MATHEMATICS CURRICULUM

ASSESSMENT POLICY STATEMENTORIENTATION WORKSHOP FOR TEACHERS

GR EDWARDS

22 OCTOBER 2011

ADMINISTRATION Attendance register

Claim forms S & T Claim Form Z43

GENERIC TRAINING

INTRODUCTION The Curriculum and Assessment Policy Statement (CAPS) is a revision of the National

Curriculum Statement (NCS).

In developing the CAPS, a key aim has been to have just one document providing guidelines

for planning, content and assessment for each subject.

The CAPS also continue to support the key principles that underline the NCS, including: • social transformation; • high knowledge and high skills; • integration and applied competence; • progression; • articulation and portability; • human rights, inclusivity, environmental and social justice; • valuing of indigenous knowledge systems (IKS) and credibility, • quality and efficiency.

WHAT IS MATHEMATICS? Mathematics is the study of quantity, structure, space and change. Mathematicians seek out patterns, formulate new conjectures , and establish axiomatic systems by rigorous deduction from appropriately chosen axioms and definitions. Axiomatic is one approach to establishing mathematical truth. Mathematics is distinctly human activity practiced by all cultures. Mathematical problem solving enables us to understand the world around us, and most of all, to teach us to think creatively.

PRINCIPLES APPLY ACROSS ALL GRADES:

No calculators with programmable functions, graphical or symbolic facilities should be allowed. Calculators should only be used to perform standard numeric computations and to verify calculations by hand. Mathematical modelling is an important focal point of the curriculum. Real life problems should be incorporated into all sections whenever possible. Examples used should be realistic and not contrived. Investigations provide the opportunity to develop in learners the ability to be methodical, to generalize, make conjectures and try to justify or prove them. Learners need to reflect on the process and not be concerned only with getting the answer(s). Appropriate approximation and rounding skills should be taught the impression is not gained that all answers which are either irrational numbers or recurring decimals should routinely be given correct to two decimal places. Teaching should not be limited to “how” but should also feature the “when” and “why” of problem types: Finding the mean and standard deviation of a set of data has little relevance unless learners have a good grasp of why and when such calculations might be useful.

PRINCIPLES APPLY ACROSS ALL GRADES:

Mixed ability teaching requires teachers to challenge the most able learners and at the same time provide remedial support for those who are not competent yet. Teachers need to design questions to rectify misconceptions that are exposed by tests and examinations.

Problem solving and cognitive development should be central to all mathematics teaching. Learning procedures and proofs without a good understanding of why they are important will leave learners ill equipped to use their knowledge in later life.

GENERIC TRAINING Policy documents Critical outcomes Time allocation Focus of Content Areas Weighting of Content Areas Allocation of Teaching Time Programme of Assessment

Policy Documents National Curriculum and Assessment Policy Statement (CAPS) for Mathematics.

National Policy pertaining to the Programme and Promotion requirements of the National Curriculum Statements Grades R – 12(NPR).

National Protocol for Assessment Grades R – 12(NPA).

Critical Outcomes

The National Curriculum Statement Grades R - 12 aims to produce learners that are able to:

identify and solve problems and make decisions using critical and creative thinking;

work effectively as individuals and with others as members of a team;

organise and manage themselves and their activities responsibly and effectively;

collect, analyse, organise and critically evaluate information;

communicate effectively using visual, symbolic and/or language skills in various modes;

use science and technology effectively and critically showing responsibility towards the environment and the health of others; and

demonstrate an understanding of the world as a set of related systems by recognising that problem solving contexts do not exist in isolation.

Time Allocation

4½ hours per week.

Six(6) periods of 45 minutes each.

Make sure you get the time on your roster.

Focus of Content Areas

The Main Topics in the FET Mathematics Curriculum

1. Functions 2. Number Patterns, Sequences, Series 3. Finance, growth and decay 4. Algebra 5. Differential Calculus

The Main Topics in the FET Mathematics Curriculum

6. Probability 7. Euclidean Geometry and Measurement 8. Analytical Geometry 9. Trigonometry 10. Statistics

The Main Topics in the FET Mathematics Curriculum

OUT : Linear Programming Transformations

IN : Probability Euclidean Geometry

Weighting of Content Areas

CAPS p. 12

Content clarification with teaching guidelines

Allocation of Teaching Time Sequencing and Pacing of Topics• Grade 10• Grade 11• Grade 12

Daily Lesson Plan

Example

Voorbeeld

Programme of Assessment

WHAT IS ASSESSMENTAssessment is a planned process of• identifying (selecting learner response items)• gathering (learner responses)• interpreting (marking learner responses)information about the knowledge and skills demonstrated by learners.

