical Mathematics is to apply the Science of

Mathematics to the Technical field where the emphasis is on APPLICATION and not on abstract ideas.

2.2 Specific Aims of Technical Mathematics

1. TO APPLY MATHEMATICAL PRINCIPLES

1. To develop fluency in computation skills with the usage of calculators .

2. Mathematical modeling is an important focal point of the curriculum. Real life technical problems should

be incorporated into all sections whenever appropriate. Examples used should be realistic and not

contrived. Contextual problems should include issues relating to health, social, economic, cultural,

scientific, political and environmental issues whenever possible.

3. To provide the opportunity to develop in learners the ability to be methodical, to generalize and to be

skillful users of the Science of Mathematics.


FET MATHEMATICS


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4. To be able to understand and work with number system.

5. To promote accessibility of Mathematical content to all learners . It could be achieved by catering for

learners with different needs. e.g. TECHNICAL NEEDS .

6. To develop problem-solving and cognitive skills. Teaching should not be limited to how but should

rather feature the when and why of problem types.

7. To provide learners at Technical schools an alternative and value adding substitute to Mathematical

Literacy.

8. To support and sustain technical subjects at Technical schools.

9. To provide a vocational route aligned with the expectations of labour in order to ensure direct access to

learnership / apprenticeship.

10. To create the opportunity for learners to further their studies at F.E.T Colleges at an entrance level of N-4

and thus creating an alternative route to access other HEIs.

2.3 Specific Skills

To develop essential mathematical skills the learner should:

develop the correct use of the language of Mathematics; use mathematical process skills to identify and solve problems. use spatial skills and properties of shapes and objects to identify, pose and solve problems creatively

and critically;

participate as responsible citizens in the technical environment locally, national and global

communities; and

communicate appropriately by using descriptions in words, graphs, symbols, tables and diagrams.

2.4 Focus of Content Areas

Technical Mathematics in the FET Phase covers ten main content areas. Each content area contributes

towards the acquisition of the specific skills.

The table below shows the main topics in the FET Phase.


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The Main Topics in the FET Technical Mathematics Curriculum1. Number system2. Functions and graphs3. Finance, growth and decay4. Algebra5. Differential calculus6. Euclidean geometry7. Mensuration8. Circles, angles and angular movement9. Analytical geometry10. Trigonometry

2.5 Weighting of Content Areas

The weighting of Technical Mathematics content areas serves two primary purposes: firstly the weighting givesguidance on the amount of time needed to address adequately the content within each content area; secondly theweighting gives guidance on the spread of content in the examination (especially end of the year summativeassessment).

Weighting of Content Areas Description Grade 10 Grade 11 Grade. 12

PAPER 1

Algebra ( Expressions, equations and inequalitiesincluding nature of roots in grade 11 &12)

65 3 90 3 50 3

Functions & Graphs 20 3 45 3 35 3Finance, growth and decay 15 3 15 3 15 3Differential Calculus and Integration 50 3TOTAL 100 150 150PAPER 2 :

Description Grade 10 Grade 11 Grade 12Analytical Geometry 15 3 25 3 25 3Trigonometry 40 3 50 3 50 3Euclidean Geometry 30 3 40 3 40 3Mensuration and circles, angles and angular movement 15 3 35 3 35 3TOTAL 100 150 150

2.6 Technical Mathematics in the FET

The subject Technical Mathematics in the Further Education and Training Phase forms the link between theSenior Phase and the Higher education band. All learners passing through this phase acquire a functioningknowledge of the Technical Mathematics that empowers them to make sense of their Technical field of studyand their place in society. It ensures access to an extended study of the technical mathematical sciences and avariety of technical career paths.

In the FET Phase, learners should be exposed to technical mathematical experiences that give them manyopportunities to develop their mathematical reasoning and creative skills in preparation for more appliedmathematics in HEIs or in -job-training.


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Introduction

Chapter 3 provides teachers with: Specification of content to show progression Clarification of content with teaching guidelines Allocation of time

3.1 Specification of Content to show Progression

The specification of content shows progression in terms of concepts and skills from Grade 10 to 12 foreach content area. However, in certain topics the concepts and skills are similar in two or threesuccessive grades. The clarification of content gives guidelines on how progression should beaddressed in these cases. The specification of content should therefore be read in conjunction with theclarification of content.


FET MATHEMATICSCONTENT SPECIFICATION AND CLARIFICATION


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3.1.1 Overview of topics

1. NUMBER SYSTEM

Grade 10 Grade 11 Grade 12

(a) Identify rational numbers : and convertterminating or recurring decimals into the

formba

where Z ba , and 0b .

(b) Understand that simple surds are notrational.

Take note that there exist numbers other thanthose on the real number line, the so-called non-real numbers . It is possible to square certainnon-real numbers and obtain negative realnumbers as answers.

Binary numbers should be known.

There exist numbers other than those studiedin earlier grades called imaginary numbers andcomplex numbers.Add, subtract, divide, multiply and simplifyImaginary numbers and complex numbers.Solve equations involving complex numbers.

2. FUNCTIONS

Work with relationships between variables interms of numerical, graphical, verbal andsymbolic representations of functions andconvert flexibly between these representations(tables, graphs, words and formulae). Includelinear and some quadratic polynomialfunctions, exponential functions and somerational functions.

Extend Grade 10 work on the relationships between variables in terms of numerical,graphical, verbal and symbolic representationsof functions and convert flexibly between theserepresentations (tables, graphs, words andformulae). Include linear and quadratic

polynomial functions, exponential functions andsome rational functions.

Introduce a more formal definition of afunction and extend Grade 11 work on therelationships between variables in terms ofnumerical, graphical, verbal and symbolicrepresentations of functions and convertflexibly between these representations (tables,graphs, words and formulae). Include linear,quadratic and some cubic polynomialfunctions, exponential and some rationalfunctions.

Generate as many graphs as necessary, initially by means of point-by-point plotting, to makeand test conjectures and hence generalise theeffect of the parameter which results in avertical shift and that which results in a verticalstretch and /or a reflection about the x -axis.

Generate as many graphs as necessary, initially by means of point-by-point plotting, to make andtest conjectures and hence generalise the effectsof the parameter which results in a horizontalshift and that which results in a horizontalstretch and/or reflection about the y-axis.

Revise work studied in earlier grades.

Problem solving and graph work involving the prescribed functions.

Problem solving and graph work involving the prescribed functions. Average gradient betweentwo points.

Problem solving and graph work involving the prescribed functions.


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3. FINANCE, GROWTH AND DECAY

Use simple and compound growth formulae)1( in P A and ni P A )1( to solve

problems (including interest, hire purchase,inflation, population growth and other real life

problems).

Use simple and compound growth/decayformulae )1( in P A and ni P A )1( to solve problems (including interest, hire

purchase, inflation, population growth and otherreal life problems).

Revise Grade 11 work

The implications of fluctuating foreignexchange rates.

The effect of different periods of compoundinggrowth and decay (including effective and

nominal interest rates).

Critically analyse different loan options.

4. ALGEBRA

(a) Simplify expressions using the laws ofexponents for integral exponents.

(b) Establish between which two integers agiven simple surd lies.

(c) Round real numbers to an appropriatedegree of accuracy (to a given number ofdecimal digits).

(d) Revise scientific notation

(a) Apply the laws of exponents to expressionsinvolving rational exponents.

(b) Add, subtract, multiply and divide simplesurds.

(c) Demonstrate an understanding of thedefinition of a logarithm and any lawsneeded to solve real life problems.


