TEC
HN
ICA
L M
ATH
EM
ATI
CS
Curriculum Assessment Policy Statement
GRADES 10 – 12
National Curriculum Statement (NCS)
1CAPS TECHNICAL MATHEMATICS
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADES 10 – 12
TECHNICAL MATHEMATICS
2 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
DISCLAIMER
In view of the stringent time requirements encountered by the Department of Basic Education to effect the necessary editorial changes and layout adjustments to the Curriculum and Assessment Policy Statements and the supplementary policy documents, possible errors may have occurred in the said documents placed on the official departmental website.
If any editorial, layout, content, terminology or formulae inconsistencies are detected, the user is kindly requested to bring this to the attention of the Department of Basic Education.
E-mail: [email protected] or fax (012) 328 9828
Department of Basic Education
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Private Bag X895
Pretoria 0001
South Africa
Tel: +27 12 357 3000
Fax: +27 12 323 0601
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Cape Town 8000
South Africa
Tel: +27 21 465 1701
Fax: +27 21 461 8110
Website:http://www.education.gov.za
© 2014 Department of Basic Education
ISBN: 978- 4315-0573-9
Design and Layout by:
Printed by: Government Printing Works
3CAPS TECHNICAL MATHEMATICS
FOREWORD BY THE MINISTER
Our national curriculum is the culmination of our efforts over a period of seventeen years to transform the curriculum bequeathed to us by apartheid. From the start of democracy we have built our curriculum on the values that inspired our Constitution (Act 108 of 1996). The Preamble to the Constitution states that the aims of the Constitution are to:
• heal the divisions of the past and establish a society based on democratic values, social justice and fundamental human rights;
• improve the quality of life of all citizens and free the potential of each person;
• lay the foundations for a democratic and open society in which government is based on the will of the people and every citizen is equally protected by law; and
• build a united and democratic South Africa able to take its rightful place as a sovereign state in the family of nations.
Education and the curriculum have an important role to play in realising these aims.
In 1997 we introduced outcomes-based education to overcome the curricular divisions of the past, but the experience of implementation prompted a review in 2000. This led to the first curriculum revision: the Revised National Curriculum Statement Grades R-9 and the National Curriculum Statement Grades 10-12 (2002).
Ongoing implementation challenges resulted in another review in 2009 and we revised the Revised National Curriculum Statement (2002) to produce this document.
From 2012 the two 2002 curricula, for Grades R-9 and Grades 10-12 respectively, are combined in a single document and will simply be known as the National Curriculum Statement Grades R-12. The National Curriculum Statement for Grades R-12 builds on the previous curriculum but also updates it and aims to provide clearer specification of what is to be taught and learnt on a term-by-term basis.
The National Curriculum Statement Grades R-12 accordingly replaces the Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines with the
(a) Curriculum and Assessment Policy Statements (CAPS) for all approved subjects listed in this document;
(b) National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R – 12; and
(c) National Protocol for Assessment Grades R – 12.
MRS ANGIE MOTSHEKGA, MP
MINISTER OF BASIC EDUCATION
4 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
CONTENTS
SECTION 1: INTRODUCTION TO THE CURRICULUM AND ASSESSMENT POLICY STATEMENTS FOR TECHNICAL MATHEMATICS GRADE 10 - 12 TECHNICAL MATHEMATICS
6
1.1 Background 6
1.2 Overview 6
1.3 General Aims of the South African Curriculum 7
1.4 Time Allocation 8
1.4.1 Foundation Phase 8
1.4.2 Intermediate Phase 8
1.4.3 Senior Phase 9
1.4.4 Grades 10 – 12 9
SECTION 2: FET TECHNICAL MATHEMATICS INTRODUCTION 10
2.1 What is Technical Mathematics? 10
2.2 Specific Aims of Technical Mathematics 10
2.3 Specific Skills 11
2.4 Focus of Content Areas 11
2.5 Weighting of Content Areas 11
SECTION 3: FET TECHNICAL MATHEMATICS CONTENT SPECIFICATION AND CLARIFICATION 13
3.1 Specification of Content to show Progression 13
3.1.1 Overview of topics 14
3.2 Content Clarification with teaching guidelines 19
3.2.1 Allocation of Teaching Time 20
3.2.2 Sequencing and Pacing of Topics 22
3.2.3 Topic allocation per term and clarification 24
SECTION 4: FET TECHNICAL MATHEMATICS ASSESSMENT GUIDELINES 46
4.1 Introduction 46
4.2 Informal or Daily Assessment 47
4.3 Formal Assessment 47
5CAPS TECHNICAL MATHEMATICS
4.4 Programme of Assessment 48
4.5 Recording and Reporting 50
4.6 Moderation of Assessment 51
4.7 General 51
4.7.1 National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R – 12; and
51
4.7.2 The policy document, National Protocol for Assessment Grades R – 12. 51
6 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
SECTION 1INTRODUCTION TO THE CURRICULUM AND ASSESSMENT POLICY STATEMENTS FOR TECHNICAL
MATHEMATICS GRADE 10 - 12
TECHNICAL MATHEMATICS
1.1 Background
The National Curriculum Statement Grades R – 12 (NCS) stipulates policy on curriculum and assessment in the schooling sector. To improve its implementation, the National Curriculum Statement was amended, with the amendments coming into effect in January 2012. A single comprehensive National Curriculum and Assessment 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 National Curriculum 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 for learning and teaching in South African schools and comprises the following:
National Curriculum and Assessment Policy statements for each approved school subject as listed in the policy document, and the National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R – 12, which replaces the following policy documents:
(i) NationalSeniorCertificate:AqualificationatLevel4ontheNationalQualificationsFramework(NQF); and
(ii) Anaddendumto thepolicydocument, theNational SeniorCertificate:Aqualificationat Level4on theNationalQualificationsFramework(NQF),for learnerswithspecialneeds, published in the GovernmentGazette,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 Government Gazette, No. 29467 of 11 December 2006.
(c) The Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines for Grades R – 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. Therefore, in terms of section 6A of the South African Schools Act, 1996 (Act No. 84 of 1996,) this forms the basis on which the Minister of Basic Education will determine minimum outcomes and standards, as well as the processes and procedures for the assessment of learner achievement, that will apply in public and independent schools.
7CAPS TECHNICAL MATHEMATICS
1.3 General Aims of the South African Curriculum
(a) The National Curriculum Statement Grades R – 12 spells out what is regarded as knowledge, skills and values worth learning. It will ensure that children acquire and apply knowledge and skills in ways that are meaningful to their own lives. In this regard, the curriculum promotes the idea of grounding knowledge in local contexts, while being sensitive to global imperatives.
(b) The National Curriculum Statement Grades R – 12 serves the purposes of:
• equipping learners, irrespective of their socio-economic background, race, gender, physical ability or intellectual ability, with the knowledge, skills and values necessary for self-fulfilment, and meaningful participation in society as citizens of a free country;
• providing access to higher education;
• facilitating the transition of learners from education institutions to the workplace; and
• providing employers with a sufficient profile of a learner’s competencies.
(c) The National Curriculum Statement Grades R – 12 is based on the following principles:
• social transformation: ensuring that the educational imbalances of the past are redressed, and that equal educational opportunities are provided for all sections of our population;
• active and critical learning: encouraging an active and critical approach to learning, rather than rote and uncritical learning of given truths;
• high knowledge and high skills: specifying the minimum standards of knowledge and skills to be achieved at each grade setting high, achievable standards in all subjects;
• progression: showing progression from simple to complex in the content and context of each grade;
• human rights, inclusivity, environmental and social justice: infusing the principles and practices of social and environmental justice and human rights as defined in the Constitution of the Republic of South Africa. (The National Curriculum Statement Grades 10 – 12 (General) is sensitive to issues of diversity such as poverty, inequality, race, gender, language, age, disability and other factors.)
• value of indigenous knowledge systems: acknowledging the rich history and heritage of this country as important contributors to nurturing the values contained in the Constitution; and
• credibility, quality and efficiency: providing an education that is comparable in quality, breadth and depth to those of other countries.
(d) 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 efficiently;
• collect, analyse, organise and evaluate information critically;
• 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.
8 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
(e) Inclusivity should be a central part of the organisation, planning and teaching at each school. This can only occur if all teachers have the capacity to recognise and address barriers to learning, and to plan for diversity. The key to managing inclusivity is to ensure that barriers are identified and addressed by all the relevant support structures within the school community, including teachers, district-based support teams, institutional-level support teams, parents and Special Schools as resource centres. To address barriers in the classroom, teachers should employ various curriculum differentiation strategies such as those contained in the Department of Basic Education’s Guidelines for Inclusive Teaching and Learning (2010).
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 and a minimum of 7 hours are allocated to Home Language and a minimum of 2 hours and a maximum of 3 hours to First Additional Language in Grades R – 2. In Grade 3 a maximum of 8 hours and a minimum of 7 hours are allocated to Home Language and a minimum of 3 hours and a maximum of 4 hours to First Additional Language.
(d) To Life Skills Beginning Knowledge 1 hour is allocated in Grades R – 2 and 2 hours are allocated for Grade 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 Languageii. First Additional Languageiii. Mathematicsiv. Science and Technologyv. Social Sciences
vi. Life Skillsvii. Creative Artsviii. Physical Educationix. Religious Studies
656
3.534
1.51.51
9CAPS TECHNICAL MATHEMATICS
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)
i. Home Languageii. First Additional Languageiii. Mathematicsiv. Natural Sciencesv. Social Sciencesvi. Technologyvii. Economic Management Sciencesviii. Life Orientationix. Arts and Culture
54
4.5332222
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 Languageii. First Additional Languageiii. Mathematics iv. Technical Mathematicsv. Mathematical Literacyvi. Life Orientationvii. Three Electives
4.54.54.54.54.52
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.
10 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
Section 2Curriculum and Assessment Policy Statement (CAPS)
FET TECHNICAL MATHEMATICS
Introduction
In Section 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, teaches 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 andnot on abstract ideas.
2.2 Specific Aims of Technical Mathematics
1. To apply mathematical principles.
2. To develop fluency in computation skills with the usage of calculators.
3. Mathematical modelling 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.
4. To provide the opportunity to develop in learners the ability to be methodical, to generalize and to be skilful users of the Science of Mathematics.
5. To be able to understand and work with the number system.
6. To promote accessibility of Mathematical content to all learners. This could be achieved by catering for learners with different needs, e.g. TECHNICAL NEEDS.
7. 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.
8. To provide learners at Technical schools an alternative and value adding substitute to Mathematical Literacy.
9. To support and sustain technical subjects at Technical schools.
10. Technical Mathematics can only be taken by learners offering a Technical subject (mechanical, civil and electrical engineering).
11. To provide a vocational route aligned with the expectations of labour in order to ensure direct access to learnership/apprenticeship.
11CAPS TECHNICAL MATHEMATICS
12. To create the opportunity for learners to further their studies at FET 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, as well as in 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.
1. Number system
2. Functions and graphs
3. Finance, growth and decay
4. Algebra
5. Differential Calculus
6. Euclidean Geometry
7. Mensuration
8. Circles, angles and angular movement
9. Analytical Geometry
10. Trigonometry
Main topics for Technical Mathematics:
2.5 Weighting of Content Areas
The weighting of Technical Mathematics content areas serves two primary purposes: firstly the weighting gives guidance on the amount of time needed to address adequately the content within each content area; secondly the weighting gives guidance on the spread of content in the examination (especially end of the year summative assessment).
12 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
Weighting of Content Areas
Description Grade 10 Grade 11 Grade 12
PAPER 1
Algebra ( Expressions, equations and inequalities including nature of roots in Grades 11 & 12)
60 ± 3 90 ± 3 50 ± 3
Functions & Graphs (excluding trig. functions) 25 ± 3 45 ± 3 35 ± 3
Finance, growth and decay 15 ± 3 15 ± 3 15 ± 3
Differential Calculus and Integration 50 ± 3
TOTAL 100 150 150
PAPER 2 :
Description Grade 10 Grade 11 Grade 12
Analytical Geometry 15 ± 3 25 ± 3 25 ± 3
Trigonometry (including trig. functions) 40 ± 3 50 ± 3 50 ± 3
Euclidean Geometry 30 ± 3 40 ± 3 40 ± 3
Mensuration, circles, angles and angular movement 15 ± 3 35 ± 3 35 ± 3
TOTAL 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 the Senior Phase and the Higher Education band. All learners passing through this phase acquire a functioning knowledge of Technical Mathematics that empowers them to make sense of their Technical field of study and their place in society. It ensures access to an extended study of the technical mathematical sciences and a variety of technical career paths.
In the FET Phase, learners should be exposed to technical mathematical experiences that give them many opportunities to develop their mathematical reasoning and creative skills in preparation for more applied mathematics in HEIs or in-job training.
13CAPS TECHNICAL MATHEMATICS
Section 3Curriculum and Assessment Policy Statement (CAPS)
FET TECHNICAL MATHEMATICS Content Specification and Clarification
Introduction
Section 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 for each content area. However, in certain topics the concepts and skills are similar in two or three successive grades. The clarification of content gives guidelines on how progression should be addressed in these cases. The specification of content should therefore be read in conjunction with the clarification of content.
14 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.1.
1
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f top
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Page
17 of
59
3.1.
1 O
verv
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of t
opic
s
1. N
UM
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SYST
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10
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ade
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term
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as
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an th
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Page
17 of
59
3.1.
1 O
verv
iew
of t
opic
s
1. N
UM
BER
SYST
EM
Gr
ade
10
Gr
ade
11
Gr
ade
12
(a)
Iden
tify
ratio
nal n
umbe
rs a
nd c
onve
rt
term
inat
ing
or re
curr
ing
deci
mal
s int
o th
e fo
rm ba
whe
re
Zb
a
, a
nd
0
b.
(b) U
nder
stan
d th
at si
mpl
e su
rds a
re n
ot
ratio
nal.
Take
not
e th
at n
umbe
rs e
xist
oth
er th
an th
ose
on th
e re
al n
umbe
r lin
e, th
e so
-cal
led
non-
real
nu
mbe
rs. I
t is p
ossib
le to
squa
re c
erta
in n
on-
real
num
bers
and
obt
ain
nega
tive
real
num
bers
as
ans
wer
s.
Bina
ry n
umbe
rs sh
ould
be
know
n.
Th
ere
are
num
bers
oth
er th
an th
ose
stud
ied
in e
arlie
r gra
des c
alle
d im
agin
ary
num
bers
an
d co
mpl
ex n
umbe
rs.
Add,
subt
ract
, div
ide,
mul
tiply
and
sim
plify
im
agin
ary
num
bers
and
com
plex
num
bers
. So
lve
equa
tions
invo
lvin
g co
mpl
ex n
umbe
rs.
2. F
UN
CTIO
NS
Wor
k w
ith re
latio
nshi
ps b
etw
een
varia
bles
in
term
s of n
umer
ical
, gra
phic
al, v
erba
l and
sy
mbo
lic re
pres
enta
tions
of f
unct
ions
and
co
nver
t fle
xibl
y be
twee
n th
ese
repr
esen
tatio
ns (t
able
s, g
raph
s, w
ords
and
fo
rmul
ae).
Incl
ude
linea
r and
som
e qu
adra
tic p
olyn
omia
l fu
nctio
ns, e
xpon
entia
l fun
ctio
ns a
nd so
me
ratio
nal f
unct
ions
.
Exte
nd G
rade
10
wor
k on
the
rela
tions
hips
be
twee
n va
riabl
es in
term
s of n
umer
ical
,
grap
hica
l, ve
rbal
and
sym
bolic
repr
esen
tatio
ns
of fu
nctio
ns a
nd c
onve
rt fl
exib
ly b
etw
een
thes
e re
pres
enta
tions
(tab
les,
gra
phs,
wor
ds a
nd
form
ulae
). In
clud
e lin
ear a
nd q
uadr
atic
pol
ynom
ial
func
tions
, exp
onen
tial f
unct
ions
and
so
me
ratio
nal f
unct
ions
.
Intr
oduc
e a
mor
e fo
rmal
def
initi
on o
f a
func
tion
and
exte
nd G
rade
11
wor
k on
the
re
latio
nshi
ps b
etw
een
varia
bles
in te
rms o
f nu
mer
ical
, gra
phic
al, v
erba
l and
sym
bolic
re
pres
enta
tions
of f
unct
ions
and
con
vert
fle
xibl
y be
twee
n th
ese
repr
esen
tatio
ns
(tab
les,
gra
phs,
wor
ds a
nd fo
rmul
ae).
In
clud
e lin
ear,
quad
ratic
and
som
e cu
bic
poly
nom
ial f
unct
ions
, exp
onen
tial a
nd so
me
ratio
nal f
unct
ions
.
Ge
nera
te a
s man
y gr
aphs
as n
eces
sary
, in
itial
ly b
y m
eans
of p
oint
-by-
poin
t plo
ttin
g, to
m
ake
and
test
con
ject
ures
and
hen
ce
gene
ralis
e th
e ef
fect
of t
he p
aram
eter
whi
ch
resu
lts in
a v
ertic
al sh
ift a
nd th
at w
hich
resu
lts
in a
ver
tical
stre
tch
and/
or a
refle
ctio
n ab
out
the
x-ax
is.
Gene
rate
as m
any
grap
hs a
s nec
essa
ry, i
nitia
lly
by m
eans
of p
oint
-by-
poin
t plo
ttin
g, to
mak
e an
d te
st c
onje
ctur
es a
nd h
ence
gen
eral
ise
the
effe
cts o
f the
par
amet
er w
hich
resu
lts in
a
horiz
onta
l shi
ft an
d th
at w
hich
resu
lts in
a
horiz
onta
l str
etch
and
/or r
efle
ctio
n ab
out t
he y
-ax
is.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Prob
lem
sol
ving
and
gra
ph w
ork
invo
lvin
g th
e pr
escr
ibed
func
tions
.
Pr
oble
m s
olvi
ng a
nd g
raph
wor
k in
volv
ing
the
pres
crib
ed fu
nctio
ns. A
vera
ge g
radi
ent b
etw
een
two
poin
ts.
Prob
lem
solv
ing
and
grap
h w
ork
invo
lvin
g th
e pr
escr
ibed
func
tions
.
and
Page
17 of
59
3.1.
1 O
verv
iew
of t
opic
s
1. N
UM
BER
SYST
EM
Gr
ade
10
Gr
ade
11
Gr
ade
12
(a)
Iden
tify
ratio
nal n
umbe
rs a
nd c
onve
rt
term
inat
ing
or re
curr
ing
deci
mal
s int
o th
e fo
rm ba
whe
re
Zb
a
, a
nd
0
b.
(b) U
nder
stan
d th
at si
mpl
e su
rds a
re n
ot
ratio
nal.
Take
not
e th
at n
umbe
rs e
xist
oth
er th
an th
ose
on th
e re
al n
umbe
r lin
e, th
e so
-cal
led
non-
real
nu
mbe
rs. I
t is p
ossib
le to
squa
re c
erta
in n
on-
real
num
bers
and
obt
ain
nega
tive
real
num
bers
as
ans
wer
s.
Bina
ry n
umbe
rs sh
ould
be
know
n.
Th
ere
are
num
bers
oth
er th
an th
ose
stud
ied
in e
arlie
r gra
des c
alle
d im
agin
ary
num
bers
an
d co
mpl
ex n
umbe
rs.
Add,
subt
ract
, div
ide,
mul
tiply
and
sim
plify
im
agin
ary
num
bers
and
com
plex
num
bers
. So
lve
equa
tions
invo
lvin
g co
mpl
ex n
umbe
rs.
2. F
UN
CTIO
NS
Wor
k w
ith re
latio
nshi
ps b
etw
een
varia
bles
in
term
s of n
umer
ical
, gra
phic
al, v
erba
l and
sy
mbo
lic re
pres
enta
tions
of f
unct
ions
and
co
nver
t fle
xibl
y be
twee
n th
ese
repr
esen
tatio
ns (t
able
s, g
raph
s, w
ords
and
fo
rmul
ae).
Incl
ude
linea
r and
som
e qu
adra
tic p
olyn
omia
l fu
nctio
ns, e
xpon
entia
l fun
ctio
ns a
nd so
me
ratio
nal f
unct
ions
.
Exte
nd G
rade
10
wor
k on
the
rela
tions
hips
be
twee
n va
riabl
es in
term
s of n
umer
ical
,
grap
hica
l, ve
rbal
and
sym
bolic
repr
esen
tatio
ns
of fu
nctio
ns a
nd c
onve
rt fl
exib
ly b
etw
een
thes
e re
pres
enta
tions
(tab
les,
gra
phs,
wor
ds a
nd
form
ulae
). In
clud
e lin
ear a
nd q
uadr
atic
pol
ynom
ial
func
tions
, exp
onen
tial f
unct
ions
and
so
me
ratio
nal f
unct
ions
.
Intr
oduc
e a
mor
e fo
rmal
def
initi
on o
f a
func
tion
and
exte
nd G
rade
11
wor
k on
the
re
latio
nshi
ps b
etw
een
varia
bles
in te
rms o
f nu
mer
ical
, gra
phic
al, v
erba
l and
sym
bolic
re
pres
enta
tions
of f
unct
ions
and
con
vert
fle
xibl
y be
twee
n th
ese
repr
esen
tatio
ns
(tab
les,
gra
phs,
wor
ds a
nd fo
rmul
ae).
In
clud
e lin
ear,
quad
ratic
and
som
e cu
bic
poly
nom
ial f
unct
ions
, exp
onen
tial a
nd so
me
ratio
nal f
unct
ions
.
Ge
nera
te a
s man
y gr
aphs
as n
eces
sary
, in
itial
ly b
y m
eans
of p
oint
-by-
poin
t plo
ttin
g, to
m
ake
and
test
con
ject
ures
and
hen
ce
gene
ralis
e th
e ef
fect
of t
he p
aram
eter
whi
ch
resu
lts in
a v
ertic
al sh
ift a
nd th
at w
hich
resu
lts
in a
ver
tical
stre
tch
and/
or a
refle
ctio
n ab
out
the
x-ax
is.