ASSESSMENTINFORMAL no need to be recorded by teacher, could be marked by learner or peer, usually used to develop skills, demonstrate knowledge and skills and for learners to practice, not used for promotion purposes(solely developmental purpose)

FORMAL marked and recorded by teacher, 7 tasks in Programme of Assessment, used for promotion purposes (mainly promotion purpose)

ASSESSMENT

• Why?• What?• Who?• How?• When?• Where?

WHY?PURPOSE

Developmental: to assist learner to learn (i.e. apply knowledge, etc)

Promotion: to make a summative judgement of the learner’s ability (e.g. use the marks for any of the tasks in the programme of assessment)

WHO• Teacher (formal assessment task e.g. project, controlled test)

• Learner (informal assessment task e.g. homework, classwork)

• Peer (informal assessment task e.g. homework)

• SMT (e.g. controlled/standardised test)

• External (e.g. controlled test set by district, etc)

WHAT•What content, skills, values

•E.g. ability to collect data, make observations, one-step problem-solving, multi-step problem solving, interpreting and drawing graphs, recalling laws and definitions, converting units, etc

•Use taxonomy

HOW• FORMAL TASK:

project, investigation, assignment, controlled test, June examination, Final examination

• INFORMAL TASK:

homework, classwork, class test

When• Last 15 minutes of a period (e.g. classwork)

• First 5 minutes of a lesson (e.g. oral Q & A)

• End of term, week, year (e.g. control test, june exam, class test, final exam, project, etc)

• End of unit of work, concept, section (e.g. class test, homework)

• Weekly (e.g. assignment)

• Daily (homework)

Where

• Classroom

• Home

GRADE 10: FORMAL ASSESSMENT

Grades Formal School-based assessments End-of –Year examinations

R-3 100% n/a

4-6 75% 25%

7-9 40% 60%

10 & 11 25% including a midyear examination

75%

12 25% including midyear and trial examinations

External examination: 75%

FORMAL ASSESSMENT: GRADE 10

PROGRAMME OF ASSESSMENT FOR GRADES 10

ASSESSMENT TASKS

(25%)

END-OF- YEAR ASSESSMENT

(75%)

TERM 1 TERM 2 TERM 3 TERM 4 Type % Type % Type %

Final Examination (2 x 100 marks giving a total of 200 marks for

papers 1 and 2)

Test 10 Assignment/Test 10

Test

10

Project / investigation

20 Mid- Year Examination 30 Test

10

Total: 30 marks Total: 40 marks Total: 20 marks Total: 200 + 10 marks Total = 300 marks

FINAL MARK = 25% (ASSESSMENT TASKS) +75% (FINAL EXAM)=100%

Table 3: Assessment plan and weighting of tasks in the programme of assessment for Grades 10

Test 10%

Cognitive levels(p.55)

Knowledge 20%

Routine Procedures 35%

Complex Procedures 30%

Problem Solving 15%

Knowledge (20%) Straight recall.

Identification of correct formula on the information sheet(no changing of the subject).

Use of mathematical facts.

Appropriate use of mathematical vocabulary.

Routine Procedures (35%) Estimation and appropriate rounding of numbers. Proofs of prescribed theorems and derivation of formulae. Identification and direct use of correct formula on the information sheet(no changing of the subjects). Perform well known procedures. Simple applications and calculations 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.

Complex Procedures (30%) Problems involve complex calculations and/or higher order reasoning. There is often not an obvious route to the solution. Problems need not be based on a real world context. Could involve making significant connections between different representations. Require conceptual understanding.

Problem Solving (15%)

Non-routine problems (which are not necessarily difficult). Higher order reasoning and processes are involved. Might require the ability to break the problem down into its constituent parts.