(a) Simplify expressions using the laws ofexponents for integral exponents.

(b) Establish between which two integers agiven simple surd lies.

(c) Round real numbers to an appropriatedegree of accuracy (to a given number ofdecimal digits).

(d) Revise scientific notation

(a) Apply the laws of exponents to expressionsinvolving rational exponents.

(b) Add, subtract, multiply and divide simplesurds.

(c) Demonstrate an understanding of thedefinition of a logarithm and any lawsneeded to solve real life problems.



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Manipulate algebraic expressions by: multiplying a binomial by a trinomial; factorising common factor (revision); factorising by grouping in pairs; factorising trinomials; factorising difference of two squares

(revision); factorising the difference and sums of two

cubes; and simplifying, adding, subtracting,

multiplying and division of algebraic

fractions with numerators anddenominators limited to the polynomialscovered in factorization.

Revise factorisation from Grade 10 Take note and understand the Remainderand Factor Theorems for polynomials upto the third degree.

Factorise third-degree polynomials(including examples which require theFactor Theorem).

Solve: linear equations; quadratic equations by factorisation; literal equations (changing the subject of

formulae); exponential equations (accepting that the

laws of exponents hold for real exponentsand solutions are not necessarily integral oreven rational);

linear inequalities in one variable andillustrate the solution graphically; and

linear equations in two variablessimultaneously (algebraically andgraphically).

Solve: quadratic equations (by factorisation and by

using the quadratic formula); equations in two unknowns, one of which is

linear the other quadratic, algebraically orgraphically.

Explore the nature of roots through the valueof acb 42 .


5. DIFFERENTIAL CALCULUS

(a) An intuitive understanding of the conceptof a limit.

(b) Differentiation of specified functions fromfirst principles.

(c) Use of the specified rules ofdifferentiation.

(d) The equations of tangents to graphs.(e) The ability to sketch graphs of cubic

functions.(f) Practical problems involving optimization


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and rates of change (including thecalculus of motion ).

(g) Basic integration

6. EUCLIDEAN GEOMETRY

(a) Revise basic results established in earliergrades.

(b) Investigate and form conjectures about the properties of special triangles (scalene,isosceles, equilateral and right-angledtriangle) and quadrilaterals.

(c) Investigate alternative (but equivalent)definitions of various polygons (includingthe scalene, isosceles, equilateral and right-angled triangle, the kite, parallelogram,rectangle, rhombus, square and trapezium).

(d) Application of the theorem of Pythagoras.

(a) Investigate theorems of the geometry ofcircles assuming results from earliergrades, together with one other resultsconcerning tangents and radii of circles.

(b) Solve circle geometry problems, providingreasons for statements when required.

(a) Revise the concept of similarity and proportionality.

(b) Apply proportionality in triangles.(c) Apply mid-point theorem.

7. MENSURATION

Conversion of units, square units and cubic units (a) Solve problems involving volume and surfacearea of solids studied in earlier grades andcombinations of those objects to form morecomplex shaped solids.

(b) Determine the area of an irregular figure usingMid-ordinate Rule.


8. CIRCLES, ANGLES AND ANGULAR MOVEMENT

Define a radian Converting degrees to radians and vice versa

Angles and arcs Degrees and radians Sectors and segments Angular and circumferential velocity.


9. ANALYTICAL GEOMETRY

Represent geometric figures in a Cartesian co-ordinate system, and derive and apply, for anytwo points ( 11; y x ) and ( 22; y x ) a formula forcalculating:

the distance between the two points;

Use a Cartesian co-ordinate system todetermine :

the equation of a line through two given points;

the equation of a line through one point

Use a two-dimensional Cartesian co-ordinatesystem to determine:

the equation of a circle with centre at theorigin (centre is (0;0));

the equation of a tangent to a circle at a


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the gradient of the line segment joining the points;

the co-ordinates of the mid-point of the linesegment joining the points; and

the equation of a straight line joining thetwo points.

and parallel or perpendicular to a givenline; and

the angle of inclination of a line.

given point on the circle and point(s) of intersection of a circle and a

straight line.

10. TRIGONOMETRY

(a) Definitions of the trigonometric functions sin,cos and tan in right-angled triangles.

Take note that there are special names for thereciprocals of the trigonometric functions

sin1cos ec ;

cos1sec and

tan1cot .

(b) Extend the definitions ofsin , cos andtan to 3600 and calculatetrigonometric ratios.

(c) Simplification of trigonometric expressions/equations by making use of a calculator.

(d) Solve simple trigonometric equations for anglesbetween 00 and 900.

(a) Use the identities:

cossintan ,

1cossin 22 , 22 sectan1 ,and 22 cos1cot ec .

(b) Reduction formulae, )180( and)360( .

(c) determine the solutions of trigonometricequations for 3600 .

(d) apply sine, cosine and area rules (proofs ofthese rules will not be examined).


Solve problems in two dimensions by using the abovetrigonometric functions and by constructing and

interpreting geometric and trigonometric models.

Solve problems in 2-dimensions by using sine,cosine and area rule.

Solve problems in two dimensions by construcand interpreting geometric and trigonometric

models. Draw the graphs of sin y , cos y and tan y .

The effects of the parameters on the graphs definedby k y sin , k y cos and k y tan .The effects of p on the graphs of )(sin p y ,

)(cos p y and )tan( p y .One parameter should be tested at a given time ifexamining horizontal shift.


Rotating vectors (sine and cosine curves only)


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3.2 Content Clarification with teaching guidelines

In Chapter 3, content clarification includes:

o Teaching guidelineso Sequencing of topics per termo Pacing of topics over the year

Each content area has been broken down into topics. The sequencing of topics within terms givesan idea of how content areas can be spread and re-visited throughout the year.

The examples discussed in the Clarification Column in the annual teaching plan which follows are by no means a complete representation of all the material to be covered in the curriculum. Theyonly serve as an indication of some questions on the topic at different cognitive levels. Text booksand other resources should be consulted for a complete treatment of all the material.

The order of topics is not prescriptive, but ensure that in the first two terms, more than six topicsare covered / taught so that assessment is balanced between paper 1 and 2.

3.2.1 Allocation of Teaching Time

Time allocation for Technical Mathematics: 4 hours and 30 minutes, e.g. six forty five-minutes periods, per week in grades 10, 11 and 12.

Terms Grade 10 Grade 11 Grade 12

No. ofweeks

No. ofweeks

No. ofweeks

Term 1

Introduction

Number systemsExponentsMensurationAlgebraic Expressions

2

3213

Exponents and surds

LogarithmsEquations and inequalities(including nature of roots)Analytical geometry

3

24

2

Complex numbers

PolynomialsCalculus and Integration

3

26

Term 2

Algebraic ExpressionsEquations and InequalitiesTrigonometryMID-YEAR EXAMS

2333

Functions and graphsEuclidean GeometryMID-YEAR EXAMS

443

Analytical GeometryEuclidian GeometryRevisionMID-YEAR EXAMS

3413

Term 3

TrigonometryFunctions and GraphsEuclidean geometryAnalytical geometry

2341

Circles, angles and angularmovementTrigonometryFinance, growth and decay

4

42

Euclidian GeometryRevisionTRIAL EXAMS

22

Term 4 Analytical geometryCircles, angles and angularmovementFinance and growthRevisionEXAMS

1

1223

MensurationRevisionEXAMS

333

Revision

EXAMS

34

The detail which follows includes examples and numerical references to the Overview.