Gene
rate
as m
any
grap
hs a
s nec
essa
ry, i
nitia
lly
by m
eans
of p
oint
-by-
poin
t plo
ttin
g, to
mak
e an
d te
st c
onje
ctur
es a
nd h
ence
gen
eral
ise
the
effe
cts o
f the
par
amet
er w
hich
resu
lts in
a
horiz
onta
l shi
ft an
d th
at w
hich
resu
lts in
a
horiz
onta
l str
etch
and
/or r
efle
ctio
n ab
out t
he y
-ax
is.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Prob
lem
sol
ving
and
gra
ph w
ork
invo
lvin
g th
e pr
escr
ibed
func
tions
.
Pr
oble
m s
olvi
ng a
nd g
raph
wor
k in
volv
ing
the
pres
crib
ed fu
nctio
ns. A
vera
ge g
radi
ent b
etw
een
two
poin
ts.
Prob
lem
solv
ing
and
grap
h w
ork
invo
lvin
g th
e pr
escr
ibed
func
tions
.
.
(b) U
nder
stan
d th
at
sim
ple
surd
s ar
e no
t ra
tiona
l.
•Ta
ke n
ote
that
num
bers
exi
st o
ther
than
thos
e on
the
real
num
ber l
ine,
the
so-c
alle
d no
n-re
al n
umbe
rs. I
t is
pos
sibl
e to
squ
are
certa
in n
on-r
eal n
umbe
rs a
nd
obta
in n
egat
ive
real
num
bers
as
answ
ers.
Bin
ary
num
bers
sho
uld
be k
now
n.
•Th
ere
are
num
bers
oth
er th
an th
ose
stud
ied
in e
arlie
r gra
des
calle
d im
agin
ary
num
bers
and
com
plex
num
bers
.
Add
, sub
tract
, div
ide,
mul
tiply
and
sim
plify
imag
inar
y nu
mbe
rs a
nd c
ompl
ex n
umbe
rs.
Sol
ve e
quat
ions
invo
lvin
g co
mpl
ex
num
bers
.2.
FU
NC
TIO
NS
•W
ork
with
rela
tions
hips
bet
wee
n va
riabl
es
in te
rms
of n
umer
ical
, gra
phic
al, v
erba
l and
sy
mbo
lic re
pres
enta
tions
of f
unct
ions
and
co
nver
t flex
ibly
bet
wee
n th
ese
repr
esen
tatio
ns
(tabl
es, g
raph
s, w
ords
and
form
ulae
).
Incl
ude
linea
r and
som
e qu
adra
tic p
olyn
omia
l fu
nctio
ns, e
xpon
entia
l fun
ctio
ns a
nd s
ome
ratio
nal f
unct
ions
.
•E
xten
d G
rade
10
wor
k on
the
rela
tions
hips
betw
een
varia
bles
in te
rms
of n
umer
ical
,
grap
hica
l, ve
rbal
and
sym
bolic
repr
esen
tatio
ns
of fu
nctio
ns a
nd c
onve
rt fle
xibl
y be
twee
n th
ese
repr
esen
tatio
ns (t
able
s, g
raph
s, w
ords
and
form
ulae
).
Incl
ude
linea
r and
qua
drat
ic p
olyn
omia
l fun
ctio
ns,
expo
nent
ial f
unct
ions
and
som
e ra
tiona
l fun
ctio
ns.
Intro
duce
a m
ore
form
al d
efini
tion
of a
func
tion
and
exte
nd G
rade
11
wor
k on
the
rela
tions
hips
bet
wee
n va
riabl
es in
term
s of
nu
mer
ical
, gra
phic
al, v
erba
l and
sym
bolic
re
pres
enta
tions
of f
unct
ions
and
con
vert
flexi
bly
betw
een
thes
e re
pres
enta
tions
(ta
bles
, gra
phs,
wor
ds a
nd fo
rmul
ae).
Incl
ude
linea
r, qu
adra
tic a
nd s
ome
cubi
c po
lyno
mia
l fun
ctio
ns, e
xpon
entia
l and
so
me
ratio
nal f
unct
ions
.•
Gen
erat
e as
man
y gr
aphs
as
nece
ssar
y,
initi
ally
by
mea
ns o
f poi
nt-b
y-po
int p
lotti
ng,
to m
ake
and
test
con
ject
ures
and
hen
ce
gene
ralis
e th
e ef
fect
of t
he p
aram
eter
whi
ch
resu
lts in
a v
ertic
al s
hift
and
that
whi
ch re
sults
in
a v
ertic
al s
tretc
h an
d/or
a re
flect
ion
abou
t th
e x-
axis
.
•G
ener
ate
as m
any
grap
hs a
s ne
cess
ary,
initi
ally
by
mea
ns o
f poi
nt-b
y-po
int p
lotti
ng, t
o m
ake
and
test
co
njec
ture
s an
d he
nce
gene
ralis
e th
e ef
fect
s of
the
para
met
er w
hich
resu
lts in
a h
oriz
onta
l shi
ft an
d th
at
whi
ch re
sults
in a
hor
izon
tal s
tretc
h an
d/or
refle
ctio
n ab
out t
he y
-axi
s.
•R
evis
e w
ork
stud
ied
in e
arlie
r gra
des.
•P
robl
em s
olvi
ng a
nd g
raph
wor
k in
volv
ing
the
pres
crib
ed fu
nctio
ns.
•P
robl
em so
lvin
g an
d gr
aph
wor
k inv
olvi
ng th
e pr
escr
ibed
fu
nctio
ns. A
vera
ge g
radi
ent b
etw
een
two
poin
ts.
•P
robl
em s
olvi
ng a
nd g
raph
wor
k in
volv
ing
the
pres
crib
ed fu
nctio
ns.
15CAPS TECHNICAL MATHEMATICS
3. F
INA
NC
E, G
RO
WTH
AN
D D
ECAY
•U
se s
impl
e an
d co
mpo
und
grow
th fo
rmul
ae
3. F
INAN
CE, G
ROW
TH A
ND
DEC
AY
Use
sim
ple
and
com
poun
d gr
owth
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Use
sim
ple
and
com
poun
d gr
owth
/dec
ay
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Stre
ngth
en th
e Gr
ade
11 w
ork.
The
impl
icat
ions
of f
luct
uatin
g fo
reig
n ex
chan
ge ra
tes.
Th
e ef
fect
of d
iffer
ent p
erio
ds o
f com
poun
ding
gr
owth
and
dec
ay (i
nclu
ding
effe
ctiv
e an
d no
min
al in
tere
st ra
tes)
.
Criti
cally
ana
lyse
diff
eren
t loa
n op
tions
.
4. A
LGEB
RA
(a)
Sim
plify
exp
ress
ions
usin
g th
e la
ws o
f ex
pone
nts f
or in
tegr
al e
xpon
ents
. (b
) Est
ablis
h be
twee
n w
hich
two
inte
gers
a
give
n sim
ple
surd
lies
. (c
) Rou
nd re
al n
umbe
rs to
an
appr
opria
te
degr
ee o
f acc
urac
y (t
o a
give
n nu
mbe
r of
deci
mal
dig
its).
(d) R
evise
scie
ntifi
c no
tatio
n.
(a) A
pply
the
law
s of e
xpon
ents
to e
xpre
ssio
ns
invo
lvin
g ra
tiona
l exp
onen
ts.
(b) A
dd, s
ubtr
act,
mul
tiply
and
div
ide
simpl
e su
rds.
(c
) Dem
onst
rate
an
unde
rsta
ndin
g of
the
defin
ition
of a
loga
rithm
and
any
law
s ne
eded
to so
lve
real
life
pro
blem
s.
Appl
y an
y la
w o
f log
arith
m to
solv
e re
al li
fe
prob
lem
s.
Man
ipul
ate
alge
brai
c ex
pres
sions
by:
mul
tiply
ing
a bi
nom
ial b
y a
trin
omia
l;
fact
orisi
ng c
omm
on fa
ctor
(rev
ision
);
fact
orisi
ng b
y gr
oupi
ng in
pai
rs;
fa
ctor
ising
trin
omia
ls;
fa
ctor
ising
diff
eren
ce o
f tw
o sq
uare
s (r
evisi
on);
fa
ctor
ising
the
diffe
renc
e an
d su
ms o
f tw
o cu
bes;
and
simpl
ifyin
g, a
ddin
g, su
btra
ctin
g,
mul
tiply
ing
and
divi
sion
of a
lgeb
raic
fr
actio
ns w
ith n
umer
ator
s and
de
nom
inat
ors l
imite
d to
the
poly
nom
ials
cove
red
in fa
ctor
isatio
n.
Revi
se fa
ctor
isatio
n fr
om G
rade
10.
Ta
ke n
ote
and
unde
rsta
nd th
e Re
mai
nder
an
d Fa
ctor
The
orem
s for
pol
ynom
ials
up
to th
e th
ird d
egre
e (p
roof
s of t
he
Rem
aind
er a
nd F
acto
r the
orem
s will
not
be
exa
min
ed).
Fa
ctor
ise th
ird-d
egre
e po
lyno
mia
ls (in
clud
ing
exam
ples
whi
ch re
quire
the
Fact
or T
heor
em).
and
Page
18 of
59
3. F
INAN
CE, G
ROW
TH A
ND
DEC
AY
Use
sim
ple
and
com
poun
d gr
owth
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Use
sim
ple
and
com
poun
d gr
owth
/dec
ay
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Stre
ngth
en th
e Gr
ade
11 w
ork.
The
impl
icat
ions
of f
luct
uatin
g fo
reig
n ex
chan
ge ra
tes.
Th
e ef
fect
of d
iffer
ent p
erio
ds o
f com
poun
ding
gr
owth
and
dec
ay (i
nclu
ding
effe
ctiv
e an
d no
min
al in
tere
st ra
tes)
.
Criti
cally
ana
lyse
diff
eren
t loa
n op
tions
.
4. A
LGEB
RA
(a)
Sim
plify
exp
ress
ions
usin
g th
e la
ws o
f ex
pone
nts f
or in
tegr
al e
xpon
ents
. (b
) Est
ablis
h be
twee
n w
hich
two
inte
gers
a
give
n sim
ple
surd
lies
. (c
) Rou
nd re
al n
umbe
rs to
an
appr
opria
te
degr
ee o
f acc
urac
y (t
o a
give
n nu
mbe
r of
deci
mal
dig
its).
(d) R
evise
scie
ntifi
c no
tatio
n.
(a) A
pply
the
law
s of e
xpon
ents
to e
xpre
ssio
ns
invo
lvin
g ra
tiona
l exp
onen
ts.
(b) A
dd, s
ubtr
act,
mul
tiply
and
div
ide
simpl
e su
rds.
(c
) Dem
onst
rate
an
unde
rsta
ndin
g of
the
defin
ition
of a
loga
rithm
and
any
law
s ne
eded
to so
lve
real
life
pro
blem
s.
Appl
y an
y la
w o
f log
arith
m to
solv
e re
al li
fe
prob
lem
s.
Man
ipul
ate
alge
brai
c ex
pres
sions
by:
mul
tiply
ing
a bi
nom
ial b
y a
trin
omia
l;
fact
orisi
ng c
omm
on fa
ctor
(rev
ision
);
fact
orisi
ng b
y gr
oupi
ng in
pai
rs;
fa
ctor
ising
trin
omia
ls;
fa
ctor
ising
diff
eren
ce o
f tw
o sq
uare
s (r
evisi
on);
fa
ctor
ising
the
diffe
renc
e an
d su
ms o
f tw
o cu
bes;
and
simpl
ifyin
g, a
ddin
g, su
btra
ctin
g,
mul
tiply
ing
and
divi
sion
of a
lgeb
raic
fr
actio
ns w
ith n
umer
ator
s and
de
nom
inat
ors l
imite
d to
the
poly
nom
ials
cove
red
in fa
ctor
isatio
n.
Revi
se fa
ctor
isatio
n fr
om G
rade
10.
Ta
ke n
ote
and
unde
rsta
nd th
e Re
mai
nder
an
d Fa
ctor
The
orem
s for
pol
ynom
ials
up
to th
e th
ird d
egre
e (p
roof
s of t
he
Rem
aind
er a
nd F
acto
r the
orem
s will
not
be
exa
min
ed).
Fa
ctor
ise th
ird-d
egre
e po
lyno
mia
ls (in
clud
ing
exam
ples
whi
ch re
quire
the
Fact
or T
heor
em).
to s
olve
pro
blem
s (in
clud
ing
inte
rest
, hire
pur
chas
e, in
flatio
n,
popu
latio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
•U
se s
impl
e an
d co
mpo
und
grow
th/d
ecay
form
ulae
3. F
INAN
CE, G
ROW
TH A
ND
DEC
AY
Use
sim
ple
and
com
poun
d gr
owth
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Use
sim
ple
and
com
poun
d gr
owth
/dec
ay
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Stre
ngth
en th
e Gr
ade
11 w
ork.
The
impl
icat
ions
of f
luct
uatin
g fo
reig
n ex
chan
ge ra
tes.
Th
e ef
fect
of d
iffer
ent p
erio
ds o
f com
poun
ding
gr
owth
and
dec
ay (i
nclu
ding
effe
ctiv
e an
d no
min
al in
tere
st ra
tes)
.
Criti
cally
ana
lyse
diff
eren
t loa
n op
tions
.
4. A
LGEB
RA
(a)
Sim
plify
exp
ress
ions
usin
g th
e la
ws o
f ex
pone
nts f
or in
tegr
al e
xpon
ents
. (b
) Est
ablis
h be
twee
n w
hich
two
inte
gers
a
give
n sim
ple
surd
lies
. (c
) Rou
nd re
al n
umbe
rs to
an
appr
opria
te
degr
ee o
f acc
urac
y (t
o a
give
n nu
mbe
r of
deci
mal
dig
its).
(d) R
evise
scie
ntifi
c no
tatio
n.
(a) A
pply
the
law
s of e
xpon
ents
to e
xpre
ssio
ns
invo
lvin
g ra
tiona
l exp
onen
ts.
(b) A
dd, s
ubtr
act,
mul
tiply
and
div
ide
simpl
e su
rds.
(c
) Dem
onst
rate
an
unde
rsta
ndin
g of
the
defin
ition
of a
loga
rithm
and
any
law
s ne
eded
to so
lve
real
life
pro
blem
s.
Appl
y an
y la
w o
f log
arith
m to
solv
e re
al li
fe
prob
lem
s.
Man
ipul
ate
alge
brai
c ex
pres
sions
by:
mul
tiply
ing
a bi
nom
ial b
y a
trin
omia
l;
fact
orisi
ng c
omm
on fa
ctor
(rev
ision
);
fact
orisi
ng b
y gr
oupi
ng in
pai
rs;
fa
ctor
ising
trin
omia
ls;
fa
ctor
ising
diff
eren
ce o
f tw
o sq
uare
s (r
evisi
on);
fa
ctor
ising
the
diffe
renc
e an
d su
ms o
f tw
o cu
bes;
and
simpl
ifyin
g, a
ddin
g, su
btra
ctin
g,
mul
tiply
ing
and
divi
sion
of a
lgeb
raic
fr
actio
ns w
ith n
umer
ator
s and
de
nom
inat
ors l
imite
d to
the
poly
nom
ials
cove
red
in fa
ctor
isatio
n.
Revi
se fa
ctor
isatio
n fr
om G
rade
10.
Ta
ke n
ote
and
unde
rsta
nd th
e Re
mai
nder
an
d Fa
ctor
The
orem
s for
pol
ynom
ials
up
to th
e th
ird d
egre
e (p
roof
s of t
he
Rem
aind
er a
nd F
acto
r the
orem
s will
not
be
exa
min
ed).
Fa
ctor
ise th
ird-d
egre
e po
lyno
mia
ls (in
clud
ing
exam
ples
whi
ch re
quire
the
Fact
or T
heor
em).
and
Page
18 of
59
3. F
INAN
CE, G
ROW
TH A
ND
DEC
AY
Use
sim
ple
and
com
poun
d gr
owth
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Use
sim
ple
and
com
poun
d gr
owth
/dec
ay
form
ulae
)
1(in
PA
and
n i
PA
)1(
to
solv
e pr
oble
ms (
incl
udin
g in
tere
st, h
ire p
urch
ase,
in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe
prob
lem
s).
Stre
ngth
en th
e Gr
ade
11 w
ork.
The
impl
icat
ions
of f
luct
uatin
g fo
reig
n ex
chan
ge ra
tes.
Th
e ef
fect
of d
iffer
ent p
erio
ds o
f com
poun
ding
gr
owth
and
dec
ay (i
nclu
ding
effe
ctiv
e an
d no
min
al in
tere
st ra
tes)
.
Criti
cally
ana
lyse
diff
eren
t loa
n op
tions
.
4. A
LGEB
RA
(a)
Sim
plify
exp
ress
ions
usin
g th
e la
ws o
f ex
pone
nts f
or in
tegr
al e
xpon
ents
. (b
) Est
ablis
h be
twee
n w
hich
two
inte
gers
a
give
n sim
ple
surd
lies
. (c
) Rou
nd re
al n
umbe
rs to
an
appr
opria
te
degr
ee o
f acc
urac
y (t
o a
give
n nu
mbe
r of
deci
mal
dig
its).
(d) R
evise
scie
ntifi
c no
tatio
n.
(a) A
pply
the
law
s of e
xpon
ents
to e
xpre
ssio
ns
invo
lvin
g ra
tiona
l exp
onen
ts.
(b) A
dd, s
ubtr
act,
mul
tiply
and
div
ide
simpl
e su
rds.
(c
) Dem
onst
rate
an
unde
rsta
ndin
g of
the
defin
ition
of a
loga
rithm
and
any
law
s ne
eded
to so
lve
real
life
pro
blem
s.
Appl
y an
y la
w o
f log
arith
m to
solv
e re
al li
fe
prob
lem
s.
Man
ipul
ate
alge
brai
c ex
pres
sions
by:
mul
tiply
ing
a bi
nom
ial b
y a
trin
omia
l;
fact
orisi
ng c
omm
on fa
ctor
(rev
ision
);
fact
orisi
ng b
y gr
oupi
ng in
pai
rs;
fa
ctor
ising
trin
omia
ls;
fa
ctor
ising
diff
eren
ce o
f tw
o sq
uare
s (r
evisi
on);
fa
ctor
ising
the
diffe
renc
e an
d su
ms o
f tw
o cu
bes;
and
simpl
ifyin
g, a
ddin
g, su
btra
ctin
g,
mul
tiply
ing
and
divi
sion
of a
lgeb
raic
fr
actio
ns w
ith n
umer
ator
s and
de
nom
inat
ors l
imite
d to
the
poly
nom
ials
cove
red
in fa
ctor
isatio
n.
Revi
se fa
ctor
isatio
n fr
om G
rade
10.
Ta
ke n
ote
and
unde
rsta
nd th
e Re
mai
nder
an
d Fa
ctor
The
orem
s for
pol
ynom
ials
up
to th
e th
ird d
egre
e (p
roof
s of t
he
Rem
aind
er a
nd F
acto
r the
orem
s will
not
be
exa
min
ed).
Fa
ctor
ise th
ird-d
egre
e po
lyno
mia
ls (in
clud
ing
exam
ples
whi
ch re
quire
the
Fact
or T
heor
em).
to s
olve
pro
blem
s (in
clud
ing
inte
rest
, hire
pur
chas
e, in
flatio
n, p
opul
atio
n gr
owth
and
oth
er re
al li
fe p
robl
ems)
.
•S
treng
then
the
Gra
de 1
1 w
ork.
•Th
e im
plic
atio
ns o
f fluc
tuat
ing
fore
ign
exch
ange
rate
s.•
The
effe
ct o
f diff
eren
t per
iods
of c
ompo
undi
ng g
row
th
and
deca
y (in
clud
ing
effe
ctiv
e an
d no
min
al in
tere
st
rate
s).
•C
ritic
ally
ana
lyse
diff
eren
t loa
n op
tions
.
4. A
LGEB
RA
•(a
) Sim
plify
ex
pres
sion
s us
ing
the
law
s of
ex
pone
nts
for i
nteg
ral e
xpon
ents
.
(b) E
stab
lish
betw
een
whi
ch t
wo
inte
gers
a
give
n si
mpl
e su
rd li
es.
(c) R
ound
re
al
num
bers
to
an
ap
prop
riate
de
gree
of
accu
racy
(to
a g
iven
num
ber
of
deci
mal
dig
its).
(d) R
evis
e sc
ient
ific
nota
tion.
•(a
) App
ly th
e la
ws
of e
xpon
ents
to e
xpre
ssio
ns in
volv
ing
ratio
nal e
xpon
ents
.
(b) A
dd, s
ubtra
ct, m
ultip
ly a
nd d
ivid
e si
mpl
e su
rds.
(c) D
emon
stra
te a
n un
ders
tand
ing
of th
e de
finiti
on
of a
loga
rithm
and
any
law
s ne
eded
to s
olve
real
life
pr
oble
ms.
•A
pply
any
law
of l
ogar
ithm
to s
olve
real
lif
e pr
oble
ms.
•M
anip
ulat
e al
gebr
aic
expr
essi
ons
by:
• m
ultip
lyin
g a
bino
mia
l by
a tri
nom
ial;
• fa
ctor
isin
g co
mm
on fa
ctor
(rev
isio
n);
• fa
ctor
isin
g by
gro
upin
g in
pai
rs;
• fa
ctor
isin
g tri
nom
ials
;•
fact
oris
ing
diffe
renc
e of
two
squa
res
(rev
isio
n);
• fa
ctor
isin
g th
e di
ffere
nce
and
sum
s of
tw
o cu
bes;
and
• si
mpl
ifyin
g, a
ddin
g, s
ubtra
ctin
g,
mul
tiply
ing
and
divi
sion
of a
lgeb
raic
fra
ctio
ns w
ith n
umer
ator
s an
d de
nom
inat
ors
limite
d to
the
poly
nom
ials
co
vere
d in
fact
oris
atio
n.