Number of Assessment Tasks and Weighting

Term 1 Test

Project/Assignment

10

20

Term 2 Assignment/Test

Examination

10

30

Term 3 Test

Test

10

10

Term 4 Test 10

In general 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 learner’s report. None graphic and none programmable calculators are allowed (for example, to factorise a² - b² = (a-b)(a+b), or to find roots of equations) will not be 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.

Mark distribution for Mathematics NCS end-of-year papers: Grades 10

38

PAPER 1 Paper 2

Description Grade 10 Statistics 15 ± 3

Algebra and equations (and inequalities)

30 ± 3 Analytical Geometry 15 ± 3

Patterns and sequences 15 ± 3 Trigonometry and measurement 50 ± 3

Finance, growth and decay 10 ± 3 Euclidean Geometry 20 ± 3

Functions and graphs 30 ± 3

Probability 15 ± 3

TOTAL 100 TOTAL 100

No calculators with programmable functions, facilities or symbolic facilities (for example, to factorise, or to find roots of equations) will be allowed. Calculators should only be used to perform standard numerical computations and to verify calculations by hand.

Bookwork Grade 12

o Paper 1 – maximum 6 markso Paper 2 – Theorems and/or trigonometric proofs – maximum 12 marks

Grade 11o Paper 2 - Theorems and/or trigonometric proofs – maximum 12 marks

Grade 10o No bookwork

Annual Assessment Plan

• Annual Assessment Plan should be given to learners at beginning of year.

• Include dates, task, maximum mark

Exam and Test templates

End of year exam

Tests

The 7 Formal Assessment Tasks

• 5 tests• 1 project / investigation• June examination• Final examinationOr • 4 tests• 1 assignment• 1 project / investigation• June examination• Final examination

ModerationInternal and External

Internal• HoD (or subject head) if HoD did not specialise in

subject• All Programme of Assessment (PoA) tasks: HoD

moderates before task is given to learners• HoD moderates 10% of learner responses to

Programme of Assessment tasksExternal

• Moderation by Subject advisors

Moderation: What to look for?

• Duration, length• Weighting of content• Weighting of cognitive levels• Language – mathematical vs layman• Use of pictures, diagrams, graphs to facilitate language• Level of difficulty

CONTENT TRAINING

IN GENERAL Each content area has been broken down into topics. The sequencing of topics within terms gives an 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. They only serve as an indication of some questions on the topic at different cognitive levels. Text books and 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 topics are covered/taught so that assessment is balanced between paper 1 and 2.

CONTENT TRAINING(p.22)

Term 1:• Algebraic expressions• Exponents• Numbers and Patterns• Equations and Inequalities• Trigonometry

ALGEBRAIC EXPRESSIONS (3 WEEKS)

Real numbersSurdsAppropriate roundingMultiply a binomial by trinomialFactorising

oTrinomialsoGrouping in pairsoSum and difference of two cubes

Use factorising to simplify algebraic fractionsAdd and subtract algebraic fractions

Different cognitive levels in factorisation

(C) )1x)(1x2)(1x2(

)2x32(4x-

)1x)(1x2)(1x2(

4x68x-

)1x)(1x2)(1x2(

1x44x9x21x32x-

simplify )1x)(1x2)(1x2(

)1x4()4x9x2()1x3(2x-

LCD ithmultiply w 1)-1)(x1)(2x-(2x

1)1)(2x-1(2x-1)4)(2x(x-1)-1)(x-(2x-

LCD the find 1x

1

)1x)(1x2(

4x

)1x2)(1x2(

1)-(2x-

rdenominato factorise x1

1

1x3x2

4x

14x

2x-1 :Simplify 2.

(R)

"factorisedfully " been has expression an henidentify w

and fractions with workto required are Learners 182

y13

2

y 1.3

(R) tests all in appears and routine is type This 3x2x 1.2

(R) square perfect

simple the recognise to able be must Learners (revision) 1m2m 1.1

:fully Factorise 1.

