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MATHEMATIC

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Curriculum and Assessment Policy Statement

Senior PhaseGrades 7-9

National Curriculum Statement (NCS)

CAPS

CurriCulum and assessment PoliCy statement Grades 7-9

matHematiCs

MATHEMATICS GRADES 7-9

CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)

disClaimer

In view of the stringent time requirements encountered by the Department of Basic Education to effect the necessary editorial changes and layout to the Curriculum and Assessment Policy Statements and the supplementary policy documents, possible errors may occur in the said documents placed on the official departmental websites.

There may also be vernacular inconsistencies in the language documents at Home-, First and Second Additional Language levels which have been translated in the various African Languages. Please note that the content of the documents translated and versioned in the African Languages are correct as they are based on the English generic language documents at all three language levels to be implemented in all four school phases.

If any editorial, layout or vernacular inconsistencies are detected, the user is kindly requested to bring this to the attention of the Department of Basic Education.

E-mail: [email protected] or fax (012) 328 9828

department of Basic education

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© 2011 department of Basic education

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Printed by: Government Printing Works


CAPS

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) and the National Curriculum Statement Grades 10-12 to produce this document.

From 2012 the two National Curriculum Statements, 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 represents a policy statement for learning and teaching in South African schools and comprises of the following:

(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


CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)


1CAPS

TABLE OF CONTENTS

seCtion 1: introduCtion and BaCKround ............................................................................... 3

1.1 Background ....................................................................................................................................................3

1.2 overview .....................................................................................................................................................3

1.3 General aims of the south african curriculum ............................................................................................4

1.4 time allocations .............................................................................................................................................6

1.4.1 Foundation Phase ..................................................................................................................................6

1.4.2 Intermediate Phase ................................................................................................................................6

1.4.3 Senior Phase..........................................................................................................................................7

1.4.4 Grades 10-12 .........................................................................................................................................7

seCtion 2: deFinition, aims, sKills and Content .................................................................... 8

2.1 introduction ....................................................................................................................................................8

2.2 What is mathematics?....................................................................................................................................8

2.3 Specificaims ..................................................................................................................................................8

2.4 Specificskills..................................................................................................................................................8

2.5 Focus of content areas ..................................................................................................................................9

mathematics content knowledge ....................................................................................................................10

2.6 Weighting of content areas .........................................................................................................................11

2.7 Specificationofcontent ..............................................................................................................................11

• Numbers, Operations and Relationships ..................................................................................................12

• Patterns, Functions and Algebra ...............................................................................................................21

• Space and Shape (Geometry) ..................................................................................................................27

• measurement ............................................................................................................................................31

• Data Handling ...........................................................................................................................................33

seCtion 3: ClariFiCation oF Content ....................................................................................... 37

3.1 introduction ..................................................................................................................................................37

3.2 allocation of teaching time .........................................................................................................................37


2 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)

3.3 Clarificationnoteswithteachingguidelines .............................................................................................38

3.3.1 Clarification of content for Grade 7 ......................................................................................................39

• Grade 7 term 1 ................................................................................................................................39

• Grade 7 term 2 ................................................................................................................................49

• Grade 7 term 3 ................................................................................................................................58

• Grade 7 term 4 ................................................................................................................................67

3.3.2 Clarification of content for Grade 8 ......................................................................................................75

• Grade 8 term 1 ................................................................................................................................75

• Grade 8 term 2 ................................................................................................................................92

• Grade 8 term 3 ..............................................................................................................................100

• Grade 8 term 4 ..............................................................................................................................113

3.3.3 Clarification of content for Grade 9...............................................................................................................119

• Grade 9 term 1 ..............................................................................................................................119

• Grade 9 term 2 ..............................................................................................................................134

• Grade 9 term 3 ..............................................................................................................................141

• Grade 9 term 4 ..............................................................................................................................147

seCtion 4: assessment ................................................................................................................ 154 4.1 introduction ...............................................................................................................................................154

4.2 types of assessment .................................................................................................................................154

4.3 informal or daily assessment ....................................................................................................................155

4.4 Formal assessment ....................................................................................................................................155

4.5 recording and reporting ...........................................................................................................................157

4.6 moderation of assessment ........................................................................................................................158

4.7 General .................................................................................................................................................158


3CAPS

SECTION 1: INTRODUCTION AND BACKGROUND

1.1 BaCKGround

The National Curriculum Statement Grades R-12 (NCS) stipulates policy on curriculum and assessment in the schooling sector.