•R
evis
e fa
ctor
isat
ion
from
Gra
de 1
0.
••
Take
not
e an
d un
ders
tand
the
Rem
aind
er a
nd F
acto
r The
orem
s fo
r pol
ynom
ials
up
to th
e th
ird
degr
ee (p
roof
s of
the
Rem
aind
er
and
Fact
or th
eore
ms
will
not
be
exam
ined
).•
Fact
oris
e th
ird-d
egre
e po
lyno
mia
ls (i
nclu
ding
exa
mpl
es
whi
ch re
quire
the
Fact
or
Theo
rem
).
16 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
4. A
LGEB
RA
•S
olve
:•
linea
r equ
atio
ns;
• qu
adra
tic e
quat
ions
by
fact
oris
atio
n;•
liter
al e
quat
ions
(cha
ngin
g th
e su
bjec
t of
form
ula)
;•
expo
nent
ial e
quat
ions
(acc
eptin
g th
at th
e la
ws
of e
xpon
ents
hol
d fo
r re
al e
xpon
ents
and
sol
utio
ns a
re n
ot
nece
ssar
ily in
tegr
al o
r eve
n ra
tiona
l);•
linea
r ine
qual
ities
in o
ne v
aria
ble
and
illus
trate
the
solu
tion
grap
hica
lly; a
nd•
linea
r equ
atio
ns in
two
varia
bles
si
mul
tane
ousl
y (a
lgeb
raic
ally
and
gr
aphi
cally
).
•S
olve
:•
quad
ratic
equ
atio
ns (b
y fa
ctor
isat
ion
and
by u
sing
th
e qu
adra
tic fo
rmul
a);
• eq
uatio
ns in
two
unkn
owns
, one
of w
hich
is li
near
th
e ot
her q
uadr
atic
, alg
ebra
ical
ly o
r gra
phic
ally.
• E
xplo
re th
e na
ture
of r
oots
thro
ugh
the
valu
e of
Page
19 of
59
Solv
e:
lin
ear e
quat
ions
;
quad
ratic
equ
atio
ns b
y fa
ctor
isatio
n;
lit
eral
equ
atio
ns (c
hang
ing
the
subj
ect o
f fo
rmul
a);
ex
pone
ntia
l equ
atio
ns (a
ccep
ting
that
the
law
s of e
xpon
ents
hol
d fo
r rea
l exp
onen
ts
and
solu
tions
are
not
nec
essa
rily
inte
gral
or
eve
n ra
tiona
l);
lin
ear i
nequ
aliti
es in
one
var
iabl
e an
d ill
ustr
ate
the
solu
tion
grap
hica
lly; a
nd
lin
ear e
quat
ions
in tw
o va
riabl
es
simul
tane
ously
(alg
ebra
ical
ly a
nd
grap
hica
lly).
Solv
e:
qu
adra
tic e
quat
ions
(by
fact
orisa
tion
and
by
usin
g th
e qu
adra
tic fo
rmul
a);
eq
uatio
ns in
two
unkn
owns
, one
of w
hich
is
linea
r the
oth
er q
uadr
atic
, alg
ebra
ical
ly o
r gr
aphi
cally
.
Expl
ore
the
natu
re o
f roo
ts th
roug
h th
e va
lue
of
acb
42
.
De
term
ine
the
natu
re o
f roo
ts a
nd th
e
cond
ition
s for
whi
ch th
e ro
ots a
re re
al,
non-
real
, equ
al, u
nequ
al, r
atio
nal a
nd
irrat
iona
l.
5. D
IFFE
REN
TIAL
CAL
CULU
S AN
D IN
TEGR
ATIO
N
(a
) An
intu
itive
und
erst
andi
ng o
f the
con
cept
of
a li
mit.
(b
) Di
ffere
ntia
tion
of sp
ecifi
ed fu
nctio
ns
from
firs
t prin
cipl
es.
(c)
Use
of t
he sp
ecifi
ed ru
les o
f di
ffere
ntia
tion.
(d
) Th
e eq
uatio
ns o
f tan
gent
s to
grap
hs.
(e)
The
abili
ty to
sket
ch g
raph
s of c
ubic
fu
nctio
ns.
(f)
Prac
tical
pro
blem
s inv
olvi
ng o
ptim
isatio
n an
d ra
tes o
f cha
nge
(incl
udin
g th
e ca
lcul
us o
f mot
ion)
. (g
) B
asic
inte
grat
ion.
6.
EU
CLID
EAN
GEO
MET
RY
(a)
Revi
se b
asic
resu
lts e
stab
lishe
d in
ear
lier
grad
es.
(b)
Inve
stig
ate
and
form
con
ject
ures
abo
ut th
e pr
oper
ties o
f spe
cial
tria
ngle
s (sc
alen
e,
isosc
eles
, equ
ilate
ral a
nd ri
ght-
angl
ed
tria
ngle
) and
qua
drila
tera
ls.
(a)
Inve
stig
ate
theo
rem
s of t
he g
eom
etry
of
circ
les a
ssum
ing
resu
lts fr
om e
arlie
r gr
ades
, tog
ethe
r with
one
oth
er re
sult
conc
erni
ng ta
ngen
ts a
nd ra
dii o
f circ
les.
(b
) So
lve
circ
le g
eom
etry
pro
blem
s, p
rovi
ding
re
ason
s for
stat
emen
ts w
hen
requ
ired.
(a)
Revi
se th
e co
ncep
t of s
imila
rity
and
prop
ortio
nalit
y.
(b)
Appl
y pr
opor
tiona
lity
in tr
iang
les.
(c
) Ap
ply
mid
-poi
nt th
eore
m.
.
••
Det
erm
ine
the
natu
re o
f roo
ts
and
the
cond
ition
s fo
r whi
ch th
e ro
ots
are
real
, non
-rea
l, eq
ual,
uneq
ual,
ratio
nal a
nd ir
ratio
nal.
5. D
IFFE
REN
TIA
L C
ALC
ULU
S A
ND
INTE
GR
ATIO
N•
••
(a) A
n in
tuiti
ve u
nder
stan
ding
of t
he
conc
ept o
f a li
mit.
(b) D
iffer
entia
tion
of s
peci
fied
func
tions
fro
m fi
rst p
rinci
ples
.
(c) U
se o
f the
spe
cifie
d ru
les
of
diffe
rent
iatio
n.
(d) T
he e
quat
ions
of t
ange
nts
to g
raph
s.
(e) T
he a
bilit
y to
ske
tch
grap
hs o
f cub
ic
func
tions
.
(f) P
ract
ical
pro
blem
s in
volv
ing
optim
isat
ion
and
rate
s of
cha
nge
(incl
udin
g th
e ca
lcul
us o
f mot
ion)
.
(g) B
asic
inte
grat
ion.
17CAPS TECHNICAL MATHEMATICS
6. E
UC
LID
EAN
GEO
MET
RY•
(a) R
evis
e ba
sic
resu
lts e
stab
lishe
d in
ear
lier g
rade
s.
(b) I
nves
tigat
e an
d fo
rm c
onje
ctur
es a
bout
the
prop
ertie
s of
spe
cial
tria
ngle
s (s
cale
ne, i
sosc
eles
, eq
uila
tera
l and
righ
t-ang
led
trian
gle)
and
qu
adril
ater
als.
(c) I
nves
tigat
e al
tern
ativ
e (b
ut e
quiv
alen
t) de
finiti
ons
of v
ario
us p
olyg
ons
(incl
udin
g th
e sc
alen
e,
isos
cele
s, e
quila
tera
l and
righ
t-ang
led
trian
gle,
th
e ki
te, p
aral
lelo
gram
, rec
tang
le, r
hom
bus,
sq
uare
and
trap
eziu
m).
(d) A
pplic
atio
n of
the
theo
rem
of P
ytha
gora
s.
•(a
) Inv
estig
ate
theo
rem
s of
the
geom
etry
of c
ircle
s as
sum
ing
resu
lts fr
om e
arlie
r gra
des,
toge
ther
w
ith o
ne o
ther
resu
lt co
ncer
ning
tang
ents
and
ra
dii o
f circ
les.
(b) S
olve
circ
le g
eom
etry
pro
blem
s, p
rovi
ding
re
ason
s fo
r sta
tem
ents
whe
n re
quire
d.
•(a
) Rev
ise
the
conc
ept o
f sim
ilarit
y an
d pr
opor
tiona
lity.
(b) A
pply
pro
porti
onal
ity in
tria
ngle
s.
(c) A
pply
mid
-poi
nt th
eore
m.
7. M
ENSU
RAT
ION
•C
onve
rsio
n of
uni
ts, s
quar
e un
its a
nd c
ubic
uni
ts.
•(a
) Sol
ve p
robl
ems
invo
lvin
g vo
lum
e an
d su
rface
ar
ea o
f sol
ids
stud
ied
in e
arlie
r gra
des
and
com
bina
tions
of t
hose
obj
ects
to fo
rm m
ore
com
plex
sha
ped
solid
s.
(b) D
eter
min
e th
e ar
ea o
f an
irreg
ular
figu
re u
sing
M
id-o
rdin
ate
Rul
e.
•R
evis
e w
ork
stud
ied
in e
arlie
r Gra
des.
8.
CIR
CLE
S, A
NG
LES
AN
D A
NG
ULA
R M
OVE
MEN
T•
• D
efine
a ra
dian
.•
Con
verti
ng d
egre
es to
radi
ans
and
vice
ver
sa.
••
Ang
les
and
arcs
• D
egre
es a
nd ra
dian
s•
Sec
tors
and
seg
men
ts•
Ang
ular
and
circ
umfe
rent
ial v
eloc
ity.
•
9. A
NA
LYTI
CA
L G
EOM
ETRY
•R
epre
sent
geo
met
ric fi
gure
s in
a C
arte
sian
co-
ordi
nate
sys
tem
, and
der
ive
and
appl
y, fo
r any
two
poin
ts (
Page
20 of
59
(c)
Inve
stig
ate
alte
rnat
ive
(but
equ
ival
ent)
de
finiti
ons o
f var
ious
pol
ygon
s (in
clud
ing
the
scal
ene,
isos
cele
s, e
quila
tera
l and
righ
t-an
gled
tria
ngle
, the
kite
, par
alle
logr
am,
rect
angl
e, rh
ombu
s, sq
uare
and
trap
eziu
m).
(d)
Appl
icat
ion
of th
e th
eore
m o
f Pyt
hago
ras.
7. M
ENSU
RATI
ON
Conv
ersio
n of
uni
ts, s
quar
e un
its a
nd c
ubic
un
its.
(a)
Solv
e pr
oble
ms i
nvol
ving
vol
ume
and
surf
ace
area
of s
olid
s stu
died
in e
arlie
r gr
ades
and
com
bina
tions
of t
hose
obj
ects
to
form
mor
e co
mpl
ex sh
aped
solid
s.
(b)
Dete
rmin
e th
e ar
ea o
f an
irreg
ular
figu
re
usin
g M
id-o
rdin
ate
Rule
.
Revi
se w
ork
stud
ied
in e
arlie
r Gra
des.
8.
CIRC
LES,
AN
GLE
S AN
D AN
GULA
R M
OVE
MEN
T
De
fine
a ra
dian
.
Conv
ertin
g de
gree
s to
radi
ans a
nd v
ice
vers
a.
An
gles
and
arc
s
Degr
ees a
nd ra
dian
s
Sect
ors a
nd se
gmen
ts
An
gula
r and
circ
umfe
rent
ial v
eloc
ity.
9. A
NAL
YTIC
AL G
EOM
ETRY
Re
pres
ent g
eom
etric
figu
res i
n a
Cart
esia
n co
-or
dina
te sy
stem
, and
der
ive
and
appl
y, fo
r any
tw
o po
ints
(1
1;y
x) a
nd (
22;
yx
) a fo
rmul
a fo
r ca
lcul
atin
g:
th
e di
stan
ce b
etw
een
the
two
poin
ts;
th
e gr
adie
nt o
f the
line
segm
ent j
oini
ng th
e po
ints
;
the
co-o
rdin
ates
of t
he m
id-p
oint
of t
he
line
segm
ent j
oini
ng th
e po
ints
; and
the
equa
tion
of a
stra
ight
line
join
ing
the
two
poin
ts.
Use
a C
arte
sian
co-o
rdin
ate
syst
em to
de
term
ine:
the
equa
tion
of a
line
thro
ugh
two
give
n po
ints
;
the
equa
tion
of a
line
thro
ugh
one
poin
t an
d pa
ralle
l or p
erpe
ndic
ular
to a
giv
en
line;
and
the
angl
e of
incl
inat
ion
of a
line
.
Use
a tw
o-di
men
siona
l Car
tesia
n co
-ord
inat
e sy
stem
to d
eter
min
e:
th
e eq
uatio
n of
a c
ircle
with
cen
tre
at
the
orig
in (c
entr
e is
(0;0
));
th
e eq
uatio
n of
a ta
ngen
t to
a ci
rcle
at a
gi
ven
poin
t on
the
circ
le; a
nd
po
int/
s of i
nter
sect
ion
of a
circ
le a
nd a
st
raig
ht li
ne.
) and
(
Page
20 of
59
(c)
Inve
stig
ate
alte
rnat
ive
(but
equ
ival
ent)
de
finiti
ons o
f var
ious
pol
ygon
s (in
clud
ing
the
scal
ene,
isos
cele
s, e
quila
tera
l and
righ
t-an
gled
tria
ngle
, the
kite
, par
alle
logr
am,
rect
angl
e, rh
ombu
s, sq
uare
and
trap
eziu
m).
(d)
Appl
icat
ion
of th
e th
eore
m o
f Pyt
hago
ras.
7. M
ENSU
RATI
ON
Conv
ersio
n of
uni
ts, s
quar
e un
its a
nd c
ubic
un
its.
(a)
Solv
e pr
oble
ms i
nvol
ving
vol
ume
and
surf
ace
area
of s
olid
s stu
died
in e
arlie
r gr
ades
and
com
bina
tions
of t
hose
obj
ects
to
form
mor
e co
mpl
ex sh
aped
solid
s.
(b)
Dete
rmin
e th
e ar
ea o
f an
irreg
ular
figu
re
usin
g M
id-o
rdin
ate
Rule
.
Revi
se w
ork
stud
ied
in e
arlie
r Gra
des.
8.
CIRC
LES,
AN
GLE
S AN
D AN
GULA
R M
OVE
MEN
T
De
fine
a ra
dian
.
Conv
ertin
g de
gree
s to
radi
ans a
nd v
ice
vers
a.
An
gles
and
arc
s
Degr
ees a
nd ra
dian
s
Sect
ors a
nd se
gmen
ts
An
gula
r and
circ
umfe
rent
ial v
eloc
ity.
9. A
NAL
YTIC
AL G
EOM
ETRY
Re
pres
ent g
eom
etric
figu
res i
n a
Cart
esia
n co
-or
dina
te sy
stem
, and
der
ive
and
appl
y, fo
r any
tw
o po
ints
(1
1;y
x) a
nd (
22;
yx
) a fo
rmul
a fo
r ca
lcul
atin
g:
th
e di
stan
ce b
etw
een
the
two
poin
ts;
th
e gr
adie
nt o
f the
line
segm
ent j
oini
ng th
e po
ints
;
the
co-o
rdin
ates
of t
he m
id-p
oint
of t
he
line
segm
ent j
oini
ng th
e po
ints
; and
the
equa
tion
of a
stra
ight
line
join
ing
the
two
poin
ts.
Use
a C
arte
sian
co-o
rdin
ate
syst
em to
de
term
ine:
the
equa
tion
of a
line
thro
ugh
two
give
n po
ints
;
the
equa
tion
of a
line
thro
ugh
one
poin
t an
d pa
ralle
l or p
erpe
ndic
ular
to a
giv
en
line;
and
the
angl
e of
incl
inat
ion
of a
line
.
Use
a tw
o-di
men
siona
l Car
tesia
n co
-ord
inat
e sy
stem
to d
eter
min
e:
th
e eq
uatio
n of
a c
ircle
with
cen
tre
at
the
orig
in (c
entr
e is
(0;0
));
th
e eq
uatio
n of
a ta
ngen
t to
a ci
rcle
at a
gi
ven
poin
t on
the
circ
le; a
nd
po
int/
s of i
nter
sect
ion
of a
circ
le a
nd a
st
raig
ht li
ne.
) a fo
rmul
a fo
r cal
cula
ting:
• th
e di
stan
ce b
etw
een
the
two
poin
ts;
• th
e gr
adie
nt o
f the
line
seg
men
t joi
ning
the
poin
ts;
• th
e co
-ord
inat
es o
f the
mid
-poi
nt o
f the
line
se
gmen
t joi
ning
the
poin
ts; a
nd•
the
equa
tion
of a
stra
ight
line
join
ing
the
two
poin
ts.
•U
se a
Car
tesi
an c
o-or
dina
te s
yste
m to
det
erm
ine:
• th
e eq
uatio
n of
a li
ne th
roug
h tw
o gi
ven
poin
ts;
• th
e eq
uatio
n of
a li
ne th
roug
h on
e po
int a
nd
para
llel o
r per
pend
icul
ar to
a g
iven
line
; and
• th
e an
gle
of in
clin
atio
n of
a li
ne.
•U
se a
two-
dim
ensi
onal
Car
tesi
an c
o-or
dina
te s
yste
m to
det
erm
ine:
• th
e eq
uatio
n of
a c
ircle
with
ce
ntre
at t
he o
rigin
(cen
tre is
(0
;0))
;•
the
equa
tion
of a
tang
ent t
o a
circ
le a
t a g
iven
poi
nt o
n th
e ci
rcle
; and
• po
int/s
of i
nter
sect
ion
of a
circ
le
and
a st
raig
ht li
ne.
18 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
10.
TRIG
ON
OM
ETRY
•(a
) Defi
nitio
ns o
f the
trig
onom
etric
func
tions
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
trian
gles
.Tak
e no
te
that
ther
e ar
e sp
ecia
l nam
es fo
r the
reci
proc
als
of th
e tri
gono
met
ric fu
nctio
ns
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
.
(b) E
xten
d th
e de
finiti
ons
of
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
to
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
calc
ulat
e tri
gono
met
ric ra
tios.
(c) S
impl
ifica
tion
of tr
igon
omet
ric e
xpre
ssio
ns/
equa
tions
by
mak
ing
use
of a
cal
cula
tor.
(d) S
olve
sim
ple
trigo
nom
etric
equ
atio
ns fo
r ang
les
betw
een
00 and
900 .
•(a
) U
se th
e id
entit
ies:
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
, and
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
(b) R
educ
tion
form
ulae
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
.
(c) D
eter
min
e th
e so
lutio
ns o
f trig
onom
etric
eq
uatio
ns fo
r
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
.
(d) A
pply
sin
e, c
osin
e an
d ar
ea ru
les
(pro
ofs
of
thes
e ru
les
will
not
be
exam
ined
).
••
App
ly tr
igon
omet
ric id
entit
ies
stud
ied
in e
arlie
r gra
des
to p
rove
th
at le
ft ha
nd s
ide
equa
ls to
righ
t ha
nd s
ide.
•S
olve
pro
blem
s in
two
dim
ensi
ons
by u
sing
the
abov
e tri
gono
met
ric fu
nctio
ns a
nd b
y co
nstru
ctin
g an
d in
terp
retin
g ge
omet
ric a
nd tr
igon
omet
ric m
odel
s.
•S
olve
pro
blem
s in
two
dim
ensi
ons
by u
sing
sin
e,
cosi
ne a
nd a
rea
rule
.•
Sol
ve p
robl
ems
in tw
o an
d th
ree
dim
ensi
ons
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
and
trig
onom
etric
m
odel
s. O
nly
angl
es a
nd n
umer
ical
di
stan
ces/
leng
ths
shou
ld b
e us
ed.
•D
raw
the
grap
hs o
f
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
.•
The
effe
cts
of th
e pa
ram
eter
s on
the
grap
hs
defin
ed b
y
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
.
The
effe
cts
of
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
on
the
grap
hs o
f
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
,
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
and
Page
21 of
59
10.
TRIG
ON
OM
ETRY
(a
) De
finiti
ons o
f the
trig
onom
etric
func
tions
si
n,
co
s a
nd
angl
edrig
htin
tan
tr
iang
les.
T
ake
note
that
ther
e ar
e sp
ecia
l nam
es fo
r
th
e re
cipr
ocal
s of t
he tr
igon
omet
ric
func
tions
sin1
cose
c;
co
s1se
c a
nd
ta
n1co
t.
(c)
Exte
nd th
e de
finiti
ons o
f
sin
,
cos
and
tan
to
36
00
and
calc
ulat
e tr
igon
omet
ric ra
tios.
(d
) Si
mpl
ifica
tion
of tr
igon
omet
ric
expr
essio
ns/e
quat
ions
by
mak
ing
use
of a
ca
lcul
ator
. (d
) So
lve
simpl
e tr
igon
omet
ric e
quat
ions
for
ang
les b
etw
een
00 and
900 .
(a)
Use
the
iden
titie
s:
cos
sin
tan
,
1co
ssi
n2
2
,
22
sec
tan
1,
and
22
cos
1co
tec
. (b
) Re
duct
ion
form
ulae
, )
180
(
and
)36
0(
. (c
) De
term
ine
the
solu
tions
of t
rigon
omet
ric
equa
tions
for
36
00
. (d
) Ap
ply
sine,
cos
ine
and
area
rule
s (pr
oofs
of
thes
e ru
les w
ill n
ot b
e ex
amin
ed).