2

2

222

222

22

2

2

2

EXPONENTS (2 WEEKS)

Revise the laws of exponents

Use the laws of exponents to simplify expressions and solve equations

0 x, 1 x(vi)

and 0, x,x

1 x(v)

definitionby also

)xy(y x)iv(

x)(x )iii(

xx x)ii(

xx x)i(

0

mm-

mmm

mnnm

nmnm

nmnm

Examples

-1 x

13x3

3

1x3

10

3

9

10x3

simplify 9

10)

3

10(x3

factorise 9

10)

3

13(x3

laws lexponentia theapply 9

1013.x313.x3

(C) 9

101x31x3 3.3

(R) 5421

2x 3.2

(R) 125,0x2 1.3

: xfor Solve .3

1x3

(P) insight requires squares of difference )1x3(

)1x3)(1x(3

a as factorised be can numerator the that spotting 1x3

12x3

taught been not has question of type this ssuming A 1x3

1x9 :Simplify 2.

NUMBERS AND PATTERNS (1 WEEK)Arithmetic sequence is done in Grade 12, hence is not used in Gr. 10

Consider the linear number pattern 3; 5; 7; 9; ....

1n2n

T term general c 1

c 2 - 3

c 2 3

c)1(21

T

difference b

term of number n

qpnn

T or cbnn

T

termnew each to 2 addingby formed is pattern the

forth so and

7 is )3(T is term third The

5 is )2(T term second The

3 is ) 1T ( term first The

Examples

143 n

715 5n

713T 7132-5n

713T )c(

2n5T (b)

30 28; 23; (a)

5 of multiple each from 2 subtract , 18 13; 8; 3; is sequence given the Since :3 Step

... 35; 30; 25; 20; 15; 10; ; 5 :)difference common the ( 5 of multiples the List :2 Step

5 is difference common the case, this In .difference common the Find :1 Step

:Solution

713? of value a has sequence this in term h Whic)c(

T term, n the for formula the down Write(b)

sequence the in terms three next the

down write way,this in continues pattern the If (a)

... 18; 13; 8; 3; sequence the at Look .1

143

n

n

nth

EQUATIONS AND INEQUALITIES (2 WEEKS)

Solve linear equationsSolve quadratic equationsSolve simultaneous equationsSolve literal equationsSolve linear inequalities

EXAMPLES

1 x -2

2- x or x 1 2- x 1

6 3x - or3x - 3 - 3- withdivide 6 3x - 3 -

2 - 8 3x - or3x - 2-1- 28x32 - 1 -

8 3x -2 or3x -2 1- (b) or 8 3x-21- (a)

(C) 83x- 21- : xfor Solve 5.

(R) hr V :h and V,of terms in r for Solve 4.

(C) 12

y

3

x ; 12y x: yand x for Solve 3.

(R) 1m2m :m for Solve 2.

(R) 6

x2x3

3

3-2x : xfor Solve .1

2

2

TRIGONOMETRY (3 WEEKS)

Special angles and reciprocal ratios

θ ε { 0º; 30º; 45º; 60º; 90º} without using a calculator

cosec θ; sec θ ; cot θ (examined only in gr. 10)

Solving right angles triangles

Solving simple trigonometric equations (angles between 0º and 90º )

Defining and plotting trigonometric functions

adj

opptan

hyp

adjcos

hyp

oppsin

EXAMPLESSimilarity of triangles is fundamental to the trigonometric ratios sinθ; cosθ and tanθ;

125

25

25

9

25

16

5

3

5

4cossin

3- x quadrant third the for but

3 x

9 x

1625 x

(pyth) )5(x(-4)

r

y

hyp

opp

5

4- sin

-45sin

045sin

.calculator a using

withoutcossin of value the calculate ,2700 and 04sin 5 If

2222

2

222

22

Triangles for special angles

CONTENT TRAINING

Term 2:• Functions

Activity

• Euclidean Geometry Activity (Revision)

FUNCTIONS (4 WEEKS)The concept of a function

Plot basic graphs defined by:

Investigate the effect of a and q on the graphs defined by

Investigate the effect of a and q on

the trigonometric graphs

Sketch, use and interpret the graphs

0b and 0 b , b y; x

1 y; xy x2

0b 0,b ,bf(x) and x

1f(x) ,xf(x) x,f(x)

where, qa(fx)y

x2

Functions

Activity

Euclidean Geometry(3 weeks)

Activity

Euclidean Geometry(3 weeks)Similar and congruent triangles

Detailed revision of basic geometryRevision of triangles and parallel linesConcepts of similarity and congruence