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3.2.2 Sequencing and Pacing of Topics

MATHEMATICS: GRADE 10 PACE SETTERTerm 1 TERM 1Weeks WEEK 1 WEEK 2 WEEK 3 WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK

10WEE

Topics Introduction Number systems Exponents Mensuration Algebraic Expressions

ssessment Investigation or project TestDate

ompletedTerm 2 TERM 2Weeks WEEK 1 WEEK 2 WEEK 3 WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK

8WEEK

9WEEK 10 WE

Topics Algebraic Expressions Equations and Inequalities Trigonometryssessment Assignment / Test MID-YEAR EXAMINATION

DateompletedTerm 3 TERM 3Weeks WEEK

1WEEK 2 WEEK 3 WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10

Topics Trigonometry Functions and graphs Euclidean Geometry AnalyticalGeomet

ssessment Test Test

DateompletedTerm 4 TERM 4 Paper 1 : 2 hours Paper 2: 2 hours

Weeks WEEK 1 WEEK 2 WEEK 3

WEEK 4

WEEK 5

WEEK6

WEEK7

WEEK8

WEEK9

WEEK10

Algebraicexpressionsand equations (andinequalities)Finance and growth ExponentsFunctions andgraphs (excludingtrig. functions)

45

10

1530

Euclidean geometryand measurementAnalyticalgeometryTrigonometryCircles, angles andangular movement

Topics Analyticalgeometry

Circles,angles andangularmovement

Finance andgrowth

Revision Admin

ssessment Test Examinations

Dateompleted

Total marks 100 Total marks


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MATHEMATICS: GRADE 11 PACE SETTERTerm 1 TERM 1Weeks WEEK

1WEEK 2 WEEK 3 WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 WEEK 11

Topics Exponents and surds Logarithms Equations and inequalities Nature of roots Analytical Geometry

Assessment Investigation or project TestDate

completedTerm 2 TERM 2Weeks WEEK

1WEEK

2WEEK 3 WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 WEEK 11

Topics Functions and graphs Euclidean Geometry Trigonometry (reduction formulae, graphs,equations)

Assessment Assignment / Test MID-YEAR EXAMINATIONDate

completedTerm 3 TERM 3Weeks WEEK 1 WEEK 2 WEEK 3 WEEK 4 WEEK 5 WEEK 6 WEEK7 WEEK 8 WEEK9 WEEK 10Topics Circles, angles and angular movement Trigonometry (sine, cosine and area rules) Finance, growth and decay

Assessment Test Test

Datecompleted

Term 4 TERM 4 Paper 1: 3 hours Paper 2: 3 hours

Weeks WEEK 1

WEEK 2

WEEK 3

WEEK4

WEEK5

WEEK6

WEEK7

WEEK8

WEEK9

WEEK10

Algebraic expressions,equations, inequalitiesand nature of rootsFunctions and graphs

(excluding trig.functions)Finance, growth anddecay

90

45

15

EuclideangeometryAnalyticalgeometry

TrigonometryMensuration andCircles, anglesand angularmovement

Topics Mensuration Revision FINAL EXAMINATION Admin

Assessment Test

Datecompleted

Total marks 150 Total marks 15


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GRADE 10: TERM 1

Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested:knowledge (K), routine procedure (R), complex procedure (C) orproblem-solving (P).

2 Introduction

Mathematical language and conceptused in previous years are revised.Teach learners to use the calculatoreffectively and accurately.

All basic algebra must be revised.

3 Numbersystems

Understand that real numbers can benatural whole, integers , rational,irrational

Introduce imaginary , binary andcomplex numbers

Round real numbers off to asignificant/appropriate degree ofaccuracy.

Convert rational numbers intodecimal numbers and vice versa.

Determine between which twointegers a given simple surd lies.

Set builder notation, interval notationand number lines.

Use real numbers as the set of points and explain each set ofnumbers on a line. (Natural, whole, integers , rational, irrational)

Deal with imaginary ,binary and complex numbersBinary numbers - basic operationsComplex numbers define only

Learners must be able to round off to one, two or three decimals.

Binary numbers system consists of two numbers, namely 0 and 1.

Examples:1. 712121111 12 2.

2 Exponents

1. Revise laws of exponents studied inGrade 9 where

0, y x and Z nm , . nmnm x x x nmnm x x x mnnm x x )( mmm xy y x )(

Also by definition:

nn

x x 1 , 0 x , and 10 x ,

0 x .

Comment: Revise prime base numbers and prime factorization

Examples :1. 32 22

2. 11

39

x

x

Solve for x :3. 273 x 4. 366 x

1100

101111


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2. Use the laws of exponents to simplifyexpressions and solve easyexponential equations(the exponentsmay only be whole numbers)

3. Revise scientific notation.

Comment: Make sure to do examples of very big and very smallnumbers

1 Mensuration

1. Conversion of units and square unitsand cubic units.All conversions should be doneboth ways.

2. Applications in technical fields.

Units of length: (mm, cm, m, km) e.g. 1000m = 1km Units of area: (mm 2, cm 2, m 2) e.g. 1 m 2 = 1 000 000 mm 2 Units of volume: (m l =cm 3, m 3 , dm 3=1l ). Typical practical examples on the learners level should be

introduced.

(3+2)= 5

Algebraicexpressions

1. Revise notation (interval, set builder, number line, sets)

2. Adding and subtracting ofalgebraic terms.

3. Multiplication of a binomial by a binomial

4. Multiplication of a binomial by atrinomial.

5. Determine the HCF and LCM ofnot more than three numerical ormonomial algebraic expressions bymaking use of factorization.

6. Factorization of the followingtypes:

Common factors Grouping in pairs. difference of two

squares addition/subtraction of

two cubes trinomials

7. Do addition, subtraction,multiplication and division ofalgebraic fractions usingfactorisation.

Examples1. Subtract ba 84 from ba 105

Remove brackets2. 22 x x 3. 2)54( y x 4. y x y x 32 5. 1232 2 aaa Factorise6. xy xy y x 642 22 7. cbacab 22 8. 22 82 ba 9. 33 8 y x 10. 4472 x x


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8. Numerators and denominatorsshould be limited to the polynomialscovered in factorisation.

3Equations

andInequalities

1.1 Revise notation( interval, set builder, number line, sets)

1.2 Solve linear equations.1.3 Solve equation with fractions

2. Solve quadratic equations byfactorization(coefficient of 2 x must be 1.)

3. Solve simultaneous linear equationswith two variables ( one of thecoefficients of the variables of one ofthe equations must be 1 )

4.1 Do basic grade 8&9 word problems4.2 Solve word problems involving

linear, quadratic or simultaneouslinear equations.

5. Solve simple linear inequalities (andshow solution graphically).

6. Manipulation of formulae (technicalrelated).

ExamplesSolve for x : 1. 843 x

2. 2133

x x

3. 0242 x x

Solve for xand y :4. 42 y x and 42 y x

5. at uv ( change the subject of the formula to a .)

6. 221 mv E (change the subject of the formula to v .)

(3+2)= 5 Trigonometry

1. Know definitions of thetrigonometric ratios sin , cos andtan , using right-angled triangles forthe domain.

2. Introduction of the reciprocals of the3 basic trigonometric ratios: for

sin1cos ec ,

cos1sec and

tan1cot .

3. Trigonometric ratios in each of thequadrants are calculated where oneratio in the quadrant is given. Makinguse of diagrams.

1. Determine the values of cos and tan if53sin and

36090 .

Making use of Pythagoras theorem.

Y

531416

x0


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4. Practise the use of a calculator forquestions applicable to trigonometry.