Ap
ply
trig
onom
etric
iden
titie
s stu
died
in
earli
er g
rade
s to
prov
e th
at le
ft ha
nd
side
equa
ls to
righ
t han
d sid
e.
Sol
ve p
robl
ems i
n tw
o di
men
sions
by
usin
g th
e ab
ove
trig
onom
etric
func
tions
and
by
cons
truc
ting
and
inte
rpre
ting
geom
etric
and
tr
igon
omet
ric m
odel
s.
Solv
e pr
oble
ms i
n tw
o di
men
sions
by
usin
g sin
e, c
osin
e an
d ar
ea ru
le.
Solv
e pr
oble
ms i
n tw
o an
d th
ree
dim
ensio
ns
by c
onst
ruct
ing
and
inte
rpre
ting
geom
etric
an
d tr
igon
omet
ric m
odel
s. O
nly
angl
es a
nd
num
eric
al d
istan
ces/
leng
ths s
houl
d be
use
d.
Dra
w th
e gr
aphs
of
sin
y,
cos
y a
nd
tan
y.
The
effe
cts o
f the
par
amet
ers o
n th
e gr
aphs
de
fined
by
ky
sin
,
k
yco
s a
nd
ky
tan
.
The
effe
cts o
f p
on
the
grap
hs o
f )
(si
np
y
,
)(
cos
py
a
nd
)ta
n(p
y
. O
ne p
aram
eter
shou
ld b
e te
sted
at a
giv
en
time
if ex
amin
ing
horiz
onta
l shi
ft.
Revi
se w
ork
stud
ied
in e
arlie
r gra
des.
Ro
tatin
g ve
ctor
s (sin
e an
d co
sine
curv
es o
nly)
.
.
One
par
amet
er s
houl
d be
test
ed a
t a g
iven
tim
e if
exam
inin
g ho
rizon
tal s
hift.
•R
evis
e w
ork
stud
ied
in e
arlie
r gra
des.
Rot
atin
g ve
ctor
s (s
ine
and
cosi
ne c
urve
s on
ly).
19CAPS TECHNICAL MATHEMATICS
3.2 Content Clarification with teaching guidelines
In Section 3, content clarification includes:
◦ Teaching guidelines
◦ Sequencing of topics per term
◦ Pacing of topics over the year
• 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 ensures that in the first two terms, more than six topics are covered/taught so that assessment is balanced between paper 1 and 2.
20 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.2.1 Allocation of Teaching Time
Time allocation for Technical Mathematics: 4 hours and 30 minutes, e.g. six forty-five-minute periods per week in Grades 10, 11 and 12.
Terms Grade 10 Grade 11 Grade 12No. ofweeks
No. of weeks
No. of weeks
Term 1IntroductionNumber systemsExponentsMensurationAlgebraic Expressions
23213
Exponents and surdsLogarithmsEquations and inequalities(including nature of roots)Analytical Geometry
324
2
Complex numbersPolynomialsDifferential Calculus
326
Term 2Algebraic ExpressionsEquations and inequalitiesTrigonometryMID-YEAR EXAMS
23
33
Functions and graphsEuclidean GeometryMID-YEAR EXAMS
443
IntegrationAnalytical GeometryEuclidean GeometryMID-YEAR EXAMS
3233
Term 3TrigonometryFunctions and graphsEuclidean GeometryAnalytical Geometry
2341
Circles, angles and angular movementTrigonometryFinance, growth and decay
4
42
Euclidean GeometryTrigonometryRevisionTRIAL EXAMS
2314
Term 4Analytical GeometryCircles, angles and angular movementFinance and growthRevisionEXAMS
11
223
MensurationRevisionEXAMS
333
RevisionEXAMS
35
21CAPS TECHNICAL MATHEMATICS
The
deta
il w
hich
follo
ws
incl
udes
exa
mpl
es a
nd n
umer
ical
refe
renc
es to
the
Ove
rvie
w.
3.
2.2
Sequ
enci
ng a
nd P
acin
g of
Top
ics
M
ATH
EMAT
ICS:
GR
AD
E 10
PA
CE
SETT
ER
Term
1
TER
M 1
W
eeks
W
EEK
1
WEE
K 2
W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K 8
W
EEK
9
WEE
K
10
WEE
K 1
1
Topi
cs
Intro
duct
ion
N
umbe
r sys
tem
s
Expo
nent
s M
ensu
ratio
n
Alge
brai
c Ex
pres
sion
s
Ass
essm
ent
Inve
stig
atio
n or
pro
ject
Te
st
Dat
e co
mpl
eted
Term
2
TER
M 2
W
eeks
W
EEK
1
WEE
K 2
W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K
8 W
EEK
9
W
EEK
10
WEE
K
11
Topi
cs
Alge
brai
c Ex
pres
sion
s Eq
uatio
ns a
nd In
equa
litie
s Tr
igon
omet
ry
Ass
essm
ent
Assi
gnm
ent /
Tes
t
MID
-YEA
R E
XA
MIN
ATIO
N
Dat
e co
mpl
eted
Term
3
TER
M 3
W
eeks
W
EEK
1
WEE
K 2
W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K 8
W
EEK
9
WEE
K 1
0
Topi
cs
Trig
onom
etry
Fu
nctio
ns a
nd g
raph
s
Eucl
idea
n G
eom
etry
An
alyt
ical
G
eom
etry
A
sses
smen
t Te
st
Test
Dat
e co
mpl
eted
Term
4
TER
M 4
Pa
per 1
: 2 h
ours
Pape
r 2: 2
hou
rs
Wee
ks
WEE
K 1
W
EEK
2
WEE
K
3 W
EEK
4
WEE
K
5 W
EEK
6
WEE
K
7 W
EEK
8
WEE K 9
W
EEK
10
Al
gebr
aic
ex
pres
sion
s
and
equa
tions
(in
clud
ing
ineq
ualit
ies
and
expo
nent
s )
Fina
nce
and
grow
th
Func
tions
and
gr
aphs
(exc
ludi
ng
trig.
func
tions
)
60
15
25
Eucl
idea
n G
eom
etry
An
alyt
ical
G
eom
etry
Tr
igon
omet
ry
Men
sura
tion,
ci
rcle
s, a
ngle
s an
d an
gula
r m
ovem
ent
30
15
40
15
Topi
cs
Anal
ytic
al
Geo
met
ry
Circ
les,
an
gles
an
d
angu
lar
mov
emen
t
Fina
nce
and
gr
owth
R
evis
ion
Ad
min
Ass
essm
ent
Test
Ex
amin
atio
ns
Dat
e co
mpl
eted
To
tal m
arks
10
0 To
tal m
arks
10
0
22 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.2.
2 Se
quen
cing
and
Pac
ing
of T
opic
s
MAT
HEM
ATIC
S: G
RA
DE
11 P
AC
E SE
TTER
Te
rm 1
TE
RM
1
Wee
ks
WEE
K
1 W
EEK
2
WEE
K 3
W
EEK
4
WEE
K 5
W
EEK
6
WEE
K 7
W
EEK
8
WEE
K 9
W
EEK
10
WEE
K 1
1
Topi
cs
Expo
nent
s an
d su
rds
Loga
rithm
s Eq
uatio
ns a
nd in
equa
litie
s N
atur
e of
ro
ots
Anal
ytic
al G
eom
etry
Ass
essm
ent
Inve
stig
atio
n or
pro
ject
Te
st
Dat
e co
mpl
eted
Term
2
TER
M 2
W
eeks
W
EEK
1
WEE
K
2 W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K 8
W
EEK
9
WEE
K 1
0 W
EEK
11
Topi
cs
Func
tions
and
gra
phs
Eucl
idea
n G
eom
etry
Trig
onom
etry
Ass
essm
ent
Assi
gnm
ent /
Tes
t
MID
-YEA
R E
XA
MIN
ATIO
N
Dat
e co
mpl
eted
Term
3
TER
M 3
W
eeks
W
EEK
1
WEE
K 2
W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K 8
W
EEK
9
WEE
K 1
0 To
pics
C
ircle
s, a
ngle
s an
d an
gula
r mov
emen
t Tr
igon
omet
ry
Fina
nce,
gro
wth
and
de
cay
A
sses
smen
t Te
st
Test
Dat
e co
mpl
eted
Term
4
TER
M 4
Pa
per 1
: 3 h
ours
Pape
r 2: 3
hou
rs
Wee
ks
WEE
K
1 W
EEK
2
WEE
K
3 W
EEK
4
WEE
K5
WEE
K
6 W
EEK
7
WEE
K
8 W
EEK
9
WEE
K
10
Alge
brai
c ex
pres
sion
s,
equa
tions
, in
equa
litie
s
and
natu
re o
f roo
ts
Func
tions
and
gr
aphs
(e
xclu
ding
trig
. fu
nctio
ns)
Fina
nce,
gro
wth
and
de
cay
90
45
15
Eucl
idea
n
Geo
met
ry
Anal
ytic
al
Geo
met
ry
Trig
onom
etry
a
nd
Men
sura
tion,
ci
rcle
s,
angl
es
and
angu
lar
m
ovem
ent
40
25
50
35
Topi
cs
Men
sura
tion
Rev
isio
n
FIN
AL E
XA
MIN
ATIO
N
Adm
in
Ass
essm
ent
Test
Dat
e co
mpl
eted
To
tal m
arks
15
0 To
tal m
arks
15
0
23CAPS TECHNICAL MATHEMATICS
M
ATH
EMAT
ICS:
GR
AD
E 12
PA
CE
SETT
ER
Term
1
TER
M 1
W
eeks
W
EEK
1
WEE
K 2
W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K 8
W
EEK
9
WEE
K 1
0 W
EEK
11
Topi
cs
Com
plex
num
bers
Func
tions
: P
olyn
omia
ls
Diff
eren
tial C
alcu
lus
Ass
essm
ent
Test
In
vest
igat
ion
or p
roje
ct
Ass
ignm
ent /
Tes
t D
ate
com
plet
ed
Term
2
TER
M 2
W
eeks
W
EEK
1
WEE
K 2
W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K 8
W
EEK
9
WEE
K
10
WEE
K 1
1
Topi
cs
Inte
grat
ion
Ana
lytic
al G
eom
etry
E
uclid
ean
Geo
met
ry
Ass
essm
ent
Test
MID
-YE
AR
EX
AM
INA
TIO
N
Dat
e co
mpl
eted
Term
3
TER
M 3
W
eeks
W
EEK
1
WEE
K 2
W
EEK
3
WEE
K 4
W
EEK
5
WEE
K 6
W
EEK
7
WEE
K 8
W
EEK
9
WEE
K 1
0 To
pics
E
uclid
ean
Geo
met
ry
Tr
igon
omet
ry
Rev
isio
n
Ass
essm
ent
Test
TRIA
L EX
AMIN
ATI
ON
Dat
e co
mpl
eted
Term
4
TER
M 4
P
aper
1: 3
hou
rs
Pap
er 2
: 3 h
ours
WEE
K
1 W
EEK
2
WEE
K
3 W
EEK
4
WEE
K
5 W
EEK
6
WEE
K
7 W
EEK
8
WEE
K
9 W
EEK
10
A
lgeb
raic
exp
ress
ions
an
d eq
uatio
ns (a
nd
ineq
ualit
ies,
logs
and
co
mpl
ex n
umbe
rs)
Func
tions
and
gra
phs
(exc
ludi
ng tr
ig.
func
tions
) Fi
nanc
e, g
row
th a
nd
deca
y D
iffer
entia
l Cal
culu
s an
d in
tegr
atio
n
50
35
15
50
Euc
lidea
n G
eom
etry
A
naly
tical
G
eom
etry
Tr
igon
omet
ry
and
M
ensu
ratio
n,
circ
les,
ang
les
and
angu
lar
mov
emen
t
40
25
50
35
Rev
isio
n FI
NA
L E
XA
MIN
ATIO
N
A
dmin
Dat
e
com
plet
ed
Tota
l mar
ks
150
Tota
l mar
ks
150
24 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.2.3 Topicallocationpertermandclarification
GRADE 10: TERM 1Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 IntroductionMathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Numbersystems
• Understand that real numbers can be natural, whole, integers, rational, irrational.
• Introduce binary and complex numbers.
• Round real numbers off to a significant/appropriate degree of accuracy.
• Convert rational numbers into decimal numbers and vice versa.
• Determine between which two integers a given simple surd lies.
• Set builder notation, interval notation and number lines.
• Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
• Deal with imaginary, binary and complex numbers• Binary numbers – basic operations including
addition, subtraction, multiplication and division of whole numbers)
• Complex 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.
Page 27 of 59
3.2.3 Topic allocation per term and clarification
GRADE 10: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Introduction Mathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Number systems
Understand that real numbers can be natural, whole, integers, rational, irrational.
Introduce imaginary, binary and complex numbers.
Round real numbers off to a significant/appropriate degree of accuracy.
Convert rational numbers into decimal numbers and vice versa.
Determine between which two integers a given simple surd lies.
Set builder notation, interval notation and number lines.
Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
Deal with imaginary, binary and complex numbers Binary numbers – basic operations including addition,
subtraction, multiplication and division of whole numbers) Complex 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 (R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
0, yx and Znm , .
nmnm xxx nmnm xxx mnnm xx )( mmm xyyx )(
Also by definition:
nn
xx 1 , 0x , and 10 x ,
Comment: Revise prime base numbers and prime factorisation Examples: 1. 32 22 (K)
2. 1
1
39
x
x (R)
Solve for x : 3. 273 x (K) 4. 366 x (K)
1100101
111
(R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
Page 27 of 59
3.2.3 Topic allocation per term and clarification
GRADE 10: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Introduction Mathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Number systems
Understand that real numbers can be natural, whole, integers, rational, irrational.
Introduce imaginary, binary and complex numbers.
Round real numbers off to a significant/appropriate degree of accuracy.
Convert rational numbers into decimal numbers and vice versa.
Determine between which two integers a given simple surd lies.
Set builder notation, interval notation and number lines.
Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
Deal with imaginary, binary and complex numbers Binary numbers – basic operations including addition,
subtraction, multiplication and division of whole numbers) Complex 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 (R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
0, yx and Znm , .
nmnm xxx nmnm xxx mnnm xx )( mmm xyyx )(
Also by definition:
nn
xx 1 , 0x , and 10 x ,
Comment: Revise prime base numbers and prime factorisation Examples: 1. 32 22 (K)
2. 1
1
39
x
x (R)
Solve for x : 3. 273 x (K) 4. 366 x (K)
1100101
111
and
Page 27 of 59
3.2.3 Topic allocation per term and clarification
GRADE 10: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Introduction Mathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Number systems
Understand that real numbers can be natural, whole, integers, rational, irrational.
Introduce imaginary, binary and complex numbers.
Round real numbers off to a significant/appropriate degree of accuracy.
Convert rational numbers into decimal numbers and vice versa.
Determine between which two integers a given simple surd lies.
Set builder notation, interval notation and number lines.
Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
Deal with imaginary, binary and complex numbers Binary numbers – basic operations including addition,
subtraction, multiplication and division of whole numbers) Complex 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 (R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
0, yx and Znm , .
nmnm xxx nmnm xxx mnnm xx )( mmm xyyx )(
Also by definition:
nn
xx 1 , 0x , and 10 x ,
Comment: Revise prime base numbers and prime factorisation Examples: 1. 32 22 (K)
2. 1
1
39
x
x (R)
Solve for x : 3. 273 x (K) 4. 366 x (K)
1100101
111
.
• nmnm xxx +=×
• nmnm xxx −=÷
•
Page 27 of 59
3.2.3 Topic allocation per term and clarification
GRADE 10: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Introduction Mathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Number systems
Understand that real numbers can be natural, whole, integers, rational, irrational.
Introduce imaginary, binary and complex numbers.
Round real numbers off to a significant/appropriate degree of accuracy.
Convert rational numbers into decimal numbers and vice versa.
Determine between which two integers a given simple surd lies.
Set builder notation, interval notation and number lines.
Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
Deal with imaginary, binary and complex numbers Binary numbers – basic operations including addition,
subtraction, multiplication and division of whole numbers) Complex 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 (R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
0, yx and Znm , .
nmnm xxx nmnm xxx mnnm xx )( mmm xyyx )(
Also by definition:
nn
xx 1 , 0x , and 10 x ,
Comment: Revise prime base numbers and prime factorisation Examples: 1. 32 22 (K)
2. 1
1
39
x
x (R)
Solve for x : 3. 273 x (K) 4. 366 x (K)
1100101
111
•
Page 27 of 59
3.2.3 Topic allocation per term and clarification
GRADE 10: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Introduction Mathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Number systems
Understand that real numbers can be natural, whole, integers, rational, irrational.
Introduce imaginary, binary and complex numbers.
Round real numbers off to a significant/appropriate degree of accuracy.
Convert rational numbers into decimal numbers and vice versa.
Determine between which two integers a given simple surd lies.
Set builder notation, interval notation and number lines.
Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
Deal with imaginary, binary and complex numbers Binary numbers – basic operations including addition,
subtraction, multiplication and division of whole numbers) Complex 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 (R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
0, yx and Znm , .
nmnm xxx nmnm xxx mnnm xx )( mmm xyyx )(
Also by definition:
nn
xx 1 , 0x , and 10 x ,
Comment: Revise prime base numbers and prime factorisation Examples: 1. 32 22 (K)
2. 1
1
39
x
x (R)
Solve for x : 3. 273 x (K) 4. 366 x (K)
1100101
111
Also by definition:
• nn
xx 1=− , 0≠x ,and
10 =x , 0≠x . 2. Use the laws of exponents
to simplify expressions and solve easy exponential equations (the exponents may only be whole numbers).
3. Revise scientific notation.
Comment: Revise prime base numbers and prime factorisation
Examples:1. 32 22 × (K)
2. 1
1
39
−
+
x
x(R)
Solve for x :3.
Page 27 of 59
3.2.3 Topic allocation per term and clarification
GRADE 10: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Introduction Mathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Number systems
Understand that real numbers can be natural, whole, integers, rational, irrational.
Introduce imaginary, binary and complex numbers.
Round real numbers off to a significant/appropriate degree of accuracy.
Convert rational numbers into decimal numbers and vice versa.
Determine between which two integers a given simple surd lies.
Set builder notation, interval notation and number lines.
Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
Deal with imaginary, binary and complex numbers Binary numbers – basic operations including addition,
subtraction, multiplication and division of whole numbers) Complex 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 (R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
0, yx and Znm , .
nmnm xxx nmnm xxx mnnm xx )( mmm xyyx )(
Also by definition:
nn
xx 1 , 0x , and 10 x ,
Comment: Revise prime base numbers and prime factorisation Examples: 1. 32 22 (K)
2. 1
1
39
x
x (R)
Solve for x : 3. 273 x (K) 4. 366 x (K)
1100101
111
(K)4.
Page 27 of 59
3.2.3 Topic allocation per term and clarification
GRADE 10: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Introduction Mathematical language and concepts used in previous years are revised.
All basic algebra must be revised.
3 Number systems
Understand that real numbers can be natural, whole, integers, rational, irrational.
Introduce imaginary, binary and complex numbers.
Round real numbers off to a significant/appropriate degree of accuracy.
Convert rational numbers into decimal numbers and vice versa.
Determine between which two integers a given simple surd lies.
Set builder notation, interval notation and number lines.
Use real numbers as the set of points and explain each set of numbers on a line. (Natural, whole, integers, rational, irrational).
Deal with imaginary, binary and complex numbers Binary numbers – basic operations including addition,
subtraction, multiplication and division of whole numbers) Complex 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 (R) 2. (R)
2 Exponents
1. Revise laws of exponents studied in Grade 9 where
0, yx and Znm , .
nmnm xxx nmnm xxx mnnm xx )( mmm xyyx )(
Also by definition:
nn
xx 1 , 0x , and 10 x ,
Comment: Revise prime base numbers and prime factorisation Examples: 1. 32 22 (K)
2. 1
1
39
x
x (R)
Solve for x : 3. 273 x (K) 4. 366 x (K)
1100101
111
(K)
Comment: Make sure to do examples of very big and very small numbers.
1 Mensuration
1. Conversion of units and square units and cubic units.
All conversions should be done both ways.
2. Applications in technical fields.
• Units of length: (mm, cm, m, km) e.g. 1000m = 1km
• Units of area: (mm2, cm2, m2) e.g. 1 m2 = 1 000 000 mm2
• Units of volume: (ml = cm3, m3, dm3 = 1l).• Typical practical examples on the learners’ level
should be introduced.
25CAPS TECHNICAL MATHEMATICS
GRADE 10: TERM 1Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Algebraic expressions
1. Revise notation (interval, set builder, number line, sets).
2. Adding and subtracting of algebraic terms.
3. Multiplication of a binomial by a binomial.
4. Multiplication of a binomial by a trinomial.
5. Determine the HCF and LCM of not more than three numerical or monomial algebraic expressions by making use of factorisation.
6. Factorisation of the following types:
• common factors• grouping in pairs• difference of two squares• addition/subtraction of two
cubes• trinomials
Examples 1. Subtract ba 84 + from
Page 28 of 59
0x .
2. Use the laws of exponents to simplify expressions and solve easy exponential equations (the exponents may only be whole numbers).
3. Revise scientific notation.
Comment: Make sure to do examples of very big and very small numbers.
1
Mensuration
1. Conversion of units and square units and cubic units.
All conversions should be done both ways. 2. Applications in technical fields.
Units of length: (mm, cm, m, km) e.g. 1000m = 1km Units of area: (mm2, cm2, m2) e.g. 1 m2 = 1 000 000 mm2 Units of volume: (ml = cm3, m3, dm3 = 1l). Typical practical examples on the learners’ level should be
introduced.
3
Algebraic expressions
1. Revise notation (interval, set builder, number line, sets).
2. Adding and subtracting of algebraic terms.
3. Multiplication of a binomial by a binomial.
4. Multiplication of a binomial by a trinomial.
5. Determine the HCF and LCM of not more than three numerical or monomial algebraic expressions by making use of factorisation.