Investigate special polygons

Properties of quadrilateralsMaking conjectures and then proving them

CONTENT TRAININGTerm 3:• Analytical Geometry• Finance and growth• Statistics• Euclidean Geometry• Trigonometry• Measurement

Analytical geometry(2 weeks)The distance between two points

The gradient of a line segment

The midpoint of a line segment

212

212 )yy()x(xd

lines larperpendicu 1mm

lines parallel mm

xx

yym

21

21

12

12

2

yy y;

2

xxx 12

m12

m

Examples

rr

rr

rr

rpm

rpm

22

22

2pq

2pq

y 5- ; x 4-

y 5 0 ; x 22-

R(-4;-5) 2

y50 ;

2

x21-

2

yyy ;

2

xx x1.2

units 41

1625

)4((-5)

)51()23(

)yy()x(x PQ 1.1

:Solution

(R) not? why or Why rectangle? a PQRS Is 1.4

(C) ramparallelog a is PQRS if S of scoordinate the Determine 1.3

(R) PR of point-mid the is M(-1;0) if R of scoordinate the Find 1.2

(K) PQ distance the Calculate 1.1

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

Examples

s y 1- ; sx 1

sy10 ; sx-32-

S(1;-1) 2

sy10 ;

2

sx3- 1-

2

syqym y;

2

sxqxm x

QS of point-mid the is M

other each bisect ramparallelog a of Diagonals 3.1

rectangle a no is PQRS therefore angle, right no

RHSLHS 5

24

1

6

5

4

43

51

32

1-5 :LHS

90Q then 1mm if

not? why or Why rectangle? a PQRS Is 1.4

QRPQ

equal are diagonals the and angle rigth a withramparallelog a is rectangleA

:rectangle of Definition

Finance and Growth(2 weeks)Simple and compound growth

Foreign exchange rates

n)i1(PA

)in1(PA

STATISTICS (2.5 WEEKS)Measures of central tendencygrouped and ungrouped data

Measure of dispersion

Five number summaries and box and whiskers diagram

Analysing and interpreting statistical summaries

Euclidean Geometry(1 week)

Solve problems and prove riders using the properties of parallel lines,

triangles and quadrilaterals.

Trigonometry(2 weeks)

Problems in two dimensions

ExampleAt the base of an electricity pylon, two poles and the cross wire are positioned as in the diagram:

(a) What angle does each wire make with the ground?

(b) What is the length of each wire?

(c) At what height above the ground do the two wires intersect?

Platinum Mathematics Gr. 10 ,Maskew Miller Longman, J Campbell S McPetrie

Example(cont.)

DB m 10 AC m 16,16

DB26 28 AC26215

2)DB(2)6(2(8) :BCD In 2)AC(2)6(2(15) :ABC In

wiresthe of length the find to Pythagoras of Theorem the Use

wire?each of length the is What)b

53,1 CBD 68,2 BC A

6

8 CBD tan : BCD In

6

15BC Atan : ABC In

ground? the withmake wirethe does angle What)a

m 21 5, x

68,2 tan 2,085 x

(i) into back Substitute

m 2,085 y

7,991 (3,832) y

53,1 tan 6 53,1 tan y 68,2 tan y

53,1 tany - 53,1 tan 6 68,2 tan y

53,1 tan y)-(668,2 tan y

(ii) (i) settingby usly simultaneo Solve

......(ii) 53,1 tan y)-(6 x ......(i)68,2 tan y x

y-6

x53,1 tan

y

x68,2 tan

y-6 QC and y BQ and x PQ heigt Let BC. to larperpendicu PQ

line construct intersect, wirestwo the whereheight the find To

intersect? wirestwo the do ground the above height what Atc)

Measurement(1 week)Revise the volume and surface area of right-prisms and cylinders

Study the effect on volume and surface area when multiplying any dimension by a constant factor k

Calculate the volume and surface area of sphere, right pyramids and right cones

(In case of pyramids, bases must either be equilateral triangle or

a square. Problem types must include composite figures)

VOLUME AND SURFACE AREA

VOLUME AND SURFACE

VOLUME AND SURFACE

CONTENT TRAINING

Term 4:

• Probability

Probability(2 weeks)

Use of probability models.

Use of Venn diagrams to solve probability problems.

THAT’S IT

TOWARDS GREATER PERFORMANCE IN MATHEMATICS