5. Solve simple trigonometric equationsfor angles

6. between 0 0 and 90 0.4. Solve two-dimensional problems

involving right-angled triangles.5.Trigonometry graphs

sina y , cosa y and

tana y for 3600 . qa y sin and

qa y cos for 3600 .

Draw the graph by the use of a table and point by point plotting andidentify the following :

- asymptotes- axes of symmetry- the domain and range

Note:- It is very important to be able to read off values from the graph.- Investigate the effect of a and q etc on the graph.

Assessment Term 1 :

1. Investigation or project (only one project a year) (at least 50 marks)Example of an investigation:Imagine a cube of white wood which is dipped into red paint so that the surface is red, but the inside still white. If one cut is made, parallel toeach face of the cube (and through the centre of the cube), then there will be 8 smaller cubes. Each of the smaller cubes will have 3 red facesand 3 white faces. Investigate the number of smaller cubes which will have 3, 2, 1 and 0 red faces if 2/3/4// n equally spaced cuts are made

parallel to each face. This task provides the opportunity to investigate, tabulate results, make conjectures and justify or prove them.2. Test (at least 50 marks and 1 hour). Make sure all topics are tested.

Two or three tests of at least 40 minutes would probably be better. Care needs to be taken to set questions on all four cognitive levels:approximately 20% knowledge, approximately 40% routine procedures, 30% complex procedures and 10% problem-solving.


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GRADE 10: TERM 2


3 Graphs

1. Functional notation2. Generate graphs by means of point- by

point plotting supported by availabletechnology.

3. Drawing of the following functions: Linear function - cmx y

(revise) Quadratic: qax y 2 Hyperbola:

xa y

Exponential xba y .

and 1b

Investigate the way(unique) output values depend on how the inputvalues vary.The terms independent (input) and dependent ( output)variables might be usefull.

qax y 2 - Draw the parabola for 1a only. Use the table method only.

- Identify the following characteristics:* y-intercept* x-intercepts* the turning points* axes of symmetry* the roots of the parabola (x-intercepts)* the domain( input values) and range( output values)* Very important to be able to read off values from

thegraphs.

* Investigate the effect of a and q on the graph.* asymptotes where applicable

NOTE: Use this approach to engage with all the other graphs.

4EuclideanGeometry

1. Revise basic geometry done in grade 8and grade9.Lines and parallel lines, angles, trianglescongruency and similarity.

No formal proves are required, only calculation of unknowns.

Know the properties of the following kinds of triangles scalene isosceles triangle equiangular triangle


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2. Apply the properties of line segments joining the mid-points of two sides of atriangle. Do practical problems

3. Know the features of the following specialquadrilaterals: the kite, parallelogram,rectangle, rhombus, square and trapezium(apply to practical problems).

4. Pythagoras theor em Calculate the unknown side of a

right- angled triangle.

Congruency:Investigate the conditions for two triangles to be congruent:

3 sides side, angle side angle, angle, corresponding side right angle, hypotenuse and side

Similarity:Investigate the condition for two triangles to be similar:

Give drawings of similar triangles, write down the correspondingangles and calculate the ratio of the corresponding sides.

Investigate: line segment joining the mid-points of two sides of a triangle, isparallel to the third side and half of the length of the third side.

Investigate and make conjectures about the properties of the sides, anglesand

diagonals of these quadrilaterals.

Calculation of one unknown side of a right-angled triangle, using Pythagorastheorem.

3 Mid-yearexaminations

Assessment Term 2 :

1. Assignment / test (at least 50 marks) 2. Mid-year examination (at least 100 marks)

One paper of 2 hours (100 marks) or Two papers - one, 1 hour (50 marks) and the other, 1 hour (50 marks)


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GRADE 10: TERM 3


1AnalyticalGeometry

(check term 4)

Represent geometric figures on aCartesian co-ordinate system.Apply for any two points );(

11 y x and

);( 22 y x formulae for determining the:1. distance between the two points;2. gradient of the line segment

connecting the two points (and fromthat identify parallel and

perpendicular lines);3. coordinates of the mid-point of the

line segment joining the two points;and.

4. the equation of a straight line passingthrough two points.

cmx y

Example :Consider the points P(2 ; 5) and Q(-3 ; 1) in the Cartesian plane.

1.1. Calculate the distance PQ.1.2 Find the gradient of PQ.1.3 Find the mid-point of PQ.1.4 Determine the equation of PQ.

3Functions

and Graphs

1. Functional notation2. Generate graphs by means of point- by

point plotting supported by availabletechnology.

3. Drawing of the following functions: Linear function - cmx y

(revise) Quadratic: qax y 2

Investigate the way(unique) output values depend on how the input valuesvary. The terms independent (input) and dependent ( output) variablesmight be useful.

qax y 2 - Draw the parabola for 1a only. Use the table method only.

- Identify the following characteristics:* y-intercept* x-intercepts* the turning points* axes of symmetry* the roots of the parabola (x-intercepts)* the domain( input values) and range( output values)* Very important to be able to read off values from the

graphs.* Investigate the effect of a and q on the graph.


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Hyperbola: xa y

Exponential xba y . and 1b

* asymptotes where applicable

NOTE: Use this approach to engage with all the other graphs.

4EuclideanGeometry

1. Revise basic geometry done in grade 8and grade9.Lines and parallel lines, angles, trianglescongruency and similarity.

2. Apply the properties of line segments joining the mid-points of two sides of atriangle. Do practical problems

3. Know the features of the following specialquadrilaterals: the kite, parallelogram,rectangle, rhombus, square and trapezium(apply to practical problems).

4. Pythagoras theorem Calculate the unknown side of a

right- angled triangle.

No formal proves are required, only calculation of unknowns.

Know the properties of the following kinds of triangles scalene isosceles triangle equiangular triangle

Congruency:Investigate the conditions for two triangles to be congruent:

3 sides side, angle side angle, angle, corresponding side right angle, hypotenuse and side

Similarity:Investigate the condition for two triangles to be similar:

Give drawings of similar triangles, write down the correspondingangles and calculate the ratio of the corresponding sides.

Investigate: line segment joining the mid-points of two sides of a triangle, isparallel to the third side and half of the length of the third side.

Investigate and make conjectures about the properties of the sides, anglesand diagonals of these quadrilaterals.

Calculation of one unknown side of a right-angled triangle, using Pythagorastheorem.

Assessment Term 3 : Two (2) Tests (at least 50 marks and 1 hour) covering all topics in approximately the ratio of the allocated teaching time.


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GRADE 10: TERM 4


1 AnalyticalGeometryContinuation from term 3.

1

Circles,

angles and

angular

movement

1. Define a radian

2. Indicate the relationship betweendegrees and radians, convert radiansto degrees or degrees to radians,convert degrees and minutes toradians and radians to degrees andminutes.

Example: Revise circle terminology: chord, secant, segment, sector,

diameter, radius, arc, etc. Rediscover unique ratio between circumference and diameter

represented by . Show that any circle 2360 radians and consequently that

283,6360 radians and that 1 radian 3,57 . Revise degrees, minutes and seconds. Learners should know to convert radians to degrees and to convert

degrees to radians Examples should include the following:

convert degrees to radians: "6'24,110;'12,300;3,133;213 etc.convert radians to degrees: 1,5 rad; 65,98 rad; 16,25 rad;etc.(answers in degrees, minutes and seconds).

Practical application in the technical field:

-

Calculate the angle in radians through which a pulley witha diameter of 0,6m will rotate if a length of belt of 120m passes over the pulley.