To improve implementation, the National Curriculum Statement was amended, with the amendments coming into effect in January 2012. A single comprehensive Curriculum and Assessment Policy document was developed for each subject to replace Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines in Grades R-12.

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:

(i) Curriculum and Assessment Policy Statements for each approved school subject;

(ii) The policy document, National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R-12; and

(iii) The policy document, National Protocol for Assessment Grades R-12 (January 2012).

(b) The National Curriculum Statement Grades R-12 (January 2012) replaces the two current national curricula statements, namely the

(i) Revised National Curriculum Statement Grades R-9, Government Gazette No. 23406 of 31 May 2002, and

(ii) National Curriculum Statement Grades 10-12 Government Gazettes, No. 25545 of 6 October 2003 and No. 27594 of 17 May 2005.

(c) The national curriculum statements contemplated in subparagraphs b(i) and (ii) comprise the following policy documents which will be incrementally repealed by the National Curriculum Statement Grades R-12 (January 2012) during the period 2012-2014:

(i) The Learning Area/Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines for Grades R-9 and Grades 10-12;

(ii) The policy document, National Policy on assessment and qualifications for schools in the GeneralEducation and Training Band, promulgated in Government Notice No. 124 in Government Gazette No. 29626 of 12 February 2007;

(iii) The policy document, the National Senior Certificate: A qualification at Level 4 on the NationalQualificationsFramework(NQF),promulgatedinGovernmentGazetteNo.27819of20July2005;



(iv) The policy document, An addendum to the policy document, the National Senior Certificate: AqualificationatLevel4ontheNationalQualificationsFramework(NQF),regardinglearnerswithspecialneeds, published in Government Gazette, No.29466 of 11 December 2006, is incorporated in the policy document, National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R-12; and

(v) The policy document, An addendum to the policy document, the National Senior Certificate: AqualificationatLevel4ontheNationalQualificationsFramework(NQF),regardingtheNationalProtocolfor Assessment (Grades R-12), promulgated in Government Notice No.1267 in Government Gazette No. 29467 of 11 December 2006.

(d) The policy document, National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R-12, and the sections on the Curriculum and Assessment Policy as contemplated in Chapters 2, 3 and 4 of this document constitute the norms and standards of the National Curriculum Statement Grades R-12. It will therefore, in terms of section 6A of the South African Schools Act, 1996(ActNo.84of1996,) form the basis for the minister of Basic Education to determine minimum outcomes and standards, as well as the processes and procedures for the assessment of learner achievement to be applicable to public and independent schools.

1.3 General aims oF tHe soutH aFriCan CurriCulum

(a) The National Curriculum Statement Grades R-12 gives expression to the knowledge, skills and values worth learning in South African schools. This curriculum aims to ensure that children acquire and apply knowledge and skills in ways that are meaningful to their own lives. In this regard, the curriculum promotes 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 competences.

(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 the 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: the minimum standards of knowledge and skills to be achieved at each grade are specified and set high, achievable standards in all subjects;

• Progression: content and context of each grade shows progression from simple to complex;


5CAPS

• 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 R-12 is sensitive to issues of diversity such as poverty, inequality, race, gender, language, age, disability and other factors;

• Valuing 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 effectively;

• collect, analyse, organise and critically evaluate information;

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

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

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

(e) Inclusivity should become a central part of the organisation, planning and teaching at each school. This can only happen if all teachers have a sound understanding of how to recognise and address barriers to learning, and how to plan for diversity.