6. Factorisation of the following types:
common factors grouping in pairs difference of two
squares addition/subtraction of
two cubes trinomials
Examples 1. Subtract ba 84 from ba 105 (K) Remove brackets 2. 22 xx (K)
3. 2)54( yx (K) 4. yxyx 32 (K)
5. 1232 2 aaa (R) Factorise 6. xyxyyx 642 22 (K) 7. cbacab 22 (R) 8. 22 82 ba (K) 9. 33 8yx (R)
10. 4472 xx (R)
(K)
Remove brackets
2. ( )( ) =−+ 22 xx (K)
3. =− 2)54( yx (K)
4. ( )( ) =−− yxyx 32 (K)
5. ( )( ) =−−+ 1232 2 aaa (R)
Factorise
6.
Page 28 of 59
0x .
2. Use the laws of exponents to simplify expressions and solve easy exponential equations (the exponents may only be whole numbers).
3. Revise scientific notation.
Comment: Make sure to do examples of very big and very small numbers.
1
Mensuration
1. Conversion of units and square units and cubic units.
All conversions should be done both ways. 2. Applications in technical fields.
Units of length: (mm, cm, m, km) e.g. 1000m = 1km Units of area: (mm2, cm2, m2) e.g. 1 m2 = 1 000 000 mm2 Units of volume: (ml = cm3, m3, dm3 = 1l). Typical practical examples on the learners’ level should be
introduced.
3
Algebraic expressions
1. Revise notation (interval, set builder, number line, sets).
2. Adding and subtracting of algebraic terms.
3. Multiplication of a binomial by a binomial.
4. Multiplication of a binomial by a trinomial.
5. Determine the HCF and LCM of not more than three numerical or monomial algebraic expressions by making use of factorisation.
6. Factorisation of the following types:
common factors grouping in pairs difference of two
squares addition/subtraction of
two cubes trinomials
Examples 1. Subtract ba 84 from ba 105 (K) Remove brackets 2. 22 xx (K)
3. 2)54( yx (K) 4. yxyx 32 (K)
5. 1232 2 aaa (R) Factorise 6. xyxyyx 642 22 (K) 7. cbacab 22 (R) 8. 22 82 ba (K) 9. 33 8yx (R)
10. 4472 xx (R)
(K)7.
Page 28 of 59
0x .
2. Use the laws of exponents to simplify expressions and solve easy exponential equations (the exponents may only be whole numbers).
3. Revise scientific notation.
Comment: Make sure to do examples of very big and very small numbers.
1
Mensuration
1. Conversion of units and square units and cubic units.
All conversions should be done both ways. 2. Applications in technical fields.
Units of length: (mm, cm, m, km) e.g. 1000m = 1km Units of area: (mm2, cm2, m2) e.g. 1 m2 = 1 000 000 mm2 Units of volume: (ml = cm3, m3, dm3 = 1l). Typical practical examples on the learners’ level should be
introduced.
3
Algebraic expressions
1. Revise notation (interval, set builder, number line, sets).
2. Adding and subtracting of algebraic terms.
3. Multiplication of a binomial by a binomial.
4. Multiplication of a binomial by a trinomial.
5. Determine the HCF and LCM of not more than three numerical or monomial algebraic expressions by making use of factorisation.
6. Factorisation of the following types:
common factors grouping in pairs difference of two
squares addition/subtraction of
two cubes trinomials
Examples 1. Subtract ba 84 from ba 105 (K) Remove brackets 2. 22 xx (K)
3. 2)54( yx (K) 4. yxyx 32 (K)
5. 1232 2 aaa (R) Factorise 6. xyxyyx 642 22 (K) 7. cbacab 22 (R) 8. 22 82 ba (K) 9. 33 8yx (R)
10. 4472 xx (R)
(R)
8. 22 82 ba − (K)
9. 33 8yx − (R) 10.
a
a is not restricted to in grade 11.
872 xx
sin1 cosec
a for parabola graphs only in grade 10
22 cosec1cot
(R)
Assessment Term 1:
1. Investigation or project (only one project or one investigation 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 to each 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 faces and 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.
Care needs to be taken to set questions on all four cognitive levels: approximately 20% knowledge, approximately 35% routine procedures, 30% complex procedures and 15% problem-solving.
26 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 10: TERM 2
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 AlgebraicExpressions
(continue)
7. Do addition, subtraction, multiplication and division of algebraic fractions using factorisation (numerators and denominators should be limited to the polynomials covered in factorisation).
Simplify
11.
Page 30 of 59
GRADE 10: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2
Algebraic Expressions (continue)
7. Do addition, subtraction, multiplication and division of algebraic fractions using factorisation (numerators and denominators should be limited to the polynomials covered in factorisation).
Simplify
11. xybybxxyx
bybxxyyx
222
(R)
12. 4
51
423
322
aaaa
a (R)
3 Equations
and Inequalities
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 by
factorisation 3. Solve simultaneous linear equations
with two variables 4.1 Do basic Grade 8 & 9 word
problems. 4.2 Solve word problems involving
linear, quadratic or simultaneous linear equations.
5. Solve simple linear inequalities (and show solution graphically).
6. Manipulation of formulae (technical related).
Examples Solve for x : 1. 843 x (K)
2. 2133
x
x (R)
3. 0242 xx (K) Solve for x and y : 4. 43 yx and 62 yx (R) 5. atuv (Change the subject of the formula to a ) (R)
6. 221 mvE (Change the subject of the formula to v ) (R)
3 Trigonometry 1. Know definitions of the trigonometric ratios sin , cos and
1. Determine the values of cos and tan if 53sin and
(R)
12. 4
51
423
322 −
−+
+++
−aaaa
a (R)
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 by factorisation
3. Solve simultaneous linear equations with two variables
4.1 Do basic Grade 8 & 9 word problems.
4.2 Solve word problems involving linear, quadratic or simultaneous linear equations.
5. Solve simple linear inequalities (and show solution graphically).
6. Manipulation of formulae (technical related).
ExamplesSolve for x :1. 843 =+x (K)
2. 2133 =+
−x
x (R)
3. 0242 =+− xx (K)
Solve for x and y :4. 43 =+ yx and 62 =+ yx (R)
5.
Page 30 of 59
GRADE 10: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2
Algebraic Expressions (continue)
7. Do addition, subtraction, multiplication and division of algebraic fractions using factorisation (numerators and denominators should be limited to the polynomials covered in factorisation).
Simplify
11. xybybxxyx
bybxxyyx
222
(R)
12. 4
51
423
322
aaaa
a (R)
3 Equations
and Inequalities
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 by
factorisation 3. Solve simultaneous linear equations
with two variables 4.1 Do basic Grade 8 & 9 word
problems. 4.2 Solve word problems involving
linear, quadratic or simultaneous linear equations.
5. Solve simple linear inequalities (and show solution graphically).
6. Manipulation of formulae (technical related).
Examples Solve for x : 1. 843 x (K)
2. 2133
x
x (R)
3. 0242 xx (K) Solve for x and y : 4. 43 yx and 62 yx (R) 5. atuv (Change the subject of the formula to a ) (R)
6. 221 mvE (Change the subject of the formula to v ) (R)
3 Trigonometry 1. Know definitions of the trigonometric ratios sin , cos and
1. Determine the values of cos and tan if 53sin and
(Change the subject of the formula to a ) (R)6.
Page 30 of 59
GRADE 10: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2
Algebraic Expressions (continue)
7. Do addition, subtraction, multiplication and division of algebraic fractions using factorisation (numerators and denominators should be limited to the polynomials covered in factorisation).
Simplify
11. xybybxxyx
bybxxyyx
222
(R)
12. 4
51
423
322
aaaa
a (R)
3 Equations
and Inequalities
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 by
factorisation 3. Solve simultaneous linear equations
with two variables 4.1 Do basic Grade 8 & 9 word
problems. 4.2 Solve word problems involving
linear, quadratic or simultaneous linear equations.
5. Solve simple linear inequalities (and show solution graphically).
6. Manipulation of formulae (technical related).
Examples Solve for x : 1. 843 x (K)
2. 2133
x
x (R)
3. 0242 xx (K) Solve for x and y : 4. 43 yx and 62 yx (R) 5. atuv (Change the subject of the formula to a ) (R)
6. 221 mvE (Change the subject of the formula to v ) (R)
3 Trigonometry 1. Know definitions of the trigonometric ratios sin , cos and
1. Determine the values of cos and tan if 53sin and
(Change the subject of the formula to v ) (R)
27CAPS TECHNICAL MATHEMATICS
GRADE 10: TERM 2
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Trigonometry
1. Know definitions of the trigonometric ratios sinθ , θcos and tanθ , using right-angled triangles for
°≤θ≤° 3600 . 2. Introduce the
reciprocals of the 3basic trigonometric ratios, sinθ , θcos and tanθ :
a
a is not restricted to in grade 11.
872 xx
sin1 cosec
a for parabola graphs only in grade 10
22 cosec1cot
,
θ=θ cos1sec and
θ=θ tan
1cot .
3. Trigonometric ratios in each of the quadrants are calculated where one ratio in the quadrant is given by making use of diagrams.
4. Practise the use of a calculator for questions applicable to trigonometry.
5. Solve simple trigonometric equations for angles between 00 and 900.
6. Solve two-dimensional problems involving right-angled triangles.
7. Trigonometry graphs
• θ= sinay , θ= cosay and θ= tanay for
°≤θ≤° 3600 .
• qay +θ= sin and qay +θ= cos for °≤θ≤° 3600 .
1. Determine the values of θcos and θtan if 53sin =θ and
Page 31 of 59
tan , using right-angled triangles for 3600 .
2. Introduce the reciprocals of the 3basic trigonometric ratios, sin ,
cos and tan : sin1cosec ,
cos1sec and
tan
1cot .
3. Trigonometric ratios in each of the quadrants are calculated where one ratio in the quadrant is given by making use of diagrams.
4. Practise the use of a calculator for questions applicable to trigonometry.
5. Solve simple trigonometric equations for angles between 00 and 900.
6. Solve two-dimensional problems involving right-angled triangles.
7. Trigonometry graphs sinay , cosay and
tanay for 3600 . qay sin and
qay cos for 3600 .
36090 . (C)
Making use of Pythagoras’ theorem.
2. Draw the graph of sin3y for 3600 by making the use of a table, point by point plotting and identify the following: - asymptotes - axes of symmetry - the domain and range. (C) Note: - It is very important to be able to read off values from the graph. - Investigate the effect of a and q on the graph.
3 Mid-year examinations
Assessment Term 2:
1. Assignment / test (at least 50 marks). Note that the weight of a test is equal to the weight of an assignment. 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)
x
y
5 31416
0
(C)
Page 31 of 59
tan , using right-angled triangles for 3600 .
2. Introduce the reciprocals of the 3basic trigonometric ratios, sin ,
cos and tan : sin1cosec ,
cos1sec and
tan
1cot .
3. Trigonometric ratios in each of the quadrants are calculated where one ratio in the quadrant is given by making use of diagrams.
4. Practise the use of a calculator for questions applicable to trigonometry.
5. Solve simple trigonometric equations for angles between 00 and 900.
6. Solve two-dimensional problems involving right-angled triangles.
7. Trigonometry graphs sinay , cosay and
tanay for 3600 . qay sin and
qay cos for 3600 .
36090 . (C)
Making use of Pythagoras’ theorem.
2. Draw the graph of sin3y for 3600 by making the use of a table, point by point plotting and identify the following: - asymptotes - axes of symmetry - the domain and range. (C) Note: - It is very important to be able to read off values from the graph. - Investigate the effect of a and q on the graph.
3 Mid-year examinations
Assessment Term 2:
1. Assignment / test (at least 50 marks). Note that the weight of a test is equal to the weight of an assignment. 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)
x
y
5 31416
0
Making use of Pythagoras’ theorem.
2. Draw the graph of θ= sin3y for °≤θ≤° 3600 by making the use of a table, point by point plotting and identify the following:
• asymptotes• axes of symmetry• the domain and range. (C)
Note:• It is very important to be able to read off values from
the graph.• Investigate the effect of a and q on the graph.
3 Mid-yearexaminations
Assessment Term 2: 1. Assignment / test (at least 50 marks). Note that the weight of a test is equal to the weight of an assignment.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)
28 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 10: TERM 3
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1AnalyticalGeometry
Represent geometric figures on a Cartesian co-ordinate system.
Apply for any two points );( 11 yx
and );( 22 yx formulae for determining the: 1. distance between the two
points;2. gradient of the line segment
connecting the two points (and from that 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 passing through two points.
Page 32 of 59
GRADE 10: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1
Analytical Geometry
Represent geometric figures on a Cartesian co-ordinate system. Apply for any two points );( 11 yx and
);( 22 yx formulae for determining the: 1. distance between the two points; 2. gradient of the line segment
connecting the two points (and from that 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 passing through two points.
cmxy
Example: Consider the points P(2 ; 5) and Q(-3 ; 1) in the Cartesian plane. (R) 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.
3
Functions
and
Graphs
1. Functional notation 2. Generate graphs by means of point-
by-point plotting supported by available technology.
3. Drawing of the following functions:
Linear function: cmxy (revise)
Investigate the way (unique) output values depend on how the input values vary. The terms independent (input) and dependent (output) variables might be useful.
7.1.1.1 qaxy 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)
Example: Consider the points P(2 ; 5) and Q(-3 ; 1) in the Cartesian plane. (R)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 available technology.
3. Drawing of the following functions:
• Linear function:
Page 32 of 59
GRADE 10: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1
Analytical Geometry
Represent geometric figures on a Cartesian co-ordinate system. Apply for any two points );( 11 yx and
);( 22 yx formulae for determining the: 1. distance between the two points; 2. gradient of the line segment
connecting the two points (and from that 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 passing through two points.
cmxy
Example: Consider the points P(2 ; 5) and Q(-3 ; 1) in the Cartesian plane. (R) 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.
3
Functions
and
Graphs
1. Functional notation 2. Generate graphs by means of point-
by-point plotting supported by available technology.
3. Drawing of the following functions:
Linear function: cmxy (revise)
Investigate the way (unique) output values depend on how the input values vary. The terms independent (input) and dependent (output) variables might be useful.
7.1.1.1 qaxy 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)
(revise)• Quadratic:
• Hyperbola: xay =
• Exponential: xbay .= where 1≠b and 0>b
Note: qpmcba ,,,,, ℝ
1±=a for parabola graphs only.
Investigate the way (unique) output values depend on how the input values vary. The terms independent (input) and dependent (output) variables might be useful.Draw the parabola
Page 32 of 59
GRADE 10: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1
Analytical Geometry
Represent geometric figures on a Cartesian co-ordinate system. Apply for any two points );( 11 yx and
);( 22 yx formulae for determining the: 1. distance between the two points; 2. gradient of the line segment
connecting the two points (and from that 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 passing through two points.
cmxy
Example: Consider the points P(2 ; 5) and Q(-3 ; 1) in the Cartesian plane. (R) 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.
3
Functions
and
Graphs
1. Functional notation 2. Generate graphs by means of point-
by-point plotting supported by available technology.
3. Drawing of the following functions:
Linear function: cmxy (revise)
Investigate the way (unique) output values depend on how the input values vary. The terms independent (input) and dependent (output) variables might be useful.
7.1.1.1 qaxy 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)
for 1±=a 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 q on the graph.
◦ asymptotes where applicable NOTE: Use this approach to engage with all the other graphs.
Page 32 of 59
GRADE 10: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1
Analytical Geometry
Represent geometric figures on a Cartesian co-ordinate system. Apply for any two points );( 11 yx and
);( 22 yx formulae for determining the: 1. distance between the two points; 2. gradient of the line segment
connecting the two points (and from that 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 passing through two points.
cmxy
Example: Consider the points P(2 ; 5) and Q(-3 ; 1) in the Cartesian plane. (R) 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.
3
Functions
and
Graphs
1. Functional notation 2. Generate graphs by means of point-
by-point plotting supported by available technology.
3. Drawing of the following functions:
Linear function: cmxy (revise)
Investigate the way (unique) output values depend on how the input values vary. The terms independent (input) and dependent (output) variables might be useful.
7.1.1.1 qaxy 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)
29CAPS TECHNICAL MATHEMATICS
GRADE 10: TERM 3
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
4EuclideanGeometry
1. Revise basic geometry done in Grades 8 and 9.
Lines and parallel lines, angles, triangles congruency and similarity.
2. Apply the properties of line segments joining the mid-points of two sides of a triangle. Do practical problems.
3. Know the features of the following special quadrilaterals: 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 corresponding angles and calculate the ratio of the corresponding sides.
Investigate: line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of the length of the third side.
Investigate and make conjectures about the properties of the sides, angles and diagonals of these quadrilaterals.
Calculation of one unknown side of a right-angled triangle, using Pythagoras’ theorem.
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.
30 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 10: TERM 4
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1 AnalyticalGeometry Continuation from term 3.
1
Circles,angles and
angularmovement
1. Define a radian2. Indicate the relationship
between degrees and radians, convert radians to degrees or degrees to radians, convert degrees and minutes to radians and radians to degrees and minutes.
Example: (K)• Revise circle terminology: chord, secant,
segment, sector, diameter, radius, arc, etc.• Rediscover unique ratio between circumference
and diameter represented by π .
• Show that any circle π=° 2360 radians and consequently that 283,6360 =° radians and that 1 radian
Page 35 of 59
GRADE 10: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1 Analytical Geometry Continuation from term 3.
1
Circles, angles and
angular movement
1. Define a radian
2. Indicate the relationship between
degrees and radians, convert radians to degrees or degrees to radians, convert degrees and minutes to radians and radians to degrees and minutes.
Example: (K) 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. (K)
(answers in degrees, minutes and seconds). Practical application in the technical field:
- Calculate the angle in radians through which a pulley with a 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 an
angle of 150 . Calculate the distance moved by a point on the tyre tread of the wheel. (C)
Also ensure the following are covered : (R)
- Add 32
4 radians and covert to degrees.
- Simplify: sin2cos0cos90sin
.• 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:
Page 35 of 59
GRADE 10: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1 Analytical Geometry Continuation from term 3.
1
Circles, angles and
angular movement
1. Define a radian
2. Indicate the relationship between
degrees and radians, convert radians to degrees or degrees to radians, convert degrees and minutes to radians and radians to degrees and minutes.
Example: (K) 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. (K)
(answers in degrees, minutes and seconds). Practical application in the technical field:
- Calculate the angle in radians through which a pulley with a 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 an
angle of 150 . Calculate the distance moved by a point on the tyre tread of the wheel. (C)
Also ensure the following are covered : (R)
- Add 32
4 radians and covert to degrees.
- Simplify: sin2cos0cos90sin
etc. • convert radians to degrees: 1,5 rad; 65,98 rad;
16,25 rad; etc. (K) (answers in degrees, minutes and seconds).
• Practical application in the technical field:
◦ Calculate the angle in radians through which a pulley with a 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 an angle of °150 . Calculate the distance moved by a point on the tyre tread of the wheel. (C)
• Also ensure the following are covered : (R)
◦ Add 32
4π+π+π radians and covert to
degrees.
◦ Simplify:
Page 35 of 59
GRADE 10: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
1 Analytical Geometry Continuation from term 3.
1
Circles, angles and
angular movement
1. Define a radian
2. Indicate the relationship between
degrees and radians, convert radians to degrees or degrees to radians, convert degrees and minutes to radians and radians to degrees and minutes.
Example: (K) 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. (K)
(answers in degrees, minutes and seconds). Practical application in the technical field:
- Calculate the angle in radians through which a pulley with a 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 an
angle of 150 . Calculate the distance moved by a point on the tyre tread of the wheel. (C)
Also ensure the following are covered : (R)
- Add 32
4 radians and covert to degrees.
- Simplify: sin2cos0cos90sin
◦ Calculate 4cos2sin π+π
◦ Determine the value of the following: 577,0tan866,0cos5,0sin 111 −−− ++ (answer in
radians).
31CAPS TECHNICAL MATHEMATICS
GRADE 10: TERM 4
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Finance andgrowth
Use the simple and compound growth formulae
3. FINANCE, GROWTH AND DECAY Use simple and compound growth formulae
)1( inPA and niPA )1( to solve problems (including interest, hire purchase, inflation, population growth and other real life problems).
Use simple and compound growth/decay formulae )1( inPA and niPA )1( to solve problems (including interest, hire purchase, inflation, population growth and other real life problems).
Strengthen the Grade 11 work.
The implications of fluctuating foreign exchange rates.
The effect of different periods of compounding growth and decay (including effective and nominal interest rates).
Critically analyse different loan options.
4. ALGEBRA
(a) Simplify expressions using the laws of exponents for integral exponents.
(b) Establish between which two integers a given simple surd lies.
(c) Round real numbers to an appropriate degree of accuracy (to a given number of decimal digits).
(d) Revise scientific notation.
(a) Apply the laws of exponents to expressions involving rational exponents.
(b) Add, subtract, multiply and divide simple surds.
(c) Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real life problems.
Apply any law of logarithm to solve real life problems.
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 and denominators limited to the polynomials covered in factorisation.
Revise factorisation from Grade 10.
Take note and understand the Remainder and Factor Theorems for polynomials up to the third degree (proofs of the Remainder and Factor theorems will not be examined).
Factorise third-degree polynomials (including examples which require the Factor Theorem).
and niPA )1( +=
to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 RevisionAssessment Term 4: 1. Test (at least 50 marks)2. ExaminationPaper 1: 2 hours (100 marks) made up as follows:
Page 36 of 59
- Calculate 4cos2sin
- Determine the value of the following: 577,0tan866,0cos5,0sin 111 (answer in radians).
