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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
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