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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)

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ISBN: 978- 4315-0573-9

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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


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


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


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.


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.


(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.


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


1.4.3 Senior Phase

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


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:


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.


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.


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).


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 :


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.



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.


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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.


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

).


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.


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.


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).


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.


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


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

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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


2 Introduction Mathematical language and concepts used in previous years are 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


Page 27 of 59


GRADE 10: TERM 1





3 Number systems










2 Exponents


0, yx and Znm , .


Also by definition:

nn

xx 1 , 0x , and 10 x ,


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


GRADE 10: TERM 1





3 Number systems










2 Exponents


0, yx and Znm , .


Also by definition:

nn

xx 1 , 0x , and 10 x ,


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


GRADE 10: TERM 1





3 Number systems










2 Exponents


0, yx and Znm , .


Also by definition:

nn

xx 1 , 0x , and 10 x ,


2. 1

1

39

x

x (R)

Solve for x : 3. 273 x (K) 4. 366 x (K)

1100101

111

•

Page 27 of 59


GRADE 10: TERM 1





3 Number systems










2 Exponents


0, yx and Znm , .


Also by definition:

nn

xx 1 , 0x , and 10 x ,


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


GRADE 10: TERM 1





3 Number systems










2 Exponents


0, yx and Znm , .


Also by definition:

nn

xx 1 , 0x , and 10 x ,


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


GRADE 10: TERM 1





3 Number systems










2 Exponents


0, yx and Znm , .


Also by definition:

nn

xx 1 , 0x , and 10 x ,


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.




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).



1

Mensuration


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







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 .




1

Mensuration




introduced.

3












3. 2)54( yx (K) 4. yxyx 32 (K)


10. 4472 xx (R)

(K)7.

Page 28 of 59

0x .




1

Mensuration




introduced.

3












3. 2)54( yx (K) 4. yxyx 32 (K)


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.


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



2

Algebraic Expressions (continue)


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


1.2 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.



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



2



Simplify

11. xybybxxyx

bybxxyyx

222

(R)

12. 4

51

423

322

aaaa

a (R)

3 Equations

and Inequalities










2. 2133

x

x (R)





(Change the subject of the formula to a ) (R)6.

Page 30 of 59

GRADE 10: TERM 2



2



Simplify

11. xybybxxyx

bybxxyyx

222

(R)

12. 4

51

423

322

aaaa

a (R)

3 Equations

and Inequalities










2. 2133

x

x (R)





(Change the subject of the formula to v ) (R)


GRADE 10: TERM 2


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


872 xx

sin1 cosec


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 .





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 .





7. Trigonometry graphs sinay , cosay and

tanay for 3600 . qay sin and

qay cos for 3600 .

36090 . (C)


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.


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


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)


GRADE 10: TERM 3


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



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




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.


• Linear function:

Page 32 of 59

GRADE 10: TERM 3



1

Analytical Geometry






cmxy


3

Functions

and

Graphs








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



1

Analytical Geometry






cmxy


3

Functions

and

Graphs








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



1

Analytical Geometry






cmxy


3

Functions

and

Graphs








values)


GRADE 10: TERM 3


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.


GRADE 10: TERM 4


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



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




1

Circles, angles and

angular movement

1. Define a radian












- Add 32



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




1

Circles, angles and

angular movement

1. Define a radian












- Add 32



◦ Calculate 4cos2sin π+π

◦ Determine the value of the following: 577,0tan866,0cos5,0sin 111 −−− ++ (answer in

radians).


GRADE 10: TERM 4


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,


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








Assessment Term 4:


on functions and graphs (trig. functions will be examined in paper 2) and

Page 36 of 59








Assessment Term 4:


on finance and growth. Paper 2: 2 hours (100 marks) made up as follows:

Page 36 of 59








Assessment Term 4:


on trigonometry,

Page 36 of 59








Assessment Term 4:


on Analytical Geometry,

Page 36 of 59








Assessment Term 4:


on Euclidean Geometry and

Page 36 of 59








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.




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





;q pqp

xx 0;0 qx .

2. Add, subtract, multiply and divide simple surds.

3. Solve exponential equations.



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





;q pqp

xx 0;0 qx .






5. 22

3 6

9168

xxx

(R)

6. 3 233

851255

6255

(C)

Solve:

7. 1284 25x (R)




2

Logarithms

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





;q pqp

xx 0;0 qx .