- A road wheel with a diameter of 560mm turns through anangle of .150 Calculate the distance moved by a point onthe tyre tread of the wheel.

Also ensure the following are covered :- add

32

4radians and covert to degrees.

- simplify: sin2

cos0cos90sin

- calculate4

cos2

sin


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- determine the value of the following:577,0tan866,0cos5,0sin 111 (answer in radians).

2 Finance andgrowth

Use the simple and compoundgrowth formulae )1( in P A and

ni P A )1( to solve problems,

including interest, hire purchase,inflation, population growth and otherreal-life problems. Understand theimplication of fluctuating foreignexchange rates (e.g. on the petrol price,imports, exports, overseas travel).

2 RevisionAssessment Term 4 :

1. Test (at least 50 marks)2. Examination

Paper 1: 2 hours (100 marks made up as follows: 355 on algebraic expressions, equations and inequalities , 330 on functions and graphs,315 on finance and 315 on exponents.

Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry andand 315 on circles, angles and angular movement.


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GRADE 11: TERM 1


2 Exponents andsurds

1. Simplify expressions and solveequations using the laws ofexponents for rational exponentswhere

;q pq p

x x 0;0 q x .

2. Add, subtract, multiply and dividesimple surds.

3. solve exponential equations

.Example :

1. Determine the value of 23

9 .2. Simplify: )23)(23( .3 Revise grade 10 exponential equations.4. Revise all exponential laws from grade 10.

Examples should include but not be limited to

5.22

3 6

9168

x x x

6. 3 233

851255

6255

Solve:

7. 1284 25

x 8 . 32 525 x x 9. Practical examples from the technical field must be included e.g.

The formulaV QC is given with 4106 Q C and

volt V 200 .Determine C without using a calculator.

3 Logarithms

Laws of logarithms y x xy aaa logloglog y x xy aaa logloglog

1. Explain relationship between logs and exponents and showconversion from log-form to exponential form and vice versa.

2. Application of laws of logarithms:


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y x y x

aaa logloglog

xn x an

a loglog Simplify applying the law:

ab

bc

ca log

loglog

Solving of logarithmic equations

- Simplify: 2.149log

343log

2.201,0log

25log4log

2.3 2log33log20log26log

- Prove: 2.421

1024log8log4log64log

x

x x x

- Simplify : 2.5 64log 32

3. Show application of logs to solve 35 x 4. Solving of log equations limited to the following:

4.1 x81log 2

4.2 900log2log2 x 4.3 3)4(log)3(log 22 x x

4 Equations

Solve quadratic equations by means of:

1. Factorization2. Quadratic formula

3. Determine the nature of roots andthe conditions for which the rootsare real, non-real, equal, unequal,rational and irrational.

4. Equations in two unknowns, one ofwhich is linear and the otherquadratic

5. Word problems

6. Manipulating formulae

- When explaining the quadratic formula it is important to showwhere it comes from although learners do not need to know the

proof.- Use the formula to introduce the discriminant and how it determines

the nature of the roots.- Show by means of sketches how the discriminant affects the

parabola.- Learners should only be able to apply the quadratic formula and to

derive from acb 42 the nature of the roots.- Word problems should focus and be related to the technical field and

should include examples similar to the following:- The length of a rectangle is 4m longer than the breadth. Determine

the measurements of the rectangle if the area equals 621m 2.- With manipulation of formulae the fundamentals of changing of the

subject should be emphasized and consolidated..- All related formulae from the technical fields should be covered.

2 AnalyticalGeometry

Determine1. the equation of a line through two

given points;2. the equation of a line through one

point and parallel or perpendicular to

1. Revision of linear equation from grade 10.2. Showing the influence of the gradient and consequently the

relationshipwhen lines are parallel and perpendicular.


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a given line; and3. the inclination )( of a line, where

t anm is the gradient of the lineand 1800 .

3. Example: Given )4;2( A , )6;2( B and )2;3( C .

4. Determine : 4.1 Equation of line passing through point C and parallel to line AC.

4.2 Equation of line passing through point B and perpendicular to line AC.

5. Introducing practical examples e.g.:A ladder leans against the wall on a straight line defined by

.42y x Determine the length of a ladder; how far it reaches up the wall

and the ladders inclination with the floor.

Assessment Term 1 :

1. An investigation or a project (a maximum of one project in a year) (at least 50 marks)2. Test (at least 50 marks) Make sure all topics are tested.

Care needs to be taken to ask questions on all four cognitive levels: approximately 20% knowledge, approximately 35% routine procedures,30% complex procedures and 15% problem-solving.


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GRADE 11: TERM 2


4Functions

andGraphs

1. Revise the effect of the parameters a and q and p on the graphs of thefunctions defined by:1.1. q p xa x f y 2)()(

1.2. cbxax x f y 2)(

1.3 xa y

1.4 xba x f a y ..)(. , 0b and1b

2. 222 r y x 22 xr y 22 xr y 22 xr y

Comment:

Learners should gain confidence and get a grasp of graphsfrom tabling and dotting and joining points to draw the graphs.

They should understand the influence of variables on the formand calculate critical points to draw graphs.

The concept of asymptotes should be clear. They should be able to deduce the equations when critical

points are given. Analyze the information from graphs (Graphs given)

4 EuclideanGeometry

Accept results established in earliergrades as axioms and also that atangent to a circle is perpendicular tothe radius, drawn to the point ofcontact.Then investigate and apply thetheorems of the geometry of circles:

The line drawn from the centre of acircle perpendicular to a chord

bisects the chord; The perpendicular bisector of a chord

passes through the centre of the

Comments:Proofs of theorems and their converses will not be examined.But proofs of theorems and their converses will applied in solvingriders.

The focus of all questions will be on applications and calculations.


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circle; The angle subtended by an arc at the

centre of a circle is double the size ofthe angle subtended by the same arcat the circle (on the same side of thechord as the centre);

Angles subtended by a chord of thecircle, on the same side of the chord,are equal;

The opposite angles of a cyclic

quadrilateral are supplementary; Exterior angle of cyclic quad. Isequal to opposite interior angle;

Two tangents drawn to a circle fromthe same point outside the circle areequal in length;

Radius is perpendicular to thetangent.

The angle between the tangent to acircle and the chord drawn from the

point of contact is equal to the anglein the alternate segment.

Make use of problems taken from the technical fields.

3 Mid-yearexaminationsAssessment Term 2 :

1. Assignment / test (at least 50 marks) 2. Mid-year examination (at least 200 marks)Paper 1: 2 hours (100 marks made up as follows: General algebra ( 335 ) , equations, inequalities and nature of roots ( 340 ), function ( 325 )Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry ( 340 ) and Euclidean Geometry ( 360 )


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GRADE 11: TERM 3


4

Circles, anglesand

Angularmovement

3.Circle

3.1 222 r y x , centre (0,0 ) only3.2 Angles and arcs3.3 degrees and radians

3.4 sectors and segments

Pay attention to the following : diameter, radius, chord, segment, arc,secant, tangent.

3.1 If point on circumference is given how to determine r and theequation of the circle

3.2 What is an arc, central angle, inscribed angle3.3 A radian is the angle at the centre of a circle subtended by an arc

of the same length as the radiusComplete revolution : 360 0 = 2 radians ; 180 0 = radians

- If an arc of length s is intercepted by a central angle ( inradians)the angle subtended will be equal to the arc divided bythe radius.