2 Finance and growth
Use the simple and compound growth formulae )1( inPA and
niPA )1( to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 Revision Assessment Term 4:
Assessment Term 4:
1. Test (at least 50 marks) 2. Examination Paper 1: 2 hours (100 marks made up as follows: 360 on algebraic expressions, equations, inequalities and exponents, 325 on functions and graphs (trig. functions will be examined in paper 2) and 315 on finance and growth. Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry and 315 on mensuration, circles, angles and angular movement.
on algebraic expressions, equations, inequalities and exponents,
Page 36 of 59
- Calculate 4cos2sin
- Determine the value of the following: 577,0tan866,0cos5,0sin 111 (answer in radians).
2 Finance and growth
Use the simple and compound growth formulae )1( inPA and
niPA )1( to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 Revision Assessment Term 4:
Assessment Term 4:
1. Test (at least 50 marks) 2. Examination Paper 1: 2 hours (100 marks made up as follows: 360 on algebraic expressions, equations, inequalities and exponents, 325 on functions and graphs (trig. functions will be examined in paper 2) and 315 on finance and growth. Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry and 315 on mensuration, circles, angles and angular movement.
on functions and graphs (trig. functions will be examined in paper 2) and
Page 36 of 59
- Calculate 4cos2sin
- Determine the value of the following: 577,0tan866,0cos5,0sin 111 (answer in radians).
2 Finance and growth
Use the simple and compound growth formulae )1( inPA and
niPA )1( to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 Revision Assessment Term 4:
Assessment Term 4:
1. Test (at least 50 marks) 2. Examination Paper 1: 2 hours (100 marks made up as follows: 360 on algebraic expressions, equations, inequalities and exponents, 325 on functions and graphs (trig. functions will be examined in paper 2) and 315 on finance and growth. Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry and 315 on mensuration, circles, angles and angular movement.
on finance and growth. Paper 2: 2 hours (100 marks) made up as follows:
Page 36 of 59
- Calculate 4cos2sin
- Determine the value of the following: 577,0tan866,0cos5,0sin 111 (answer in radians).
2 Finance and growth
Use the simple and compound growth formulae )1( inPA and
niPA )1( to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 Revision Assessment Term 4:
Assessment Term 4:
1. Test (at least 50 marks) 2. Examination Paper 1: 2 hours (100 marks made up as follows: 360 on algebraic expressions, equations, inequalities and exponents, 325 on functions and graphs (trig. functions will be examined in paper 2) and 315 on finance and growth. Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry and 315 on mensuration, circles, angles and angular movement.
on trigonometry,
Page 36 of 59
- Calculate 4cos2sin
- Determine the value of the following: 577,0tan866,0cos5,0sin 111 (answer in radians).
2 Finance and growth
Use the simple and compound growth formulae )1( inPA and
niPA )1( to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 Revision Assessment Term 4:
Assessment Term 4:
1. Test (at least 50 marks) 2. Examination Paper 1: 2 hours (100 marks made up as follows: 360 on algebraic expressions, equations, inequalities and exponents, 325 on functions and graphs (trig. functions will be examined in paper 2) and 315 on finance and growth. Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry and 315 on mensuration, circles, angles and angular movement.
on Analytical Geometry,
Page 36 of 59
- Calculate 4cos2sin
- Determine the value of the following: 577,0tan866,0cos5,0sin 111 (answer in radians).
2 Finance and growth
Use the simple and compound growth formulae )1( inPA and
niPA )1( to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 Revision Assessment Term 4:
Assessment Term 4:
1. Test (at least 50 marks) 2. Examination Paper 1: 2 hours (100 marks made up as follows: 360 on algebraic expressions, equations, inequalities and exponents, 325 on functions and graphs (trig. functions will be examined in paper 2) and 315 on finance and growth. Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry and 315 on mensuration, circles, angles and angular movement.
on Euclidean Geometry and
Page 36 of 59
- Calculate 4cos2sin
- Determine the value of the following: 577,0tan866,0cos5,0sin 111 (answer in radians).
2 Finance and growth
Use the simple and compound growth formulae )1( inPA and
niPA )1( to solve problems,
including interest, hire purchase, inflation, population growth and other real life problems. Understand the implication of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).
2 Revision Assessment Term 4:
Assessment Term 4:
1. Test (at least 50 marks) 2. Examination Paper 1: 2 hours (100 marks made up as follows: 360 on algebraic expressions, equations, inequalities and exponents, 325 on functions and graphs (trig. functions will be examined in paper 2) and 315 on finance and growth. Paper 2: 2 hours (100 marks made up as follows: 340 on trigonometry, 315 on Analytical Geometry, 330 on Euclidean Geometry and 315 on mensuration, circles, angles and angular movement. on mensuration, circles, angles and angular movement.
32 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 11: TERM 1Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Exponents and surds
1. Simplify expressions and solve equations using the laws of exponents for rational exponents where
;q pqp
xx = 0;0 >> qx .2. Add, subtract, multiply and
divide simple surds.3. Solve exponential equations.
Example: Without the use of a calculator,
1. Determine the value of 23
9 .(K)
2. Simplify:
Page 37 of 59
GRADE 11: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Exponents and surds
1. Simplify expressions and solve equations using the laws of exponents for rational exponents where
;q pqp
xx 0;0 qx .
2. Add, subtract, multiply and divide simple surds.
3. Solve exponential equations.
Example: Without the use of a calculator,
1. Determine the value of 23
9 . (K) 2. Simplify: )23)(23( . (R) 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
xxx
(R)
6. 3 233
851255
6255
(C)
Solve:
7. 1284 25x (R)
8. 32 525 xx (R) 9. Practical examples from the technical field must be included e.g.
The formula VQC is given with 4106 Q C and
voltV 200 . Determine C without using a calculator. (R)
2
Logarithms
Laws of logarithms
yxxy aaa logloglog yxxy aaa logloglog
1. Explain relationship between logs and exponents and show
conversion from log-form to exponential form and vice versa. 2. Application of laws of logarithms:
.(R) 3. Revise Grade 10 exponential equations.4. Revise all exponential laws from Grade 10.
Examples should include but not be limited to
5.
Page 37 of 59
GRADE 11: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Exponents and surds
1. Simplify expressions and solve equations using the laws of exponents for rational exponents where
;q pqp
xx 0;0 qx .
2. Add, subtract, multiply and divide simple surds.
3. Solve exponential equations.
Example: Without the use of a calculator,
1. Determine the value of 23
9 . (K) 2. Simplify: )23)(23( . (R) 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
xxx
(R)
6. 3 233
851255
6255
(C)
Solve:
7. 1284 25x (R)
8. 32 525 xx (R) 9. Practical examples from the technical field must be included e.g.
The formula VQC is given with 4106 Q C and
voltV 200 . Determine C without using a calculator. (R)
2
Logarithms
Laws of logarithms
yxxy aaa logloglog yxxy aaa logloglog
1. Explain relationship between logs and exponents and show
conversion from log-form to exponential form and vice versa. 2. Application of laws of logarithms:
(R)
6. 3 233
851255
6255−+
×
×
(C)
Solve:
7. 1284 25=x (R)
8.
Page 37 of 59
GRADE 11: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Exponents and surds
1. Simplify expressions and solve equations using the laws of exponents for rational exponents where
;q pqp
xx 0;0 qx .
2. Add, subtract, multiply and divide simple surds.
3. Solve exponential equations.
Example: Without the use of a calculator,
1. Determine the value of 23
9 . (K) 2. Simplify: )23)(23( . (R) 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
xxx
(R)
6. 3 233
851255
6255
(C)
Solve:
7. 1284 25x (R)
8. 32 525 xx (R) 9. Practical examples from the technical field must be included e.g.
The formula VQC is given with 4106 Q C and
voltV 200 . Determine C without using a calculator. (R)
2
Logarithms
Laws of logarithms
yxxy aaa logloglog yxxy aaa logloglog
1. Explain relationship between logs and exponents and show
conversion from log-form to exponential form and vice versa. 2. Application of laws of logarithms:
(R) 9. Practical examples from the technical field must be included e.g.
The formula VQC = is given with
Page 37 of 59
GRADE 11: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Exponents and surds
1. Simplify expressions and solve equations using the laws of exponents for rational exponents where
;q pqp
xx 0;0 qx .
2. Add, subtract, multiply and divide simple surds.
3. Solve exponential equations.
Example: Without the use of a calculator,
1. Determine the value of 23
9 . (K) 2. Simplify: )23)(23( . (R) 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
xxx
(R)
6. 3 233
851255
6255
(C)
Solve:
7. 1284 25x (R)
8. 32 525 xx (R) 9. Practical examples from the technical field must be included e.g.
The formula VQC is given with 4106 Q C and
voltV 200 . Determine C without using a calculator. (R)
2
Logarithms
Laws of logarithms
yxxy aaa logloglog yxxy aaa logloglog
1. Explain relationship between logs and exponents and show
conversion from log-form to exponential form and vice versa. 2. Application of laws of logarithms:
C and
voltV 200= .Determine C without using a calculator. (R)
2 Logarithms
Laws of logarithms•
Page 37 of 59
GRADE 11: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Exponents and surds
1. Simplify expressions and solve equations using the laws of exponents for rational exponents where
;q pqp
xx 0;0 qx .
2. Add, subtract, multiply and divide simple surds.
3. Solve exponential equations.
Example: Without the use of a calculator,
1. Determine the value of 23
9 . (K) 2. Simplify: )23)(23( . (R) 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
xxx
(R)
6. 3 233
851255
6255
(C)
Solve:
7. 1284 25x (R)
8. 32 525 xx (R) 9. Practical examples from the technical field must be included e.g.
The formula VQC is given with 4106 Q C and
voltV 200 . Determine C without using a calculator. (R)
2
Logarithms
Laws of logarithms
yxxy aaa logloglog yxxy aaa logloglog
1. Explain relationship between logs and exponents and show
conversion from log-form to exponential form and vice versa. 2. Application of laws of logarithms: •
Page 37 of 59
GRADE 11: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Exponents and surds
1. Simplify expressions and solve equations using the laws of exponents for rational exponents where
;q pqp
xx 0;0 qx .
2. Add, subtract, multiply and divide simple surds.
3. Solve exponential equations.
Example: Without the use of a calculator,
1. Determine the value of 23
9 . (K) 2. Simplify: )23)(23( . (R) 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
xxx
(R)
6. 3 233
851255
6255
(C)
Solve:
7. 1284 25x (R)
8. 32 525 xx (R) 9. Practical examples from the technical field must be included e.g.
The formula VQC is given with 4106 Q C and
voltV 200 . Determine C without using a calculator. (R)
2
Logarithms
Laws of logarithms
yxxy aaa logloglog yxxy aaa logloglog
1. Explain relationship between logs and exponents and show
conversion from log-form to exponential form and vice versa. 2. Application of laws of logarithms:
• yxy
xaaa logloglog −=
• xnx an
a loglog =• Simplify applying the law:
•
abb
c
ca log
loglog =
Solving of logarithmic equations.
1. Explain relationship between logs and exponents and show conversion from log-form to exponential form and vice versa.
2. Application of laws of logarithms:
- Simplify without the aid of a calculator:
2.1
Page 38 of 59
yxyx
aaa logloglog
xnx an
a loglog Simplify applying the law:
abb
c
ca log
loglog
Solving of logarithmic equations.
- Simplify without the aid of a calculator:
2.1 49log343log (R)
2.2 01,0log25log4log
2.3 2log33log20log26log (R)
- Prove: 2.4 21
1024log8log4log64log
x
xxx (C)
- Simplify: 2.5 64log32
3. Show application of logs to solve 35 x 4. Solving of log equations limited to the following:
4.1 x81log2 (K)
4.2 900log2log2 x (K) 4.3 3)4(log)3(log 22 xx (K)
4 Equations
1. Solve quadratic equations by means of: 1.1 Factorisation 1.2 Quadratic formula.
2. Quadratic inequalities (interpret
solutions graphically). 3. Determine the nature of roots and
the conditions for which the roots are real, non-real, equal, unequal, rational and irrational.
4. Equations in two unknowns, one of
which is linear and the other quadratic.
5. Word problems. 6. Manipulating formulae (Make a
variable the subject of the formula).
- When explaining the quadratic formula it is important to show where 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: e.g. The length of a rectangle is 4m longer than the breadth. Determine the measurements of the rectangle if the area equals 621m2. - 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.
(R)
2.2
Page 38 of 59
yxyx
aaa logloglog
xnx an
a loglog Simplify applying the law:
abb
c
ca log
loglog
Solving of logarithmic equations.
- Simplify without the aid of a calculator:
2.1 49log343log (R)
2.2 01,0log25log4log
2.3 2log33log20log26log (R)
- Prove: 2.4 21
1024log8log4log64log
x
xxx (C)
- Simplify: 2.5 64log32
3. Show application of logs to solve 35 x 4. Solving of log equations limited to the following:
4.1 x81log2 (K)
4.2 900log2log2 x (K) 4.3 3)4(log)3(log 22 xx (K)
4 Equations
1. Solve quadratic equations by means of: 1.1 Factorisation 1.2 Quadratic formula.
2. Quadratic inequalities (interpret
solutions graphically). 3. Determine the nature of roots and
the conditions for which the roots are real, non-real, equal, unequal, rational and irrational.
4. Equations in two unknowns, one of
which is linear and the other quadratic.
5. Word problems. 6. Manipulating formulae (Make a
variable the subject of the formula).
- When explaining the quadratic formula it is important to show where 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: e.g. The length of a rectangle is 4m longer than the breadth. Determine the measurements of the rectangle if the area equals 621m2. - 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.
2.3
Page 38 of 59
yxyx
aaa logloglog
xnx an
a loglog Simplify applying the law:
abb
c
ca log
loglog
Solving of logarithmic equations.
- Simplify without the aid of a calculator:
2.1 49log343log (R)
2.2 01,0log25log4log
2.3 2log33log20log26log (R)
- Prove: 2.4 21
1024log8log4log64log
x
xxx (C)
- Simplify: 2.5 64log32
3. Show application of logs to solve 35 x 4. Solving of log equations limited to the following:
4.1 x81log2 (K)
4.2 900log2log2 x (K) 4.3 3)4(log)3(log 22 xx (K)
4 Equations
1. Solve quadratic equations by means of: 1.1 Factorisation 1.2 Quadratic formula.
2. Quadratic inequalities (interpret
solutions graphically). 3. Determine the nature of roots and
the conditions for which the roots are real, non-real, equal, unequal, rational and irrational.
4. Equations in two unknowns, one of
which is linear and the other quadratic.
5. Word problems. 6. Manipulating formulae (Make a
variable the subject of the formula).
- When explaining the quadratic formula it is important to show where 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: e.g. The length of a rectangle is 4m longer than the breadth. Determine the measurements of the rectangle if the area equals 621m2. - 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.
(R)
- Prove: 2.4
Page 38 of 59
yxyx
aaa logloglog
xnx an
a loglog Simplify applying the law:
abb
c
ca log
loglog
Solving of logarithmic equations.
- Simplify without the aid of a calculator:
2.1 49log343log (R)
2.2 01,0log25log4log
2.3 2log33log20log26log (R)
- Prove: 2.4 21
1024log8log4log64log
x
xxx (C)
- Simplify: 2.5 64log32
3. Show application of logs to solve 35 x 4. Solving of log equations limited to the following:
4.1 x81log2 (K)
4.2 900log2log2 x (K) 4.3 3)4(log)3(log 22 xx (K)
4 Equations
1. Solve quadratic equations by means of: 1.1 Factorisation 1.2 Quadratic formula.
2. Quadratic inequalities (interpret
solutions graphically). 3. Determine the nature of roots and
the conditions for which the roots are real, non-real, equal, unequal, rational and irrational.
4. Equations in two unknowns, one of
which is linear and the other quadratic.
5. Word problems. 6. Manipulating formulae (Make a
variable the subject of the formula).
- When explaining the quadratic formula it is important to show where 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: e.g. The length of a rectangle is 4m longer than the breadth. Determine the measurements of the rectangle if the area equals 621m2. - 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.
(C)
- Simplify: 2.5
Page 38 of 59
yxyx
aaa logloglog
xnx an
a loglog Simplify applying the law:
abb
c
ca log
loglog
Solving of logarithmic equations.
- Simplify without the aid of a calculator:
2.1 49log343log (R)
2.2 01,0log25log4log
2.3 2log33log20log26log (R)
- Prove: 2.4 21
1024log8log4log64log
x
xxx (C)
- Simplify: 2.5 64log32
3. Show application of logs to solve 35 x 4. Solving of log equations limited to the following:
4.1 x81log2 (K)
4.2 900log2log2 x (K) 4.3 3)4(log)3(log 22 xx (K)
4 Equations
1. Solve quadratic equations by means of: 1.1 Factorisation 1.2 Quadratic formula.
2. Quadratic inequalities (interpret
solutions graphically). 3. Determine the nature of roots and
the conditions for which the roots are real, non-real, equal, unequal, rational and irrational.
4. Equations in two unknowns, one of
which is linear and the other quadratic.
5. Word problems. 6. Manipulating formulae (Make a
variable the subject of the formula).
- When explaining the quadratic formula it is important to show where 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: e.g. The length of a rectangle is 4m longer than the breadth. Determine the measurements of the rectangle if the area equals 621m2. - 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.
3. Show application of logs to solve 35 =x 4. Solving of log equations limited to the following:
4.1 x=81log2 (K)
4.2 900log2log2 =+x (K)
4.3 3)4(log)3(log 22 =−++ xx (K)
33CAPS TECHNICAL MATHEMATICS
GRADE 11: TERM 1Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
4 Equations
1. Solve quadratic equations by means of:
1.1 Factorisation1.2 Quadratic formula.2. Quadratic inequalities
(interpret solutions graphically).
3. Determine the nature of roots and the conditions for which the roots are real, non-real, equal, unequal, rational and irrational.
4. Equations in two unknowns, one of which is linear and the other quadratic.
5. Word problems.6. Manipulating formulae (Make
a variable the subject of the formula).
• When explaining the quadratic formula it is important to show where 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: e.g. The length of a rectangle is 4m longer than the breadth. Determine the measurements of the rectangle if the area equals 621m2.
• 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 Analytical Geometry
1. Revise to find the equation of a line through two given points;
Determine:2. the equation of a line through
one point and parallel or perpendicular to a given line; and
3. the inclination )(θ of a line, where θ= tanm is the gradient of the line and
°≤θ≤° 1800
1. Revision of linear equation from Grade 10.2. Showing the influence of the gradient and
consequently the relationship when lines are parallel and perpendicular.
3. Example: Given )4;2(−A , )6;2(B and )2;3( −C Determine:
3.1 Equation of line passing through point C and parallel to line AC. (R)
3.2 Equation of line passing through point B and perpendicular to line AC. (R)
4. Example: A ladder leans against a straight line wall defined by
.42y += x4.1 Determine the length of a ladder. (P)4.2 How far it reaches up the wall and the
ladder’s inclination with the floor. (C)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.
34 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 11: TERM 2Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
4Functions
and graphs
1. Revise the effect of the parameters and q on the graphs. a
a is not restricted to in grade 11.
872 xx
sin1 cosec
a for parabola graphs only in grade 10
22 cosec1cot
1
Investigate the effect of p on the graphs of the functions defined by:
1.1. qpxaxfy ++== 2)()(
1.2.
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GRADE 11: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
4 Functions
and graphs
1. Revise the effect of the parameters a and q on the graphs. Investigate the effect of p on the graphs of the functions defined by: 1.1. qpxaxfy 2)()(
1.2. cbxaxxfy 2)(
1.3. qxay
1.4. qbaxfay x ..)(. , 0b and 1b
2. 222 ryx
22 xry 22 xry 22 xry
Comment:
Learners should gain confidence and get a grasp of graphs from tabling and dotting and joining points to draw the graphs.
They should understand the influence of variables on the form and 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. Analyse the information from graphs (graphs given).
4 Euclidean Geometry
Accept results established in earlier grades as axioms and also that a tangent to a circle is perpendicular to the radius, drawn to the point of contact.
Then investigate and apply the theorems of the geometry of circles: The line drawn from the centre of a
circle perpendicular to a chord bisects the chord;
The perpendicular bisector of a
Comments: Proofs of theorems and their converses will not be examined. But proofs of theorems and their converses will applied in solving riders. The focus of all questions will be on applications and calculations.
1.3. qxay +=
1.4. qbaxfay x +== ..)(. , 0>b and 1≠b
2. 222 ryx =+
22 xry −±=
22 xry −+=
22 xry −−=
Comment:
• Learners should gain confidence and get a grasp of graphs from tabling and dotting and joining points to draw the graphs.
• They should understand the influence of variables on the form and 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.• Analyse the information from graphs
(graphs given).
35CAPS TECHNICAL MATHEMATICS
GRADE 11: TERM 2Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
4 EuclideanGeometry
Accept results established in earlier grades as axioms and also that a tangent to a circle is perpendicular to the radius, drawn to the point of contact.Then investigate and apply the theorems of the geometry of circles:• The line drawn from
the centre of a circle perpendicular to a chord bisects the chord;
• The perpendicular bisector of a chord passes through the centre of the circle;
• The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
• Angles subtended by a chord of the circle, on the same side of the chord, are equal;
• The opposite angles of a cyclic quadrilateral are supplementary;
• Exterior angle of cyclic quad. is equal to opposite interior angle;
• Two tangents drawn to a circle from the same point outside the circle are equal in length;
• Radius is perpendicular to the tangent; and
• The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
Comments:Proofs of theorems and their converses will not be examined.But proofs of theorems and their converses will applied in solving riders.
The focus of all questions will be on applications and calculations.
Make use of problems taken from the technical fields.
3 Mid-yearexaminations
Assessment 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 (
Page 41 of 59
chord passes through the centre of the circle;
The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
Angles subtended by a chord of the circle, on the same side of the chord, are equal;
The opposite angles of a cyclic quadrilateral are supplementary;
Exterior angle of cyclic quad. is equal to opposite interior angle;
Two tangents drawn to a circle from the same point outside the circle are equal in length;
Radius is perpendicular to the tangent; and
The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
Make use of problems taken from the technical fields.