5. 22

3 6

9168

xxx

(R)

6. 3 233

851255

6255

(C)

Solve:

7. 1284 25x (R)




2

Logarithms

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





;q pqp

xx 0;0 qx .






5. 22

3 6

9168

xxx

(R)

6. 3 233

851255

6255

(C)

Solve:

7. 1284 25x (R)




2

Logarithms

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





;q pqp

xx 0;0 qx .






5. 22

3 6

9168

xxx

(R)

6. 3 233

851255

6255

(C)

Solve:

7. 1284 25x (R)




2

Logarithms

Laws of logarithms



conversion from log-form to exponential form and vice versa. 2. Application of laws of logarithms: •

Page 37 of 59

GRADE 11: TERM 1





;q pqp

xx 0;0 qx .






5. 22

3 6

9168

xxx

(R)

6. 3 233

851255

6255

(C)

Solve:

7. 1284 25x (R)




2

Logarithms

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



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


abb

c

ca log

loglog



2.1 49log343log (R)



- Prove: 2.4 21


x

xxx (C)



4.1 x81log2 (K)


4 Equations










2.3

Page 38 of 59

yxyx

aaa logloglog

xnx an


abb

c

ca log

loglog



2.1 49log343log (R)



- Prove: 2.4 21


x

xxx (C)



4.1 x81log2 (K)


4 Equations










(R)

- Prove: 2.4

Page 38 of 59

yxyx

aaa logloglog

xnx an


abb

c

ca log

loglog



2.1 49log343log (R)



- Prove: 2.4 21


x

xxx (C)



4.1 x81log2 (K)


4 Equations










(C)

- Simplify: 2.5

Page 38 of 59

yxyx

aaa logloglog

xnx an


abb

c

ca log

loglog



2.1 49log343log (R)



- Prove: 2.4 21


x

xxx (C)



4.1 x81log2 (K)


4 Equations










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)




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.




4Functions

and graphs

1. Revise the effect of the parameters and q on the graphs. a


872 xx

sin1 cosec


22 cosec1cot

1

Investigate the effect of p on the graphs of the functions defined by:

1.1. qpxaxfy ++== 2)()(

1.2.

Page 40 of 59

GRADE 11: TERM 2



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).




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.


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.



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











Assessment Term 2:



), function and graphs (

Page 41 of 59











Assessment Term 2:



).Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry (

Page 41 of 59











Assessment Term 2:


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











Assessment Term 2:


graphs ( 325 ). Paper 2: 2 hours (100 marks made up as follows: Analytical Geometry ( 340 ) and Euclidean Geometry ( 360 ). ).




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


52,60 = 1806,52












4 Trigonometry







= 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




Page 43 of 59

2. Angular and


52,60 = 1806,52












4 Trigonometry







; 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


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


52,60 = 1806,52












4 Trigonometry







,

Page 43 of 59

2. Angular and


52,60 = 1806,52












4 Trigonometry







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


872 xx

sin1 cosec


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.




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.


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.


GRADE 11: TERM 4


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



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.


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,

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GRADE 11: TERM 4



3 Mensuration









1oom



Assessment Term 4:



on functions and graphs (excluding trigonometric functions) and

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GRADE 11: TERM 4



3 Mensuration









1oom



Assessment Term 4:



on finance growth and decay.Paper 2: 3 hours (150 marks made up as follows:

Page 46 of 59

GRADE 11: TERM 4



3 Mensuration









1oom



Assessment Term 4:



on trigonometry (including trigonometric functions),

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GRADE 11: TERM 4



3 Mensuration









1oom



Assessment Term 4:



on Analytical Geometry,

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GRADE 11: TERM 4



3 Mensuration









1oom



Assessment Term 4:


)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



3 Mensuration









1oom



Assessment Term 4:


)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.


GRADE 12: TERM 1


3 Complex numbers

• Define a complex number,

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GRADE 12: TERM 1


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:

,

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GRADE 12: TERM 1




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, xfdxdDx

Examples:

. • Learners should know:

◦ the conjugate of

Page 47 of 59

GRADE 12: TERM 1




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, 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.

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GRADE 12: TERM 1




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, xfdxdDx

Examples:

(R)

2.

Page 47 of 59

GRADE 12: TERM 1




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, xfdxdDx

Examples:

(K)

3.