The key to managing inclusivity is ensuring 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 use various curriculum differentiation strategies such as those included 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 in the Foundation Phase is as follows:

suBJeCtGrade r (Hours)

Grades 1-2 (Hours)

Grade 3 (Hours)

Home Language 10 8/7 8/7

First Additional Language 2/3 3/4

mathematics 7 7 7

Life Skills

• Beginning Knowledge

• Creative Arts

• Physical Education

• Personal and Social Well-being

6

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(2)

(2)

(1)

6

(1)

(2)

(2)

(1)

7

(2)

(2)

(2)

(1)

total 23 23 25

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

(c) Ten hours are allocated for languages in Grades R-2 and 11 hours in Grade 3. A maximum of 8 hours and a minimum of 7 hours are allocated for Home Language and a minimum of 2 hours and a maximum of 3 hours for Additional Language in Grades 1-2. In Grade 3 a maximum of 8 hours and a minimum of 7 hours are allocated for Home Language and a minimum of 3 hours and a maximum of 4 hours for First Additional Language.

(d) In Life Skills Beginning Knowledge is allocated 1 hour in Grades R – 2 and 2 hours as indicated by the hours in brackets for Grade 3.

1.4.2 intermediate Phase

(a) The instructional time in the Intermediate Phase is as follows:

suBJeCt Hours

Home Language 6

First Additional Language 5

mathematics 6

Natural Sciences and Technology 3,5

Social Sciences 3

Life Skills

• Creative Arts

• Physical Education

• Personal and Social Well-being

4

(1,5)

(1)

(1,5)

total 27,5


7CAPS

1.4.3 senior Phase

(a) The instructional time in the Senior Phase is as follows:

suBJeCt Hours

Home Language 5

First Additional Language 4

mathematics 4,5

Natural Sciences 3

Social Sciences 3

Technology 2

Economic management Sciences 2

Life Orientation 2

Creative Arts 2

total 27,5

1.4.4 Grades 10-12

(a) The instructional time in Grades 10-12 is as follows:

suBJeCt time alloCation Per WeeK (Hours)

Home Language 4.5

First Additional Language 4.5

mathematics 4.5

Life Orientation 2

A minimum of any three subjects selected from Group B Annexure B, Tables B1-B8 of the policy document, National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R-12, subject to the provisos stipulated in paragraph 28 of the said policy document.

12 (3x4h)

total 27,5

The allocated time per week may be utilised only for the minimum 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 must be allocated for the offering of these subjects.



SECTION 2: DEFINITION, AIMS, SKILLS AND CONTENT

2.1 introduCtion

In Section 2, the Senior Phase Mathematics Curriculum and Assessment Policy Statement (CAPS) provides teachers with a definition of mathematics, specific aims, specific skills, focus of content areas, weighting of content areas and content specification.

2.2 WHat is matHematiCs?

mathematics is a language that makes use of symbols and notations to describe numerical, geometric and graphical relationships. It is a human activity that involves observing, representing and investigating patterns and quantitative 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.

2.3 sPeCiFiC aims

The teaching and learning of mathematics aims to develop

• a critical awareness of how mathematical relationships are used in social, environmental, cultural and economic relations

• confidence and competence to deal with any mathematical situation without being hindered by a fear of mathematics

• an appreciation for the beauty and elegance of mathematics

• a spirit of curiosity and a love for mathematics

• recognition that mathematics is a creative part of human activity

• deep conceptual understandings in order to make sense of mathematics

• acquisition of specific knowledge and skills necessary for:

- the application of mathematics to physical, social and mathematical problems

- the study of related subject matter (e.g. other subjects)

- further study in Mathematics.

2.4 sPeCiFiC sKills

To develop essential mathematical skills the learner should

• develop the correct use of the language of mathematics

• develop number vocabulary, number concept and calculation and application skills


9CAPS

• learn to listen, communicate, think, reason logically and apply the mathematical knowledge gained

• learn to investigate, analyse, represent and interpret information

• learn to pose and solve problems

• build an awareness of the important role that mathematics plays in real life situations including the personal development of the learner.

2.5 FoCus oF Content areas

Mathematics in the Senior Phase covers five main Content Areas.

• Numbers, Operations and Relationships;

• Patterns, Functions and Algebra;

• Space and Shape (Geometry);

• measurement; and

• Data Handling.

Each content area contributes towards the acquisition of specific skills. The table below shows the general focus of the content areas as well as the specific focus of the content areas for the Senior Phase.