- If s = r then = 1 radian- Convert degrees to radians and visa versa

Convert 2 radians to degrees 180rad

1 rad = 180

2 rad = 1802

=114,59 0

Convert 52,6 0 to radians

52,6 0 =180

6,52

= 0, 918 rad


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4. Angular andcircumferential/peripheral velocity

3.4 A sector of a circle is a plane figure bounded by an arc and tworadii.

area of a sector = r r rs22

2

=radius; s = arc length and

= central angle in radians

- A segment of a circle is the area bounded by an arc and a chord.

The relation between the diameter of a circle, a chord and theheight of the segment formed by the chord, can be set out in the

formula04 22 xdhh ; h = height of segment; d = diameter of

circle; h = length of chord3 Angular velocity ( omega) can be determined by multiplying

the radians in one revolution ( 2 with the revolutions persecond (n)

2 n = 360 0 nCircumferential velocity is the linear velocity of a point on thecircumference

n Dv with D the diameter , n is the rotation frequency

4 Trigonometry

1. Revise the trig ratios in the solving ofright-angle triangle in all 4 quadrants (Grade 10)

2. Apply the sine, cosine and area rules.

3. Solve problems in two dimensions usingthe sine, cosine and area rules.4. Draw the graphs of the functions defined

by xk y sin , xk y cos ,

)(sin kx y , )(cos kx y and x y tan 5. Draw the graphs of the functions defined

by)sin( p x y and)cos( p x y

Comment:

No proofs of the sine, cosine and area rules are required. Apply only with actual numbers, no variables.

(Acute and obtuse angled triangles)

- Learners must be able to draw all mentioned graphs and also be able todeduct important information from given sketches.

- One parameter should be tested at a given time when examining horizontalshifts

- Rotating vectors should be done on graph paper.

- Determine the solutions of equations for ]360;0[ Limited to routine procedures


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6. Rotating vectorsDeveloping the sine and cosine curve

7. Trigonometric equations

8.Introduce identities and apply

cossintan ,

1cossin 22 , 22 sectan1 and

22 cos1cot ec

- Reduction formulae, )180( and )360( .

- Examples related to the use of identities limited to routine procedures

2Finance, growth

and decay

1. Use simple and compound decayformulae:

)1( in P A and ni P A )1( to solve problems (including straightline depreciation and depreciation ona reducing balance).

2. The effect of different periods ofcompound growth and decay,including nominal and effectiveinterest rates.

Examples :1. The value of a piece of equipment depreciates from R10 000 to

R5 000 in four years. What is the rate of depreciation if calculatedon the:1.1 straight line method; and (R)1.2 reducing balance? (C)

2. Which is the better investment over a year or longer: 10,5% p.a.compounded daily or 10,55% p.a. compounded monthly? (R)

Comments :The use of a timeline to solve problems is a useful technique.

3. R50 000 is invested in an account which offers 8% p.a. interestcompounded quarterly for the first 18 months. The interest thenchanges to 6% p.a. compounded monthly. Two years after themoney is invested, R10 000 is withdrawn. How much will be inthe account after 4 years? (C)

Comment :Stress the importance of not working with rounded answers, but ofusing the maximum accuracy afforded by the calculator right to thefinal answer when rounding might be appropriate.

3 Mid-yearexaminations

Assessment Term 3 :

Two (2) tests (at least 50 marks per testand 1 hour) covering all topics in approximately the ratio of the allocated teaching time.


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GRADE 11: TERM 4


2 Mensuration

1. Surface area and volume of right prisms,cylinders, pyramids, cones and spheres,and combinations of these geometricobjects.

2. The effect on volume and surface areawhen multiplying any dimension by factork.

3. Determine the area of an irregular figureusing mid-ordinate rule.

1. Surface Area = 2 area of base + circumference of base X height( for closed right angled prism )Surface Area = area of base + circumference of base X height ( for open right angled prism ) Volume = area of base height

2.What is the effect if some of the measurements are multiplied by a factor k

3.Using the mid- ordinate rule:

AT = a( m1 + m2 + m3 + .mn) where 221

1oo

m

e tc. and n = number of ordinates - 1

3 Revision3 Examinations

Assessment Term 4 :

1. Test (at least 50 marks)2. Examination (300 marks)Paper 1: 3 hours (150 marks made up as follows: )390( on algebraic expressions, equations, inequalities and nature of roots,

)345( on functions and graphs (excluding trigonometric functions) and , )315( on finance growth and decay.Paper 2: 3 hours (150 marks made up as follows: )350( on trigonometry (including trigonometric functions), )325( on Analytical Geometry,

)340( on Euclidean Geometry, )335( on Mensuration and Circles, angles and angular movement.


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GRADE 12: TERM 1


3 Complex numbers

Define a complex number, ,bia z .

Learners should know- the conjugate of bia z .- imaginary numbers, 12i .- how to add, subtract, divide and

multiply complex numbers.- Represent complex numbers in

the Argand diagram.- argument of . z - trigonometric (polar) form of a

complex numbers. Solve equations with complex

numbers with two variables

Simplify :

1. 1416

2. 516

3.6

12.4

4. iii 5132 5. )1)(23( ii

Solve for x and : y 6. yii x 53152

2 Polynomials

Factorise third-degree polynomials.Apply the Remainder and FactorTheorems to polynomials of the thirddegree(no proofs are required).

Long division method can also be used.

Revise functional notationAny method may be used to factorise third degree polynomials but it

shouldinclude examples which require the Factor Theorem.Examples :

Solve for x : 010178 23 x x x 1. An intuitive understanding of the limit

concept, in the context ofapproximating the rate of change orgradient of a function at a point.

2. Determine the average gradient of acurve between two points.

h x f h x f m )()(

3. Determine the gradient of a tangent toa graph, which is also the gradient of

Comment:Differentiation from first principles will be examined on any of the

typesdescribed in 3.1Understand that the following notations mean the same

)(',, x f dxd D x

Examples:


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6 DifferentialCalculus

the graph at that pointIntroduce the limit-principle byshifting the secant until it becomes atangent

4. By using first principals

h x f h x f x f

h

)()(lim)('0

for

k x f )( , ax x f )( andbax x f )(

5. Use the rule 1)( nn anxaxdxd for

n

6. Find equations of tangents to graphsof functions

7. Sketch graphs of cubic polynomialfunctions using differentiation todetermine the co-ordinate ofstationary points. Also, determine the x - intercepts of the graph using thefactor theorem and other techniques.

8. Solve practical problems concerningoptimisation and rates of change,including calculus of motion.

9. Introducing Integration- Determineareas under a curve by splitting two ormore intervals when the graphintercept the x-axis.

1. In each of the following cases, find the derivative of )( x f at the point

where 1 x , using the definition of the derivative:

1.1 2)( x x f

1.2 2)( 2 x x f

3. Sketch the graph defined by x x x y 23 4 by:3.1 finding the intercepts with the axes;3.2 finding maxima, minima.

On word problems the diagram with all the measurements must begiven. Guide the learners through the question with subsections.

Refer to displacement formulae like 221 at u s .

Very simple problems.Refer to practical application in the technical field.

Assessment Term 2 :

1. Test ( at least 50 marks )2. Investigation or project3. Test ( at least 50 marks ) or Assignment (at least 50 marks)


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GRADE 12: TERM 2


3 AnalyticalGeometry

1. The equation 222 r y x defines acircle with radius r and centre (0 ; 0).

2. Find the equation of the circle when

the radius is given or a point on thecircle is given. Only circles with theorigin as centre.

3. Determination of the equation of atangent to a given circle. ( gradient or

point of contact is given)4. Find the points of intersection of the

circle and a given straight line.