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GRADE 12: TERM 1




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, xfdxdDx

Examples:

(R)4. iii 5132 −−+− (K)5.

Page 47 of 59

GRADE 12: TERM 1




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, xfdxdDx

Examples:

(K)

Solve for x and :y6.

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GRADE 12: TERM 1




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, 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




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, xfdxdDx

Examples:

(R)


GRADE 12: TERM 1


6DifferentialCalculus



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

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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

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6




hxfhxfxf

h

)()(lim)('0

for


baxxf )(




1.1 2)( xxf (R)

1.2 2)( 2 xxf (R)




21 atus .


Assessment Term 1:


for

Page 48 of 59

6




hxfhxfxf

h

)()(lim)('0

for


baxxf )(




1.1 2)( xxf (R)

1.2 2)( 2 xxf (R)




21 atus .


Assessment Term 1:


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




3 Complex numbers







Simplify : 1. 1416 (R) 2. 516 (K)

3. 6

12.4 (R)


2 Polynomials






hxfhxfm )()(



)(',, 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




hxfhxfxf

h

)()(lim)('0

for


baxxf )(




1.1 2)( xxf (R)

1.2 2)( 2 xxf (R)




21 atus .


Assessment Term 1:


.

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).


GRADE 12: TERM 2


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

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GRADE 12: TERM 2




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

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GRADE 12: TERM 2




3

Integration






1.1 10 xdx . (K)

1.2 .)32(21

23 dxxx (R)


2

Analytical Geometry







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

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GRADE 12: TERM 2




3

Integration






1.1 10 xdx . (K)

1.2 .)32(21

23 dxxx (R)


2

Analytical Geometry







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




3

Integration






1.1 10 xdx . (K)

1.2 .)32(21

23 dxxx (R)


2

Analytical Geometry







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).


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



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).


5. Plotting of the graph of ellipse,

12

2

2

2 b

yax

line with the equation 2 xy . (R)

3

Euclidean Geometry


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)


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:

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

In

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>> and

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>> . D is a point

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

so that

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

. E is a point on

Page 50 of 59




12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

so that

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

is parallel to

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

.Find the lengths of

Page 50 of 59




12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

and

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12

2

2

2 b

yax


3

Euclidean Geometry










Assessment Term 2:


>>

>>

. (R)


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.


GRADE 12: TERM 3


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.

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GRADE 12: TERM 3






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

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GRADE 12: TERM 3






3

Trigonometry







).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






3

Trigonometry







).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






3

Trigonometry







).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






3

Trigonometry







).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



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






3

Trigonometry







).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

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GRADE 12: TERM 3






3

Trigonometry







).335(

R

Q 20 150

28 m

10

P

T

R

Q 20 150

28 m

10

P

T

R

, Trigonometry

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GRADE 12: TERM 3






3

Trigonometry







).335(

R

Q 20 150

28 m

10

P

T

R

Q 20 150

28 m

10

P

T

R

, Euclidean Geometry

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GRADE 12: TERM 3






3

Trigonometry







).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






3

Trigonometry







).335(

R

Q 20 150

28 m

10

P

T

R

Q 20 150

28 m

10

P

T

R


GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4: Final examination:Paper 1: 150 marks: 3 hours Paper 2: 150 marks: 3 hoursAlgebraic expressions, equations

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GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:


Analytical Geometry

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GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:


and inequalities (nature of roots, logs and Trigonometry

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GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:


complex numbers) Euclidean Geometry

Page 52 of 59

GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:


Functions and graphs

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GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:


Mensuration, circles, angles and angular movement

Page 52 of 59

GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:


Finance, growth and decay

Page 52 of 59

GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:


Differential Calculus and Integration

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GRADE 12: TERM 4



3 Revision Revision

Assessment Term 4:




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.


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.


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 ÂOB 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 ÂOB 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)


01)30sin(2 x (Grade 11)

3. Prove that the angle ÂOBsubtended by arc AB at the centre O of a circle is double the size of the angle ÂCB 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 ÂOB subtended by arc AB at the centre O of a circle

is double the size of the angle ÂCBwhich 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



20%




vocabulary


3 2y f xx













01)30sin(2 x (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



20%




vocabulary


3 2y f xx













01)30sin(2 x (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.


(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.


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


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.


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.


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