3 Mid-year examinations
Assessment 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 and
graphs ( 325 ). Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry ( 340 ) and Euclidean Geometry ( 360 ).
), equations, inequalities and nature of roots (
Page 41 of 59
chord passes through the centre of the circle;
The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
Angles subtended by a chord of the circle, on the same side of the chord, are equal;
The opposite angles of a cyclic quadrilateral are supplementary;
Exterior angle of cyclic quad. is equal to opposite interior angle;
Two tangents drawn to a circle from the same point outside the circle are equal in length;
Radius is perpendicular to the tangent; and
The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
Make use of problems taken from the technical fields.
3 Mid-year examinations
Assessment 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 and
graphs ( 325 ). Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry ( 340 ) and Euclidean Geometry ( 360 ).
), function and graphs (
Page 41 of 59
chord passes through the centre of the circle;
The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
Angles subtended by a chord of the circle, on the same side of the chord, are equal;
The opposite angles of a cyclic quadrilateral are supplementary;
Exterior angle of cyclic quad. is equal to opposite interior angle;
Two tangents drawn to a circle from the same point outside the circle are equal in length;
Radius is perpendicular to the tangent; and
The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
Make use of problems taken from the technical fields.
3 Mid-year examinations
Assessment 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 and
graphs ( 325 ). Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry ( 340 ) and Euclidean Geometry ( 360 ).
).Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry (
Page 41 of 59
chord passes through the centre of the circle;
The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
Angles subtended by a chord of the circle, on the same side of the chord, are equal;
The opposite angles of a cyclic quadrilateral are supplementary;
Exterior angle of cyclic quad. is equal to opposite interior angle;
Two tangents drawn to a circle from the same point outside the circle are equal in length;
Radius is perpendicular to the tangent; and
The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
Make use of problems taken from the technical fields.
3 Mid-year examinations
Assessment 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 and
graphs ( 325 ). Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry ( 340 ) and Euclidean Geometry ( 360 ). ) and Euclidean Geometry (
Page 41 of 59
chord passes through the centre of the circle;
The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
Angles subtended by a chord of the circle, on the same side of the chord, are equal;
The opposite angles of a cyclic quadrilateral are supplementary;
Exterior angle of cyclic quad. is equal to opposite interior angle;
Two tangents drawn to a circle from the same point outside the circle are equal in length;
Radius is perpendicular to the tangent; and
The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
Make use of problems taken from the technical fields.
3 Mid-year examinations
Assessment 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 and
graphs ( 325 ). Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry ( 340 ) and Euclidean Geometry ( 360 ). ).
36 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 11: TERM 3Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
4
Circles, angles and
angular movement
1. Circle
1.1 222 ryx =+ , with centre (0;0 ) only
1.2 Angles and arcs1.3 Degrees and radians
1.4 Sectors and segments
2. Angular and circumferential/peripheral velocity
Pay attention to the following: diameter, radius, chord, segment, arc,secant, tangent.1.1 If a point on a circumference is given, how to
determine r and the equation of the circle?1.2 What is an arc, central angle, inscribed angle?1.3 A radian is the angle at the centre of a circle
subtended by an arc of the same length as the radius.
Complete revolution: 3600 = π2 radians; 1800 = π radians• If an arc of length s is intercepted by a central
angle (in radians) the angle subtended will be
equal to the arc divided by the radius. rs=θ .
• If s = r then = 1 radian.• Convert degrees to radians and vice versa.
Convert 2 radians to degrees
°=π 180rad
1 rad = π°180
2 rad = π°×1802
= 114,590
Convert 52,60 to radians
rad°
π=° 1801
52,60 =
Page 43 of 59
2. Angular and
circumferential/peripheral velocity
52,60 = 1806,52
= 0, 918 rad 1.3 A sector of a circle is a plane figure bounded by an arc and two radii.
area of a sector = rrrs 222
= 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 the height of the segment formed by the chord, can be set out in the formula
04 22 xdhh ; h = height of segment; d = diameter of circle;
h = length of chord.
2. Angular velocity (omega) can be determined by multiplying the radians in one revolution ( 2 ) with the revolutions per second (n)
2 n= 3600 n Circumferential velocity is the linear velocity of a point on the
circumference nDv with D the diameter, n is the rotation frequency
4 Trigonometry
1. Revise the trig ratios in the solving of right-angle triangle in all 4 quadrants (Grade 10).
2. Apply the sine, cosine and area rules. 3. Solve problems in two dimensions
using the sine, cosine and area rules 4. Draw the graphs of the functions
defined by xky sin , xky cos ,
)(sin kxy , )(cos kxy and xy tan .
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 to deduct important information from given sketches. - One parameter should be tested at a given time when examining
= 0, 918 rad1.3 A sector of a circle is a plane figure bounded by
an arc and two radii.
Area of a sector =
Page 43 of 59
2. Angular and
circumferential/peripheral velocity
52,60 = 1806,52
= 0, 918 rad 1.3 A sector of a circle is a plane figure bounded by an arc and two radii.
area of a sector = rrrs 222
= 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 the height of the segment formed by the chord, can be set out in the formula
04 22 xdhh ; h = height of segment; d = diameter of circle;
h = length of chord.
2. Angular velocity (omega) can be determined by multiplying the radians in one revolution ( 2 ) with the revolutions per second (n)
2 n= 3600 n Circumferential velocity is the linear velocity of a point on the
circumference nDv with D the diameter, n is the rotation frequency
4 Trigonometry
1. Revise the trig ratios in the solving of right-angle triangle in all 4 quadrants (Grade 10).
2. Apply the sine, cosine and area rules. 3. Solve problems in two dimensions
using the sine, cosine and area rules 4. Draw the graphs of the functions
defined by xky sin , xky cos ,
)(sin kxy , )(cos kxy and xy tan .
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 to deduct important information from given sketches. - One parameter should be tested at a given time when examining
= radius; s = arc length andθ = 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 the height of the segment formed by the chord, can be set out in the formula
37CAPS TECHNICAL MATHEMATICS
GRADE 11: TERM 3Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
Page 43 of 59
2. Angular and
circumferential/peripheral velocity
52,60 = 1806,52
= 0, 918 rad 1.3 A sector of a circle is a plane figure bounded by an arc and two radii.
area of a sector = rrrs 222
= 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 the height of the segment formed by the chord, can be set out in the formula
04 22 xdhh ; h = height of segment; d = diameter of circle;
h = length of chord.
2. Angular velocity (omega) can be determined by multiplying the radians in one revolution ( 2 ) with the revolutions per second (n)
2 n= 3600 n Circumferential velocity is the linear velocity of a point on the
circumference nDv with D the diameter, n is the rotation frequency
4 Trigonometry
1. Revise the trig ratios in the solving of right-angle triangle in all 4 quadrants (Grade 10).
2. Apply the sine, cosine and area rules. 3. Solve problems in two dimensions
using the sine, cosine and area rules 4. Draw the graphs of the functions
defined by xky sin , xky cos ,
)(sin kxy , )(cos kxy and xy tan .
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 to deduct important information from given sketches. - One parameter should be tested at a given time when examining
; h = height of segment; d = diameter of circle; h = length of chord.2. Angular velocity (omega) ωcan be determined by
multiplying the radians in one revolution ( π2 ) with the revolutions per second (n)
π=ω 2 n= 3600 nCircumferential velocity is the linear velocity of a point on the circumference
nDv π= with D the diameter, n is the rotation frequency
4 Trigonometry
1. Revise the trig ratios in the solving of right-angle triangle in all 4 quadrants (Grade 10).
2. Apply the sine, cosine and area rules.
3. Solve problems in two dimensions using the sine, cosine and area rules
4. Draw the graphs of the functions defined by
xky sin= , xky cos= ,
Page 43 of 59
2. Angular and
circumferential/peripheral velocity
52,60 = 1806,52
= 0, 918 rad 1.3 A sector of a circle is a plane figure bounded by an arc and two radii.
area of a sector = rrrs 222
= 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 the height of the segment formed by the chord, can be set out in the formula
04 22 xdhh ; h = height of segment; d = diameter of circle;
h = length of chord.
2. Angular velocity (omega) can be determined by multiplying the radians in one revolution ( 2 ) with the revolutions per second (n)
2 n= 3600 n Circumferential velocity is the linear velocity of a point on the
circumference nDv with D the diameter, n is the rotation frequency
4 Trigonometry
1. Revise the trig ratios in the solving of right-angle triangle in all 4 quadrants (Grade 10).
2. Apply the sine, cosine and area rules. 3. Solve problems in two dimensions
using the sine, cosine and area rules 4. Draw the graphs of the functions
defined by xky sin , xky cos ,
)(sin kxy , )(cos kxy and xy tan .
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 to deduct important information from given sketches. - One parameter should be tested at a given time when examining
,
Page 43 of 59
2. Angular and
circumferential/peripheral velocity
52,60 = 1806,52
= 0, 918 rad 1.3 A sector of a circle is a plane figure bounded by an arc and two radii.
area of a sector = rrrs 222
= 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 the height of the segment formed by the chord, can be set out in the formula
04 22 xdhh ; h = height of segment; d = diameter of circle;
h = length of chord.
2. Angular velocity (omega) can be determined by multiplying the radians in one revolution ( 2 ) with the revolutions per second (n)
2 n= 3600 n Circumferential velocity is the linear velocity of a point on the
circumference nDv with D the diameter, n is the rotation frequency
4 Trigonometry
1. Revise the trig ratios in the solving of right-angle triangle in all 4 quadrants (Grade 10).
2. Apply the sine, cosine and area rules. 3. Solve problems in two dimensions
using the sine, cosine and area rules 4. Draw the graphs of the functions
defined by xky sin , xky cos ,
)(sin kxy , )(cos kxy and xy tan .
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 to deduct important information from given sketches. - One parameter should be tested at a given time when examining
and xy tan= .
5. Draw the graphs of the functions defined by
)sin( pxy += and
)cos( pxy +=6. Rotating vectors Developing the sine and
cosine curve.7. Trigonometric equations.8. Introduce identities and
apply θθ=θ cos
sintan ,
1cossin 22 =θ+θ ,
θ=θ+ 22 sectan1 and
a
a is not restricted to in grade 11.
872 xx
sin1 cosec
a for parabola graphs only in grade 10
22 cosec1cot .
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 to deduct important information from given sketches.
• One parameter should be tested at a given time when examining horizontal shifts
• Rotating vectors should be done on graph paper.• Determine the solutions of equations for
]360;0[ °°∈θ • Limited to routine procedures.
• Reduction formulae, )180( θ±° and )360( θ±° . • Examples related to the use of identities limited
to routine procedures.
38 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 11: TERM 3Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2Finance, growth
and decay
1. Use simple and compound decay
formulae:
3. FINANCE, GROWTH AND DECAY Use simple and compound growth formulae
)1( inPA and niPA )1( to solve problems (including interest, hire purchase, inflation, population growth and other real life problems).
Use simple and compound growth/decay formulae )1( inPA and niPA )1( to solve problems (including interest, hire purchase, inflation, population growth and other real life problems).
Strengthen the Grade 11 work.
The implications of fluctuating foreign exchange rates.
The effect of different periods of compounding growth and decay (including effective and nominal interest rates).
Critically analyse different loan options.
4. ALGEBRA
(a) Simplify expressions using the laws of exponents for integral exponents.
(b) Establish between which two integers a given simple surd lies.
(c) Round real numbers to an appropriate degree of accuracy (to a given number of decimal digits).
(d) Revise scientific notation.
(a) Apply the laws of exponents to expressions involving rational exponents.
(b) Add, subtract, multiply and divide simple surds.
(c) Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real life problems.
Apply any law of logarithm to solve real life problems.
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 and denominators limited to the polynomials covered in factorisation.
Revise factorisation from Grade 10.
Take note and understand the Remainder and Factor Theorems for polynomials up to the third degree (proofs of the Remainder and Factor theorems will not be examined).
Factorise third-degree polynomials (including examples which require the Factor Theorem).
and niPA )1( −= to solve problems (including straight line depreciation and depreciation on a reducing balance).
2. The effect of different periods of compound growth and decay, including nominal and effective interest 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 calculated on 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. interest compounded quarterly for the first 18 months. The interest then changes to 6% p.a. compounded monthly. Two years after the money is invested, R10 000 is withdrawn. How much will be in the account after 4 years? (C)
Comment: Stress the importance of not working with rounded answers, but of using the maximum accuracy afforded by the calculator right to the final answer when rounding might be appropriate.
3 Mid-yearexaminations
Assessment Term 3: Two (2) tests (at least 50 marks per test and 1 hour) covering all topics in approximately the ratio of the allocated teaching time.
39CAPS TECHNICAL MATHEMATICS
GRADE 11: TERM 4
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 × area of base + circumference of base × height(for closed right-angled prism) Surface Area = area of base + circumference of base × 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 = )...( 321 nmmmma ++++ where
221
1oom +
= etc. and n = number of ordinates
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: (
Page 46 of 59
GRADE 11: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right
prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface
area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 area of base + circumference of base height (for closed right-angled prism) Surface Area = area of base + circumference of base 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 = )...( 321 nmmmma where 221
1oom
etc. and n = number of ordinates
3 Revision 3 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, circles, angles and angular movement.
) on algebraic expressions, equations, inequalities and nature of roots,
Page 46 of 59
GRADE 11: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right
prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface
area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 area of base + circumference of base height (for closed right-angled prism) Surface Area = area of base + circumference of base 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 = )...( 321 nmmmma where 221
1oom
etc. and n = number of ordinates
3 Revision 3 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, circles, angles and angular movement.
on functions and graphs (excluding trigonometric functions) and
Page 46 of 59
GRADE 11: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right
prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface
area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 area of base + circumference of base height (for closed right-angled prism) Surface Area = area of base + circumference of base 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 = )...( 321 nmmmma where 221
1oom
etc. and n = number of ordinates
3 Revision 3 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, circles, angles and angular movement.
on finance growth and decay.Paper 2: 3 hours (150 marks made up as follows:
Page 46 of 59
GRADE 11: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right
prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface
area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 area of base + circumference of base height (for closed right-angled prism) Surface Area = area of base + circumference of base 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 = )...( 321 nmmmma where 221
1oom
etc. and n = number of ordinates
3 Revision 3 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, circles, angles and angular movement.
on trigonometry (including trigonometric functions),
Page 46 of 59
GRADE 11: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right
prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface
area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 area of base + circumference of base height (for closed right-angled prism) Surface Area = area of base + circumference of base 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 = )...( 321 nmmmma where 221
1oom
etc. and n = number of ordinates
3 Revision 3 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, circles, angles and angular movement.
on Analytical Geometry,
Page 46 of 59
GRADE 11: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right
prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface
area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 area of base + circumference of base height (for closed right-angled prism) Surface Area = area of base + circumference of base 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 = )...( 321 nmmmma where 221
1oom
etc. and n = number of ordinates
3 Revision 3 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, circles, angles and angular movement. on Euclidean Geometry,
Page 46 of 59
GRADE 11: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Mensuration
1. Surface area and volume of right
prisms, cylinders, pyramids, cones and spheres, and combinations of these geometric objects.
2. The effect on volume and surface
area when multiplying any dimension by factor k.
3. Determine the area of an irregular figure using mid-ordinate rule.
1. Surface Area = 2 area of base + circumference of base height (for closed right-angled prism) Surface Area = area of base + circumference of base 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 = )...( 321 nmmmma where 221
1oom
etc. and n = number of ordinates
3 Revision 3 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, circles, angles and angular movement. on Mensuration, circles, angles and
angular movement.
40 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 12: TERM 1
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
• Define a complex number,
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
,
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
. • Learners should know:
◦ the conjugate of
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
. ◦ imaginary numbers,
12 −=i . ◦ how to add, subtract,
divide and multiply complex numbers.
◦ represent complex numbers in the Argand diagram.
◦ argument of .z ◦ trigonometric (polar)
form of complex numbers.
• Solve equations with complex numbers with two variables.
Simplify :
1.
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
(R)
2.
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
(K)
3.
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
(R)4. iii 5132 −−+− (K)5.
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
(K)
Solve for x and :y6.
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
(R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required).Long division method can also be used.
Revise functional notation.Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem.Example: Solve for x :
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples:
(R)
41CAPS TECHNICAL MATHEMATICS
GRADE 12: TERM 1
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
6DifferentialCalculus
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()( −+=
3. Determine the gradient of a tangent to a graph, which is also the gradient of the graph at that point. Introduce the limit-principle by shifting the secant until it becomes a tangent.
4. By using first principals for
Page 48 of 59
6
Differential Calculus
of the graph at that point. Introduce the limit-principle by shifting the secant until it becomes a tangent.
4. By using first principals
hxfhxfxf
h
)()(lim)('0
for
kxf )( , axxf )( and
baxxf )(
5. Use the rule 1)( nn anxaxdxd for
n ℝ 6. Find equations of tangents to graphs of functions 7. Sketch graphs of cubic polynomial functions using differentiation to determine the co-ordinate of stationary points. Also, determine the x - intercepts of the graph using the factor theorem and other techniques. 8. Solve practical problems concerning optimisation and rates of change, including calculus of motion.
1. In each of the following cases, find the derivative of )(xf at the point where 1x , using the definition of the derivative:
1.1 2)( xxf (R)
1.2 2)( 2 xxf (R)
2. Sketch the graph defined by xxxy 23 4 by: 2.1 finding the intercepts with the axes; 2.2 finding maxima, minima. (C)
On word problems the diagram with all the measurements must be given. Guide the learners through the question with subsections.
Refer to displacement formulae like 2
21 atus .
Very simple problems. Refer to practical application in the technical field.
Assessment Term 1:
1. Test (at least 50 marks). 2. Investigation or project. 3. Test (at least 50 marks) or Assignment (at least 50 marks).
5. Use the rule
Page 48 of 59
6
Differential Calculus
of the graph at that point. Introduce the limit-principle by shifting the secant until it becomes a tangent.
4. By using first principals
hxfhxfxf
h
)()(lim)('0
for
kxf )( , axxf )( and
baxxf )(
5. Use the rule 1)( nn anxaxdxd for
n ℝ 6. Find equations of tangents to graphs of functions 7. Sketch graphs of cubic polynomial functions using differentiation to determine the co-ordinate of stationary points. Also, determine the x - intercepts of the graph using the factor theorem and other techniques. 8. Solve practical problems concerning optimisation and rates of change, including calculus of motion.
1. In each of the following cases, find the derivative of )(xf at the point where 1x , using the definition of the derivative:
1.1 2)( xxf (R)
1.2 2)( 2 xxf (R)
2. Sketch the graph defined by xxxy 23 4 by: 2.1 finding the intercepts with the axes; 2.2 finding maxima, minima. (C)
On word problems the diagram with all the measurements must be given. Guide the learners through the question with subsections.
Refer to displacement formulae like 2
21 atus .
Very simple problems. Refer to practical application in the technical field.
Assessment Term 1:
1. Test (at least 50 marks). 2. Investigation or project. 3. Test (at least 50 marks) or Assignment (at least 50 marks).
for
Page 48 of 59
6
Differential Calculus
of the graph at that point. Introduce the limit-principle by shifting the secant until it becomes a tangent.
4. By using first principals
hxfhxfxf
h
)()(lim)('0
for
kxf )( , axxf )( and
baxxf )(
5. Use the rule 1)( nn anxaxdxd for
n ℝ 6. Find equations of tangents to graphs of functions 7. Sketch graphs of cubic polynomial functions using differentiation to determine the co-ordinate of stationary points. Also, determine the x - intercepts of the graph using the factor theorem and other techniques. 8. Solve practical problems concerning optimisation and rates of change, including calculus of motion.
1. In each of the following cases, find the derivative of )(xf at the point where 1x , using the definition of the derivative:
1.1 2)( xxf (R)
1.2 2)( 2 xxf (R)
2. Sketch the graph defined by xxxy 23 4 by: 2.1 finding the intercepts with the axes; 2.2 finding maxima, minima. (C)
On word problems the diagram with all the measurements must be given. Guide the learners through the question with subsections.
Refer to displacement formulae like 2
21 atus .
Very simple problems. Refer to practical application in the technical field.
Assessment Term 1:
1. Test (at least 50 marks). 2. Investigation or project. 3. Test (at least 50 marks) or Assignment (at least 50 marks).
6. Find equations of tangents to graphs of functions
7. Sketch graphs of cubic polynomial functions using differentiation to determine the co-ordinate of stationary points. Also, determine the x - intercepts of the graph using the factor theorem and other techniques.
8. Solve practical problems concerning optimisation and rates of change, including calculus of motion.
Comment: Differentiation from first principles will be examined on any of the types described in 3.1.Understand that the following notations mean the same
Page 47 of 59
GRADE 12: TERM 1
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Complex numbers
Define a complex number, ℂ , biaz .
Learners should know: - the conjugate of biaz . - imaginary numbers, 12 i . - how to add, subtract, divide and
multiply complex numbers. - represent complex numbers in
the Argand diagram. - argument of .z - trigonometric (polar) form of
complex numbers. Solve equations with complex
numbers with two variables.
Simplify : 1. 1416 (R) 2. 516 (K)
3. 6
12.4 (R)
4. iii 5132 (K) 5. )1)(23( ii (K) Solve for x and :y 6. yiix 53152 (R)
2 Polynomials
Factorise third-degree polynomials. Apply the Remainder and Factor Theorems to polynomials of the third degree (no proofs are required). Long division method can also be used.
Revise functional notation. Any method may be used to factorise third degree polynomials but it should include examples which require the Factor Theorem. Example:
Solve for x : 010178 23 xxx (R)
1. An intuitive understanding of the limit concept, in the context of approximating the rate of change or gradient of a function at a point.