Examples:

1.1 Determine the equation of the circle passing through the point(2;4)with centre at the origin.

1.2 Hence determine the equation of the tangent to the circle at the point (2;4).

1.3 Hence determine the points of intersection of the circle and theline with the equation 2 x y .

4EuclideanGeometry

1. Revise earlier work on the necessaryand sufficient conditions for

polygons to be similar.2. Introduce and apply the following

theorems: that a line drawn parallel to one

side of a triangle divides the othertwo sides proportionally that equiangular triangles are

similar; that triangles with sides in

proportion are similar

Example:The examples must be very easy. Only with one unknown used

once.

Start with problems like2010

3 x

Example:

In ABC , 8 AB , 5 AC and 6 BC . D is a point AB

>>

>>


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so that 4 AD . E is a point on AC so that DE is parallel to BC .Find the lengths of DE and AE .

1 Revision

3Mid-year

examinationsAssessment Term 2 :

1. Assignment / test (at least 50 marks)2. Mid-year examination (300 marks)

One paper of 3 hours (150 marks)

Two papers - one, 3 hour (150 marks)


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GRADE 12: TERM 3


2EuclideanGeometry

Continuation from term 2.

2

Revision

( Finance, growthand decay)

Revision

2Revision(Trigonometry) Revision

2 Revision Revision

3 ExaminationsAssessment Term 3 :

1. Assignment / test (at least 50 marks)2. Mid-year examination (at least 200 marks of sum of paper 1 and paper 2)

One paper of 2 hours (at least 100 marks) and Two papers - 2hour (at least 100 marks) marks)


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GRADE 12: TERM 4


3 Revision Revision

5Mid-year

examinationsAssessment Term 4 :

Final examination:Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hoursAlgebraic expressions, equations 350 Analytical Geometry 325 and inequalities (nature of roots, logs and Trigonometry 350 complex numbers) Euclidean Geometry 340 Functions and graphs 335 Mensuration and circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350


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4.1 INTRODUCTION

Assessment is a continuous planned process of identifying, gathering and interpreting information about

The performance of learners, using various forms of assessment. It involves four steps: generating and

collecting evidence of achievement; evaluating this evidence; recording the findings and using this

inform ation to understand and assist in the learners development to improve the process of learning and

teaching.

Assessment should be both informal (Assessment for Learning) and formal (Assessment of Learning). In

both cases regular feedback should be provided to learners to enhance the learning experience.

Although assessment guidelines are included in the Annual Teaching Plan at the end of each term, the

following general principles apply:

1. Tests and examinations are assessed using a marking memorandum.

2. Assignments are generally extended pieces of work completed at home. They can be collections of

past examination questions, but should focus on the more demanding aspects as any resource

material can be used, which is not the case when a task is done in class under strict supervision.

3. At most one project or assignment should be set in a year. The assessment criteria need to be

clearly indicated on the project specification. The focus should be on the mathematics involved

and not on duplicated pictures and regurgitation of facts from reference material. The collection

and display of real data, followed by deductions that can be substantiated from the data, constitute

good projects.

4. Investigations are set to develop the skills of systematic investigation into special cases with aview to observing general trends, making conjecures and proving them. To avoid having to assess

work which is copied without understanding, it is recommended that while the initial investigation

can be done at home, the final write up should be done in class, under supervision, without access

to any notes. Investigations are marked using rubrics which can be specific to the task, or generic,

listing the number of marks awarded for each skill:


FET MATHEMATICS ASSESSMENT GUIDELINES


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40% for communicating individual ideas and discoveries, assuming the reader has not come

across the text before. The appropriate use of diagrams and tables will enhance the

investigation.

35% for the effective consideration of special cases;

20% for generalising, making conjectures and proving or disproving these conjectures; and 5% for presentation: neatness and visual impact.

4.2 INFORMAL OR DAILY ASSESSMENT

The aim of assessment for learning is to colle ct continually information on a learners achievement that

can be used to improve individual learning.

Informal assessment involves daily monitoring of a learners progress. This can be done through

observations, discussions, practical demonstrations , learner-teacher conferences, informal classroom

interactions, etc.

Although informal assessment may be as simple as stopping during the lesson to observe learners or to

discuss with learners how learning is progressing. Informal assessment should be used to provide

feedback to the learners and to inform planning for teaching, it need not be recorded. This should not be

seen as separate from learning activities taking place in the classroom. Learners or teachers can evaluate

these tasks.

Self assessment and peer assessment actively involve learners in assessment. Both are important as these

allow learners to learn from and reflect on their own performance. Results of the informal daily

assessment activities are not formally recorded, unless the teacher wishes to do so. The results of daily

assessment tasks are not taken into account for promotion and/or certification purposes.

4.3 FORMAL ASSESSMENT

All assessment tasks that make up a formal programme of assessment for the year are regarded as Formal

Assessment. Formal assessment tasks are marked and formally recorded by the teacher for progress and

certification purposes. All Formal Assessment tasks are subject to moderation for the purpose of quality

assurance.


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Formal assessments provide teachers with a systematic way of evaluating how well learners are

progressing in a grade and/or in a particular subject. Examples of formal assessments include tests,

examinations, practical tasks, projects, oral presentations, demonstrations, performances, etc. Formal

assessment tasks form part of a year-long formal Programme of Assessment in each grade and subject.Formal assessments in Mathematics include tests, a June examination, a trial examination(for Grade 12),

a project or an investigation.

The forms of assessment used should be age- and developmental- level appropriate. The design of these

tasks should cover the content of the subject and include a variety of activities designed to achieve the

objectives of the subject.

Formal assessments need to accommodate a range of cognitive levels and abilities of learners as shown

below:

4.4 PROGRAMME OF ASSESSMENT

The four cognitive levels used to guide all assessment tasks are based on those suggested in the TIMSSstudy of 1999. Descriptors for each level and the approximate percentages of tasks, tests andexaminations which should be at each level are given below:

Cognitive levels Description of skills to be demonstrated ExamplesKnowledge

20%

Straight recall Identification of correct formula on the

information sheet (no changing of thesubject)Use of mathematical facts

Appropriate use of mathematicalvocabulary

1. Write down the domain of thefunction

3 2 y f x x

(Grade 10)2. The angle

AOB subtended by arc AB at the centre O of a circle .......

Routine

Procedures

35%

Estimation and appropriate roundingof numbers Proofs of prescribed theorems andderivation of formulae

Identification and direct use of correctformula on the information sheet (nochanging of the subject)

Perform well known procedures Simple applications and calculations

1. Solve for 2: 5 14 x x x (Grade 10)

2. Determine the general solution ofthe equation

01)30sin(2 x (Grade 11)

3. Prove that the angle

AOBsubtended by arc AB at the


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which might involve few steps Derivation from given information

may be involved Identification and use (after changing

the subject) of correct formula Generally similar to those encountered

in class

centre O of a circle is double thesize of the angle

ACB which thesame arc subtends at the circle.(Grade 11)

ComplexProcedures

30%

Problems involve complexcalculations and/or higher orderreasoning

There is often not an obvious route tothe solution

Problems need not be based on a realworld context

Could involve making significantconnections between differentrepresentations

Require conceptual understanding

1. What is the average speed coveredon a round trip to andfrom a destination if the averagespeed going to thedestination is hkm /100 and theaverage speed for thereturn journey is ?/80 hkm (Grade11)

2. Differentiate x

x 2)2( with respect

to x. (Grade 12)

Problem Solving

15%

Non-routine problems (which are notnecessarily difficult)

Higher order reasoning and processesare involved

Might require the ability to break the problem down into its constituent parts

Suppose a piece of wire could be tiedtightly around the earth at the equator.Imagine that this wire is thenlengthened by exactly one metre andheld so that it is still around the earth atthe equator.Would a mouse be able to crawl

between the wire and the earth? Whyor why not?