2. Determine the average gradient of a curve between two points.
hxfhxfm )()(
3. Determine the gradient of a tangent to a graph, which is also the gradient
Comment: Differentiation from first principles will be examined on any of the types described in 3.1. Understand that the following notations mean the same
)(',, xfdxdDx
Examples: Examples:1. In each of the following cases, find the derivative of
)(xf at the point where 1−=x , using the definition of the derivative:
1.1 2)( xxf = (R)
1.2 2)( 2 += xxf (R)
2. Sketch the graph defined by xxxy −+−= 23 4 by:2.1 finding the intercepts with the axes;2.2 finding maxima, minima. (C)
On word problems the diagram with all the measurements must be given. Guide the learners through the question with subsections.Refer to displacement formulae like
Page 48 of 59
6
Differential Calculus
of the graph at that point. Introduce the limit-principle by shifting the secant until it becomes a tangent.
4. By using first principals
hxfhxfxf
h
)()(lim)('0
for
kxf )( , axxf )( and
baxxf )(
5. Use the rule 1)( nn anxaxdxd for
n ℝ 6. Find equations of tangents to graphs of functions 7. Sketch graphs of cubic polynomial functions using differentiation to determine the co-ordinate of stationary points. Also, determine the x - intercepts of the graph using the factor theorem and other techniques. 8. Solve practical problems concerning optimisation and rates of change, including calculus of motion.
1. In each of the following cases, find the derivative of )(xf at the point where 1x , using the definition of the derivative:
1.1 2)( xxf (R)
1.2 2)( 2 xxf (R)
2. Sketch the graph defined by xxxy 23 4 by: 2.1 finding the intercepts with the axes; 2.2 finding maxima, minima. (C)
On word problems the diagram with all the measurements must be given. Guide the learners through the question with subsections.
Refer to displacement formulae like 2
21 atus .
Very simple problems. Refer to practical application in the technical field.
Assessment Term 1:
1. Test (at least 50 marks). 2. Investigation or project. 3. Test (at least 50 marks) or Assignment (at least 50 marks).
.
Very simple problems.Refer to practical application in the technical field.
Assessment Term 1: 1. Test (at least 50 marks).2. Investigation or project. 3. Test (at least 50 marks) or Assignment (at least 50 marks).
42 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 12: TERM 2
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3
Integration
Introduce integration.1. Understand the concept
of integration as a summation function (definite integral) and as converse of differentiation (indefinite integral).
2. Apply standard forms of integrals as a converse of differentiation.
3. Integrate the following functions:3.1
Page 49 of 59
GRADE 12: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3
Integration
Introduce integration. 1. Understand the concept of
integration as a summation function (definite integral) and as converse of differentiation (indefinite integral). 2. Apply standard forms of integrals as a converse of differentiation. 3. Integrate the following functions: 3.1 ,nkx n ℝ with 1n
3.2 xk and nxka with 0a , ak , ℝ
4. Integrate polynomials consisting of terms of the above forms (3.1 and 3.2). 5. Apply integration to determine the magnitude of an area included by a curve and the x-axis or by a curve, the x-axis and the ordinates ax and bx where ., Zba
Examples 1. Calculate the values of
1.1 10 xdx . (K)
1.2 .)32(21
23 dxxx (R)
Determine the area included by the curve of xxy 2 and the x-axis. (C)
2
Analytical Geometry
1. The equation 222 ryx defines a circle with radius r and centre (0; 0).
2. Find the equation of the circle when the radius is given or a point on the circle is given. Only circles with the origin as centre.
3. Determination of the equation of a
Examples: 1.1 Determine the equation of the circle passing through the point (2;
4) with centre at the origin. (R) 1.2 Hence determine the equation of the tangent to the circle at the
point (2; 4). (R) 1.3 Hence determine the points of intersection of the circle and the
x
y
x 1 0
xxy 2
y with
1−≠n
3.2 xk and
Page 49 of 59
GRADE 12: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3
Integration
Introduce integration. 1. Understand the concept of
integration as a summation function (definite integral) and as converse of differentiation (indefinite integral). 2. Apply standard forms of integrals as a converse of differentiation. 3. Integrate the following functions: 3.1 ,nkx n ℝ with 1n
3.2 xk and nxka with 0a , ak , ℝ
4. Integrate polynomials consisting of terms of the above forms (3.1 and 3.2). 5. Apply integration to determine the magnitude of an area included by a curve and the x-axis or by a curve, the x-axis and the ordinates ax and bx where ., Zba
Examples 1. Calculate the values of
1.1 10 xdx . (K)
1.2 .)32(21
23 dxxx (R)
Determine the area included by the curve of xxy 2 and the x-axis. (C)
2
Analytical Geometry
1. The equation 222 ryx defines a circle with radius r and centre (0; 0).
2. Find the equation of the circle when the radius is given or a point on the circle is given. Only circles with the origin as centre.
3. Determination of the equation of a
Examples: 1.1 Determine the equation of the circle passing through the point (2;
4) with centre at the origin. (R) 1.2 Hence determine the equation of the tangent to the circle at the
point (2; 4). (R) 1.3 Hence determine the points of intersection of the circle and the
x
y
x 1 0
xxy 2
y
with
0≥a , ∈ak , R4. Integrate polynomials
consisting of terms of the above forms (3.1 and 3.2).
5. Apply integration to determine the magnitude of an area included by a curve and the x-axis or by a curve, the x-axis and the ordinates
ax = and bx = where
., Zba ∈
Examples1. Calculate the values of
1.1 ∫10 xdx . (K)
1.2
Page 49 of 59
GRADE 12: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3
Integration
Introduce integration. 1. Understand the concept of
integration as a summation function (definite integral) and as converse of differentiation (indefinite integral). 2. Apply standard forms of integrals as a converse of differentiation. 3. Integrate the following functions: 3.1 ,nkx n ℝ with 1n
3.2 xk and nxka with 0a , ak , ℝ
4. Integrate polynomials consisting of terms of the above forms (3.1 and 3.2). 5. Apply integration to determine the magnitude of an area included by a curve and the x-axis or by a curve, the x-axis and the ordinates ax and bx where ., Zba
Examples 1. Calculate the values of
1.1 10 xdx . (K)
1.2 .)32(21
23 dxxx (R)
Determine the area included by the curve of xxy 2 and the x-axis. (C)
2
Analytical Geometry
1. The equation 222 ryx defines a circle with radius r and centre (0; 0).
2. Find the equation of the circle when the radius is given or a point on the circle is given. Only circles with the origin as centre.
3. Determination of the equation of a
Examples: 1.1 Determine the equation of the circle passing through the point (2;
4) with centre at the origin. (R) 1.2 Hence determine the equation of the tangent to the circle at the
point (2; 4). (R) 1.3 Hence determine the points of intersection of the circle and the
x
y
x 1 0
xxy 2
y
(R)
Determine the area included by the curve of xxy += 2 and the x-axis. (C)
Page 49 of 59
GRADE 12: TERM 2
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3
Integration
Introduce integration. 1. Understand the concept of
integration as a summation function (definite integral) and as converse of differentiation (indefinite integral). 2. Apply standard forms of integrals as a converse of differentiation. 3. Integrate the following functions: 3.1 ,nkx n ℝ with 1n
3.2 xk and nxka with 0a , ak , ℝ
4. Integrate polynomials consisting of terms of the above forms (3.1 and 3.2). 5. Apply integration to determine the magnitude of an area included by a curve and the x-axis or by a curve, the x-axis and the ordinates ax and bx where ., Zba
Examples 1. Calculate the values of
1.1 10 xdx . (K)
1.2 .)32(21
23 dxxx (R)
Determine the area included by the curve of xxy 2 and the x-axis. (C)
2
Analytical Geometry
1. The equation 222 ryx defines a circle with radius r and centre (0; 0).
2. Find the equation of the circle when the radius is given or a point on the circle is given. Only circles with the origin as centre.
3. Determination of the equation of a
Examples: 1.1 Determine the equation of the circle passing through the point (2;
4) with centre at the origin. (R) 1.2 Hence determine the equation of the tangent to the circle at the
point (2; 4). (R) 1.3 Hence determine the points of intersection of the circle and the
x
y
x 1 0
xxy 2
y
2
AnalyticalGeometry
1. The equation 222 ryx =+ defines a
circle with radius r and centre (0; 0).
2. Find the equation of the circle when the radius is given or a point on the circle is given. Only circles with the origin as centre.
3. Determination of the equation of a tangent 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.
5. Plotting of the graph of
ellipse,
12
2
2
2=+ b
yax
Examples:
1.1 Determine the equation of the circle passing through the point (2; 4) with centre at the origin. (R)
1.2 Hence determine the equation of the tangent to the circle at the point (2; 4). (R)
1.3 Hence determine the points of intersection of the circle and the line with the equation 2+= xy . (R)
x
x
xxy += 2
43CAPS TECHNICAL MATHEMATICS
3 Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
• that equiangular triangles are similar; and
• that triangles with sides in proportion are similar.
Example:The examples must be very easy. Only with one unknown used once.
Start with problems like
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
Example:
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
In
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>> and
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>> . D is a point
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
so that
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
. E is a point on
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
so that
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
is parallel to
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
.Find the lengths of
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
and
Page 50 of 59
tangent 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.
5. Plotting of the graph of ellipse,
12
2
2
2 b
yax
line with the equation 2 xy . (R)
3
Euclidean Geometry
1. Revise earlier work on the necessary and 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 other two sides proportionally;
that equiangular triangles are similar; and
that triangles with sides in proportion are similar.
Example: The examples must be very easy. Only with one unknown used once.
Start with problems like 20103 x
Example: In ABC , 8AB , 5AC and 6BC . D is a point AB so that 4AD . E is a point on AC so that DE is parallel to BC . Find the lengths of DE and AE . (R)
3 Mid-year examinations
Assessment Term 2:
1. Test (at least 50 marks) 2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
>>
>>
. (R)
3 Mid-yearexaminations
Assessment Term 2: 1. Test (at least 50 marks)2. Mid-year examination (300 marks) Paper 1: 3 hours , 150 marks. Paper 2: 3 hours , 150 marks.
44 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 12: TERM 3
Weeks Topic Curriculum statement ClarificationWhere an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 EuclideanGeometry
Continuation from term 2.
3 Trigonometry
1. Solve problems in two and three dimensions.
2. Measurements must always be given for angles and lengths of sides.
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane.From Q the angle of elevation to the top of the building
is
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
Furthermore, °= 150ˆRQP ,
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
and the distance between P and R is 28m. Find the height of the tower, TP. (C)
2 Revision Revision3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks)2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers)
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
, Functions
and graphs
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
, Finance, growth and decay (
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
) and Differential Calculus and Integration
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
. Paper 2: 150 marks: 3 hours
Analytical Geometry
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
, Trigonometry
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
, Euclidean Geometry
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
and Mensuration, circles, angles and angular movement
Page 51 of 59
GRADE 12: TERM 3
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested:
knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
2 Euclidean Geometry
Continuation from term 2.
3
Trigonometry
1. Solve problems in two and three dimensions. 2. Measurements must always be given for angles and lengths of sides.
TP is a tower. Its foot, P, and the points Q and R are on the same horizontal plane. From Q the angle of elevation to the top of the building is .20 Furthermore,
150ˆRQP , 10ˆRPQ and the distance between P and R is 28m. Find the height
of the tower, TP. (C)
2 Revision Revision 3 Trial Exam
Assessment Term 3: 1. Assignment / test (at least 50 marks) 2. Trial examination Paper 1: 150 marks: 3 hours Algebraic expressions, equations, and inequalities (nature of roots, logs and complex numbers) ),350( Functions and graphs ),335( Finance, growth and decay )335( and Differential Calculus and Integration )325( . Paper 2: 150 marks: 3 hours Analytical Geometry ),325( Trigonometry ),340( Euclidean Geometry )350( and Mensuration and circles, angles and angular movement
).335(
R
Q 20 150
28 m
10
P
T
R
Q 20 150
28 m
10
P
T
R
45CAPS TECHNICAL MATHEMATICS
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4: Final examination:Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hoursAlgebraic expressions, equations
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
Analytical Geometry
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
and inequalities (nature of roots, logs and Trigonometry
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
complex numbers) Euclidean Geometry
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
Functions and graphs
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
Mensuration, circles, angles and angular movement
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
Finance, growth and decay
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
Differential Calculus and Integration
Page 52 of 59
GRADE 12: TERM 4
Weeks Topic Curriculum statement Clarification
Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem-solving (P).
3 Revision Revision
Assessment Term 4:
Final examination: Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hours Algebraic 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, circles, angles and angular movement 335 Finance, growth and decay 315 Differential Calculus and Integration 350
46 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
Section 4Curriculum and Assessment Policy Statement (CAPS)
FET TECHNICAL MATHEMATICS ASSESSMENT GUIDELINES
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 information to understand and assist in the learner’s 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 a view to observing general trends, making conjectures 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:
• 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.
47CAPS TECHNICAL MATHEMATICS
4.2 INFORMAL OR DAILY ASSESSMENT
The aim of assessment for learning is to continually collect information on a learner’s achievement that can be used to improve individual learning.
Informal assessment involves daily monitoring of a learner’s 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, and it need not be recorded. This should however 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.
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.
48 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
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 TIMSS study of 1999. Descriptors for each level and the approximate percentages of tasks, tests and examinations which should be at each level are given below:
Cognitive levels Description of skills to be demonstrated ExamplesKnowledge20%
• Straight recall• Identification of correct formula on the
information sheet (no changing of the subject)
• Use of mathematical facts• Appropriate use of mathematical
vocabulary
1. Write down the domain of the
function ( ) 3 2y f xx
= = + (Grade 10)
2. The angle ˆAOB subtended by arc AB at the centre O of a circle .......
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 subject)
• 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
1. Solve for 2: 5 14x x x− = (Grade 10)
2. Determine the general solution of the equation
Page 55 of 59
4.4 PROGRAMME OF ASSESSMENT The four cognitive levels used to guide all assessment tasks are based on those suggested in the TIMSS study of 1999. Descriptors for each level and the approximate percentages of tasks, tests and examinations which should be at each level are given below:
Cognitive levels Description of skills to be demonstrated Examples 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
1. Write down the domain of the function
3 2y f xx
(Grade 10) 2. The angle ˆAOB subtended by arc
AB at the centre O of a circle ....... 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 subject)
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
1. Solve for 2: 5 14x x x (Grade 10)
2. Determine the general solution of the equation
01)30sin(2 x (Grade 11)
3. Prove that the angle ˆAOBsubtended by arc AB at the centre O of a circle is double the size of the angle ˆACB which the same arc subtends at the circle. (Grade 11)
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
1. What is the average speed covered on a round trip to and
from a destination if the average speed going to the
destination is hkm /100 and the average speed for the
return journey is ?/80 hkm (Grade 11)
2. Differentiate x
x 2)2( with respect
to x. (Grade 12)
(Grade 11)
3. Prove that the angle ˆAOB subtended by arc AB at the centre O of a circle
is double the size of the angle ˆACBwhich the same arc subtends at the circle. (Grade 11)
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
1. What is the average speed covered on a round trip to andfrom a destination if the average speed going to the destination is
Page 55 of 59
4.4 PROGRAMME OF ASSESSMENT The four cognitive levels used to guide all assessment tasks are based on those suggested in the TIMSS study of 1999. Descriptors for each level and the approximate percentages of tasks, tests and examinations which should be at each level are given below:
Cognitive levels Description of skills to be demonstrated Examples 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
1. Write down the domain of the function
3 2y f xx
(Grade 10) 2. The angle ˆAOB subtended by arc
AB at the centre O of a circle ....... 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 subject)
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
1. Solve for 2: 5 14x x x (Grade 10)
2. Determine the general solution of the equation
01)30sin(2 x (Grade 11)
3. Prove that the angle ˆAOBsubtended by arc AB at the centre O of a circle is double the size of the angle ˆACB which the same arc subtends at the circle. (Grade 11)
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
1. What is the average speed covered on a round trip to and
from a destination if the average speed going to the
destination is hkm /100 and the average speed for the
return journey is ?/80 hkm (Grade 11)
2. Differentiate x
x 2)2( with respect
to x. (Grade 12)
and the average speed for the return journey is
Page 55 of 59
4.4 PROGRAMME OF ASSESSMENT The four cognitive levels used to guide all assessment tasks are based on those suggested in the TIMSS study of 1999. Descriptors for each level and the approximate percentages of tasks, tests and examinations which should be at each level are given below:
Cognitive levels Description of skills to be demonstrated Examples 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
1. Write down the domain of the function
3 2y f xx
(Grade 10) 2. The angle ˆAOB subtended by arc
AB at the centre O of a circle ....... 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 subject)
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
1. Solve for 2: 5 14x x x (Grade 10)
2. Determine the general solution of the equation
01)30sin(2 x (Grade 11)
3. Prove that the angle ˆAOBsubtended by arc AB at the centre O of a circle is double the size of the angle ˆACB which the same arc subtends at the circle. (Grade 11)
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
1. What is the average speed covered on a round trip to and
from a destination if the average speed going to the
destination is hkm /100 and the average speed for the
return journey is ?/80 hkm (Grade 11)
2. Differentiate x
x 2)2( with respect
to x. (Grade 12)
(Grade 11)
2. Differentiate x
x 2)2( + with respect to x. (Grade 12)
Problem Solving15%
• 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
Suppose a piece of wire could be tied tightly around the earth at the equator. Imagine that this wire is then lengthened by exactly one metre and held so that it is still around the earth at the equator. Would a mouse be able to crawl between the wire and the earth? Why or why not? (Any grade)
The Programme of Assessment is designed to set formal assessment tasks in all subjects in a school throughout the year.
49CAPS TECHNICAL MATHEMATICS
(a) Number of Assessment Tasks and Weighting:
Learners are expected to have seven (7) formal assessment tasks for their school-based assessment (SBA). The number of tasks and their weighting are listed below:
TASKSGRADE 10 GRADE 11 GRADE 12
WEIGHT (%) TASKS WEIGHT (%) TASKS WEIGHT (%)
Scho
ol-b
ased
Ass
essm
ent
Term 1
Projector
InvestigationTest
20
10
Projector
InvestigationTest
20
10
TestProject or
InvestigationAssignment / Test
1020
10
Term 2Assignment or
TestMid-year Exam
10
30
Assignment orTest
Mid-year Exam
10
30
TestMid-year Exam
1015
Term 3TestTest
1010
TestTest
1010
TestTrial Exam
1025
Term 4 Test 10 Test 10School-basedAssessment
mark100 100 100
School-basedAssessment
mark (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 same weighting of 25% should apply in cases where a project/investigation is in term 2.• The combination (25% and 75%) of the marks must appear in the learner’s report.• Graphic and non-programmable calculators are not allowed (for example, calculators which factorise
, or find roots of equations). 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 final examinations in Grades 10 and 11. Learners can be with formula sheet in Grade 12 for tests and examinations.
• Trigonometric functions and graphs will be examined in paper 2.
(b) Examinations:
In Grades 10, 11 and 12, 25% of the final promotion mark is a year mark and 75% is an examination mark.
All assessments in Grades 10 and 11 are internal while in Grade 12 the 25% year mark assessment is internally set and marked but externally moderated and the 75% examination is externally set, marked and moderated.
50 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
Mark distribution for Technical Mathematics NCS end-of-year papers: Grades 10 - 12
Description Grade 10 Grade 11 Grade. 12
PAPER 1:
Algebra ( Expressions, equations and inequalities including nature of roots in Grades 11 & 12)
60 ± 3 90 ± 3 50 ± 3
Functions & Graphs 25 ± 3 45 ± 3 35 ± 3
Finance, growth and decay 15 ± 3 15 ± 3 15 ± 3
Differential Calculus and Integration 50 ± 3
TOTAL 100 150 150
PAPER 2 : Grades 11 and 12: theorems and/or trigonometric proofs: maximum 12 marks
Description Grade 10 Grade 11 Grade 12
Analytical Geometry 15 ± 3 25 ± 3 25 ± 3
Trigonometry 40 ± 3 50 ± 3 50 ± 3
Euclidean Geometry 30 ± 3 40 ± 3 40 ± 3
Mensuration and circles, angles and angular movement
15 ± 3 35 ± 3 35 ± 3
TOTAL 100 150 150
Note: • 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 BUT for
Grade 12 formula sheet can be provided for tests and examinations. • 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 learner’s performance in a specific assessment task.
• It indicates learner progress towards the achievement of the knowledge as prescribed in the Curriculum and Assessment Policy Statements.
• Records of learner performance should provide evidence of the learner’s conceptual progression within a grade and her/his readiness to progress or to be promoted to the next grade.
• Records of learner performance should also be used to monitor the progress made by teachers and learners in the teaching and learning process.
• Reporting is a process of communicating learner performance to learners, parents, schools and other stakeholders. Learner performance can be reported in a number of ways.
• These include report cards, parents’ meetings, school visitation days, parent-teacher conferences, 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 and their corresponding percentage bands are shown in the Table below.
51CAPS TECHNICAL MATHEMATICS
CODES AND PERCENTAGES FOR RECORDING AND REPORTING
RATING CODE DESCRIPTION OF COMPETENCE PERCENTAGE7 Outstanding achievement 80 – 1006 Meritorious achievement 70 – 795 Substantial achievement 60 – 694 Adequate achievement 50 – 593 Moderate achievement 40 – 492 Elementary achievement 30 – 391 Not achieved 0 – 29
Note: The seven-point scale should have clear descriptors that give detailed information for each level. Teachers will record actual marks for the task on a record sheet; and indicate percentages for each subject on the learners’ report cards.
4.6 MODERATION OF ASSESSMENT
Moderation refers to the process which 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 National Curriculum Statement Grades R – 12; and
4.7.2 The policy document, National Protocol for Assessment Grades R – 12.
52 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
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