(Any grade)

The Programme of Assessment is designed to set formal assessment tasks in all subjects in a school

throughout

the year.


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a) Number of Assessment Tasks and Weighting:

Learners are expected to have seven (7) formal assessment tasks for their school-based assessment. Thenumber of tasks and their weighting

are listed below:

GRADE 10 GRADE 11 GRADE 12TASKS WEIGHT

(%)TASKS WEIGHT

(%)TASKS WEIGH

T (%)

S c h o o

l - b a s e d

A s s e s s m e n

t

Term 1

ProjectorInvestigationTest

20

10

ProjectorInvestigationTest

20

10

TestProject orInvestigationAssignmentor Test

1020

10

Term 2

Assignmentor Test

Examination

10

30

Assignment orTestExamination

10

30

TestExamination

1015

Term 3TestTest

1010

TestTest

1010

TestTrialExamination

1025

Term 4 Test 10 Test 10

School-basedAssessment mark 100 100 100

School-basedAssessmentmark (as % ofpromotion mark)

25% 25% 25%

End-of-yearexaminations

75% 75%

Promotion mark 100% 100%Note: Although the project/investigation is indicated in the first term, it could be scheduled in term 2. Only ONE

project/investigation should be set per year. Tests should be at least ONE hour long and count at least 50 marks. Project or investigation must contribute 25% of term 1 marks while the test marks contribute 75% of the

term 1 marks.

The combination (25% and 75%) of the marks must appear in the learne rs report. Graphic and none programmable calculators are not allowed (for example, calculators which factorise))((22 bababa , or to find roots of equations are not allowed). Calculators should only be used to

perform standard numerical computations and to verify calculations by hand. Formula sheet must not be provided for tests and for final examinations in Grades 10 and 11. Trigonometric functions and graphs will be examined in paper 2.


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b) Examinations:

In Grades 10, 11 and 12, 25% of the final promotion mark is a year mark and 75% is an examinationmark.All assessments in Grade 10 and 11 are internal while in Grade 12 the 25% year mark assessment isinternally set and marked but externally moderatedand the 75% examination is externally set, marked andmoderated.

Mark distribution for Technical Mathematics NCS end-of-year papers: Grades 10 - 12Description Grade 10 Grade 11 Grade. 12

PAPER 1 :

Algebra ( Expressions, equations and inequalitiesincluding nature of roots in grade 11 &12)

65 3 90 3 50 3

Functions & Graphs 20 3 45 3 35 3Finance, growth and decay 15 3 15 3 15 3Differential Calculus and Integration 50 3TOTAL 100 150 150PAPER 2 : Grades 11 and 12: theorems and/or trigonometric proofs: maximum 12 marks

Description Grade 10 Grade 11 Grade 12Analytical Geometry 15 3 25 3 25 3Trigonometry 40 3 50 3 50 3Euclidean Geometry 30 3 40 3 40 3Mensuration and circles, angles and angularmovement

15 3 35 3 35 3

TOTAL 100 150 150Note: Modelling as a process should be included in all papers, thus contextual questions can be set on any

topic. Questions will not necessarily be compartmentalised in sections, as this table indicates. Various

topics can be integrated in the same question. Formula sheet must not be provided for tests and for final examinations in Grades 10 and 11. Trigonometric functions and graphs will be examined in paper 2.

4.5 RECORDING AND REPORTING

Recording is a process in which the teacher is able to document the level of a learners performance in a specific assessment task.

It indicates learner progress towards the achievement of the knowledge as prescribed in theCurriculum and Assessment Policy Statements. Records of learner performance should provide evidence of the learners conceptual

progression within a grade and her / his readiness to progress or to be promoted to the nextgrade.

Records of learner performance should also be used to monitor the progress made by teachersand learners in the teaching and learning process.

`Reporting is a process of communicating learner performance to learners, parents, schools andother stakeholders. Learner performance can be reported in a number of ways.


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These include report cards, parents meetings, school visitation days, parent -teacherconferences, phone calls, letters, class or school newsletters, etc.

Teachers in all grades report percentages for the subject. Seven levels of competence have been described for each subject listed for Grades R - 12. The individual achievement levels andtheir corresponding percentage bands are shown in the Table below.

CODES AND PERCENTAGES FOR RECORDING AND REPORTING

RATINGCODE

DESCRIPTION OF COMPETENCE RCENTAGE

7 Outstanding achievement 1006 Meritorious achievement 795 Substantial achievement 694 Adequate achievement 593 Moderate achievement 492 Elementary achievement 39

1 Not achieved - 29

Note: The seven-point scale should have clear descriptors that give detailed information for eachlevel.Teachers will record actual marks for the task on a record sheet; and indicate percentagesfor each subject on the learners report cards.

4.6 MODERATION OF ASSESSMENT

Moderation refers to the process that ensures that the assessment tasks are fair, valid and reliable.

Moderation should be implemented at school, district, provincial and national levels. Comprehensive

and

appropriate moderation practices must be in place to ensure quality

assurance for all subject assessments.

4.7 GENERAL

This document should be read in conjunction with:

4.7.1 National policy pertaining to the programme and promotion requirements of the NationalCurriculum Statement Grades R 12 ; and

4.7.2 The policy document, National Protocol for Assessment Grades R 12.


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APPENDIX A

Strategic Intention documentBackgroundFor a very long time the majority of learners in Most schools did not cope with the current difficult level of the mathematics.Because of this they either took Maths Lit or they achieved such a bad mark that they could do nothing with it.

Our aim was to put on paper a syllabi that is learner friendly for the technical learners and which will be recognised by HEInstitutions.

What is of the utmost importance is:1. Must be recognised and approved and recognised on AT LEAST the same level as N3.2. Must surely add value to the technical field of study at school level and for further study at a FET College, HEIs or

apprentiship in the workplace.

3. The learner must be able to go onto N4 WITHOUT any bridging course.

Process followedA.The following documents were used in the compilation of these syllabi:

1. Caps2. N1 to N3 curriculum document3. FET Colleges Curriculum 4. Mathematical concepts used in all the technical subjects ( Where a topic only needed in ONE subject only we did not

include it )

After we have put in all the topics we also compare it with the Cambridge Syllabi for Mathematics.

B.How did the discussion go in determining which topics must be in which grade:

1. Grade 10 must form the base for all the mathematics that follows.2. Grade 8 and 9 Mathematics, and even primary school Mathematics, must be revised and explained in detail before

we can start on the actual grade 10 works. Ample time was allocated to that.

3. No PROOFS were considered in these syllabi. We tried to move away from the ABSTRACT Mathematics to APPLIEDTECHNICAL MATHEMATICS.

4. Applications in all topics were the main focus5. Learners must be equipped with Mathematics that will take them into further study in the Technical field of their

choice.

6. Nine topics in Grade 10 and 11 each. In Grade 12 we have only put in 4 topics so to have enough time to do a lot ofrevision and practice.

7. A formula sheet must be compulsory during tests and examinations8. Examinations must consist of two question papers.

C.

We only focused on the CONTENT and not on the comments, examples etc. at this stage. This will be done at a later stage.

D

Documents

Technical CAPS Technical Mathematics Grades 10-12