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ns a

bout

situ

atio

ns•

Pro

babi

lity

of o

utco

mes

incl

ude

both

sin

gle

and

com

poun

d ev

ents

and

thei

r re

lativ

e fre

quen

cy in

sim

ple

expe

rimen

ts


11CAPS

2.6 WeiGHtinG oF Content areas

The weighting of mathematics content areas serves two primary purposes:

• guidance on the time needed to adequately address the content within each content area

• guidance on the spread of content in the examination (especially end-of-year summative assessment).

WeiGHtinG oF Content areas

Content Area Grade 7 Grade 8 Grade 9

Number, Operations and Relations 30% 25% 15%

Patterns, Functions and Algebra 25% 30% 35%

Space and Shape (Geometry) 25% 25% 30%

measurement 10% 10% 10%

Data Handling 10% 10% 10%

100% 100% 100%

2.7 sPeCiFiCation oF Content

The Specification of Content in Section 2 shows progression in terms of concepts and skills from Grades 7 - 9 for each Content Area. However, in certain topics the concepts and skills are similar in two or three successive grades. The Clarification of Content in Section 3 provides guidelines on how progression should be addressed in these cases. The Specification of Content in Section 2 should therefore be read in conjunction with the Clarification of Content in Section 3.



sPeC

iFiC

atio

n o

F C

on

ten

t (P

Ha

se o

ver

vieW

)

nu

mB

ers,

oPe

rat

ion

s a

nd

rel

atio

nsH

iPs

• P

rogr

essi

on in

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

in th

e S

enio

r Pha

se is

ach

ieve

d pr

imar

ily b

y:

-de

velo

pmen

t of c

alcu

latio

ns u

sing

who

le n

umbe

rs to

cal

cula

tions

usi

ng ra

tiona

l num

bers

, int

eger

s an

d nu

mbe

rs in

exp

onen

tial f

orm

-de

velo

pmen

t of u

nder

stan

ding

of d

iffer

ent n

umbe

r sys

tem

s fro

m n

atur

al a

nd w

hole

num

bers

to in

tege

rs a

nd ra

tiona

l num

bers

, as

wel

l as

the

reco

gniti

on o

f irr

atio

nal n

umbe

rs

-in

crea

sing

use

of p

rope

rties

of n

umbe

rs to

per

form

cal

cula

tions

-in

crea

sing

com

plex

ity o

f con

text

s fo

r sol

ving

pro

blem

s

• N

umbe

rs, O

pera

tions

and

Rel

atio

nshi

ps in

the

Sen

ior P

hase

con

solid

ates

wor

k do

ne in

the

Inte

rmed

iate

Pha

se a

nd is

gea

red

tow

ards

mak

ing

lear

ners

com

pete

nt a

nd e

ffici

ent i

n pe

rform

ing

calc

ulat

ions

par

ticul

arly

with

inte

gers

and

ratio

nal n

umbe

rs.

• R

ecog

nisi

ng a

nd u

sing

the

prop

ertie

s of

ope

ratio

ns fo

r diff

eren

t num

bers

pro

vide

s a

criti

cal f

ound

atio

n fo

r wor

k in

alg

ebra

whe

n le

arne

rs w

ork

with

var

iabl

es in

pla

ce o

f num

bers

and

m

anip

ulat

e al

gebr

aic

expr

essi

ons

and

solv

e al

gebr

aic

equa

tions

.

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.1

Who

le n

umbe

rsm

enta

l cal

cula

tions

Rev

ise

the

follo

win

g do

ne in

Gra

de 6

:

• M

ultip

licat

ion

of w

hole

num

bers

to a

t lea

st 1

2 x

12

• m

ultip

licat

ion

fact

s fo

r:

-un

its a

nd te

ns b

y m

ultip

les

of te

n

-un

its a

nd te

ns b

y m

ultip

les

of 1

00

-un

its a

nd te

ns b

y m

ultip

les

of 1

000

-un

its a

nd te

ns b

y m

ultip

les

of 1

0 00

0

• In

vers

e op

erat

ion

betw

een

mul

tiplic

atio

n an

d di

visi

on

ord

erin

g an

d co

mpa

ring

who

le n

umbe

rs

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-or

der,

com

pare

and

repr

esen

t num

bers

to a

t le

ast 9

-dig

it nu

mbe

rs

-re

cogn

ize

and

repr

esen

t prim

e nu

mbe

rs to

at

leas

t 100

-ro

und

off n

umbe

rs to

the

near

est 5

, 10,

100

or

1 00

0

men

tal c

alcu

latio

ns

• R

evis

e m

ultip

licat

ion

of w

hole

num

bers

to a

t lea

st

12 x

12

ord

erin

g an

d co

mpa

ring

who

le n

umbe

rs

• R

evis

e pr

ime

num

bers

to a

t lea

st 1

00


13CAPS

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.1

Who

le n

umbe

rsPr

oper

ties

of w

hole

num

bers

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-re

cogn

ize

and

use

the

com

mut

ativ

e;

asso

ciat

ive;

dis

tribu

tive

prop

ertie

s of

who

le

num

bers

-re

cogn

ize

and

use

0 in

term

s of

its

addi

tive

prop

erty

(ide

ntity

ele

men

t for

add

ition

)

-re

cogn

ize

and

use

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty (i

dent

ity e

lem

ent f

or m

ultip

licat

ion)

Cal

cula

tions

usi

ng w

hole

num

bers

• R

evis

e th

e fo

llow

ing

done

in G

rade

6, w

ithou

t use

of

cal

cula

tors

:

-A

dditi

on a

nd s

ubtra

ctio

n of

who

le n

umbe

rs to

at

leas

t 6-d

igit

num

bers

-M

ultip

licat

ion

of a

t lea

st w

hole

4-d

igit

by 2

-dig

it nu

mbe

rs

-D

ivis

ion

of a

t lea

st w

hole

4-d

igit

by 2

-dig

it nu

mbe

rs

-P

erfo

rm c

alcu

latio

ns u

sing

all

four

ope

ratio

ns

on w

hole

num

bers

, est

imat

ing

and

usin

g ca

lcul

ator

s w

here

app

ropr

iate

Cal

cula

tion

tech

niqu

es

• U

se a

rang

e of

stra

tegi

es to

per

form

and

che

ck

writ

ten

and

men

tal c

alcu

latio

ns o

f who

le n

umbe

rs

incl

udin

g:

-lo

ng d

ivis

ion

-ad

ding

, sub

tract

ing

and

mul

tiply

ing

in c

olum

ns

-es

timat

ion

-ro

undi

ng o

ff an

d co

mpe

nsat

ing

-us

ing

a ca

lcul

ator

Prop

ertie

s of

who

le n

umbe

rs

• R

evis

e:

-Th

e co

mm

utat

ive;

ass

ocia

tive;

dis

tribu

tive

prop

ertie

s of

who

le n

umbe

rs

-0

in te

rms

of it

s ad

ditiv

e pr

oper

ty (i

dent

ity

elem

ent f

or a

dditi

on)

-1

in te

rms

of it

s m

ultip

licat

ive

prop

erty

(ide

ntity

el

emen

t for

mul

tiplic

atio

n)

• R

ecog

nize

the

divi

sion

pro

perty

of 0

, whe

reby

any

nu

mbe

r div

ided

by

0 is

und

efine

d

Cal

cula

tions

usi

ng w

hole

num

bers

• R

evis

e ca

lcul

atio

ns u

sing

all

four

ope

ratio

ns o

n w

hole

num

bers

, est

imat

ing

and

usin

g ca

lcul

ator

s w

here

app

ropr

iate

Cal

cula

tion

tech

niqu

es

• U

se a

rang

e of

tech

niqu

es to

per

form

and

che

ck

writ

ten

and

men

tal c

alcu

latio

ns o

f who

le n

umbe

rs

incl

udin

g:

-lo

ng d

ivis

ion

-ad

ding

, sub

tract

ing

and

mul

tiply

ing

in c

olum

ns

-es

timat

ion

-ro

undi

ng o

ff an

d co

mpe

nsat

ing

-us

ing

a ca

lcul

ator

Prop

ertie

s of

num

bers

• D

escr

ibe

the

real

num

ber s

yste

m b

y re

cogn

isin

g,

defin

ing

and

dist

ingu

ishi

ng p

rope

rties

of:

-na

tura

l num

bers

-w

hole

num

bers

-in

tege

rs

-ra

tiona

l num

bers

-irr

atio

nal n

umbe

rs

Cal

cula

tions

usi

ng w

hole

num

bers

• R

evis

e ca

lcul

atio

ns u

sing

all

four

ope

ratio

ns o

n w

hole

num

bers

, est

imat

ing

and

usin

g ca

lcul

ator

s w

here

app

ropr

iate

Cal

cula

tion

tech

niqu

es

• U

se a

rang

e of

tech

niqu

es to

per

form

and

che

ck

writ

ten

and

men

tal c

alcu

latio

ns o

f who

le n

umbe

rs

incl

udin

g:

-lo

ng d

ivis

ion

-ad

ding

, sub

tract

ing

and

mul

tiply

ing

in c

olum

ns

-es

timat

ion

-ro

undi

ng o

ff an

d co

mpe

nsat

ing

-us

ing

a ca

lcul

ator



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.1

Who

le n

umbe

rsm

ultip

les

and

fact

ors

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-m

ultip

les

of 2

-dig

it an

d 3-

digi

t who

le n

umbe

rs

-fa

ctor

s of

2-d

igit

and

3-di

git w

hole

num

bers

-pr

ime

fact

ors

of n

umbe

rs to

at l

east

100

• Li

st p

rime

fact

ors

of n

umbe

rs to

at l

east

3-d

igit

who

le n

umbe

rs

• Fi

nd th

e LC

m a

nd H

CF

of n

umbe

rs to

at

leas

t 3-d

igit

who

le n

umbe

rs, b

y in

spec

tion

or

fact

oris

atio

n

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

volv

ing

who

le n

umbe

rs,

incl

udin

g

-co

mpa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd (r

atio

)

-co

mpa

ring

two

quan

titie

s of

diff

eren

t kin

ds (r

ate)

-sh

arin

g in

a g

iven

ratio

whe

re th

e w

hole

is g

iven

• S

olve

pro

blem

s th

at in

volv

e w

hole

num

bers

, pe

rcen

tage

s an

d de

cim

al fr

actio

ns in

fina

ncia

l co

ntex

ts s

uch

as:

-pr

ofit,

loss

and

dis

coun

t

-bu

dget

s

-ac

coun

ts

-lo

ans

-si

mpl

e in

tere

st

mul

tiple

s an

d fa

ctor

s

• R

evis

e:

-P

rime

fact

ors

of n

umbe

rs t

o at

lea

st 3

-dig

it w

hole

num

bers

-LC

M a

nd H

CF

of n

umbe

rs t

o at

lea

st 3

-dig

it w

hole

num

bers

, by

insp

ectio

n or

fact

oris

atio

n

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

volv

ing

who

le n

umbe

rs,

incl

udin

g

-co

mpa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd (r

atio

)

-co

mpa

ring

two

quan

titie

s of

diff

eren

t kin

ds (r

ate)

-sh

arin

g in

a g

iven

ratio

whe

re th

e w

hole

is g

iven

-in

crea

sing

or d

ecre

asin

g of

a n

umbe

r in

a gi

ven

ratio

• S

olve

pro

blem

s th

at in

volv

e w

hole

num

bers

, pe

rcen

tage

s an

d de

cim

al fr

actio

ns in

fina

ncia

l co

ntex

ts s

uch

as:

-pr

ofit,

loss

, dis

coun

t and

VAT

-bu

dget

s

-ac

coun

ts

-lo

ans

-si

mpl

e in

tere

st

-hi

re p

urch

ase

-ex

chan

ge ra

tes

mul

tiple

s an

d fa

ctor

s

• U

se p

rime

fact

oris

atio

n of

num

bers

to fi

nd L

CM

an

d H

CF

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

-ra

tio a

nd ra

te

-di

rect

and

indi

rect

pro

porti

on

• S

olve

pro

blem

s th

at in

volv

e w

hole

num

bers

, pe

rcen

tage

s an

d de

cim

al fr

actio

ns in

fina

ncia

l co

ntex

ts s

uch

as:

-pr

ofit,

loss

, dis

coun

t and

VAT

-bu

dget

s

-ac

coun

ts

-lo

ans

-S

impl

e in

tere

st

-hi

re p

urch

ase

-ex

chan

ge ra

tes

-co

mm

issi

on

-re

ntal

s

-co

mpo

und

inte

rest


15CAPS

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.2

expo

nent

sm

enta

l cal

cula

tions

• D

eter

min

e sq

uare

s to

at l

east

122 a

nd th

eir

squa

re ro

ots

• D

eter

min

e cu

bes

to a

t lea

st 6

3 and

cub

e ro

ots

Com

parin

g an

d re

pres

entin

g nu

mbe

rs in

ex

pone

ntia

l for

m

• C

ompa

re a

nd re

pres

ent w

hole

num

bers

in

expo

nent

ial f

orm

: ab =

a x

a x

a x

... fo

r b n

umbe

r of f

acto

rs

Cal

cula

tions

usi

ng n

umbe

rs in

exp

onen

tial f

orm

• R

ecog

nize

and

use

the

appr

opria

te la

ws

of

oper

atio

ns w

ith n

umbe

rs in

volv

ing

expo

nent

s an

d sq

uare

and

cub

e ro

ots

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our o

pera

tions

us

ing

num

bers

in e

xpon

entia

l for

m, l

imite

d to

ex

pone

nts

up to

5, a

nd s

quar

e an

d cu

be ro

ots

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

num

bers

in

expo

nent

ial f

orm

.

men

tal c

alcu

latio

ns

• R

evis

e:

-S

quar

es to

at l

east

122 a

nd th

eir s

quar

e ro

ots

-C

ubes

to a

t lea

st 6

3 and

thei

r cub

e ro

ots

Com

parin

g an

d re

pres

entin

g nu

mbe

rs in

ex

pone

ntia

l for

m

• R

evis

e co

mpa

re a

nd re

pres

ent w

hole

num

bers

in

expo

nent

ial f

orm

• C

ompa

re a

nd re

pres

ent i

nteg

ers

in e

xpon

entia

l fo

rm

• C

ompa

re a

nd re

pres

ent n

umbe

rs in

sci

entifi

c no

tatio

n, li

mite

d to

pos

itive

exp

onen

ts

Cal

cula

tions

usi

ng n

umbe

rs in

exp

onen

tial f

orm

• E

stab

lish

gene

ral l

aws

of e

xpon

ents

, lim

ited

to:

-na

tura

l num

ber e

xpon

ents

-am

x a

n = a

m +

n

-am

÷ a

n = a

m –

n, i

f m>n

-(a

m)n =

am

x n

-(a

x t)

n = a

n x

tn

-a0 =

1

• R

ecog

nize

and

use

the

appr

opria

te la

ws

of

oper

atio

ns u

sing

num

bers

invo

lvin

g ex

pone

nts

and

squa

re a

nd c

ube

root

s

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our o

pera

tions

w

ith n

umbe

rs th

at in

volv

e th

e sq

uare

s, c

ubes

, sq

uare

root

s an

d cu

be ro

ots

of in

tege

rs

• C

alcu

late

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

ratio

nal n

umbe

rs

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

num

bers

in

expo

nent

ial f

orm

Com

parin

g an

d re

pres

entin

g nu

mbe

rs in

ex

pone

ntia

l for

m

• R

evis

e co

mpa

re a

nd re

pres

ent i

nteg

ers

in

expo

nent

ial f

orm

-co

mpa

re a

nd r

epre

sent

num

bers

in

scie

ntifi

c no

tatio

n

• E

xten

d sc

ient

ific

nota

tion

to in

clud

e ne

gativ

e ex

pone

nts

Cal

cula

tions

usi

ng n

umbe

rs in

exp

onen

tial f

orm

• R

evis

e th

e fo

llow

ing

gene

ral l

aws

of e

xpon

ents

:

-am

x a

n = a

m +

n

-am

÷ a

n = a

m –

n, i

f m>n

-(a

m)n =

am

x n

-(a

x t)

n = a

n x

tn

-a0 =

1

• E

xten

d th

e ge

nera

l law

s of

exp

onen

ts to

incl

ude:

-in

tege

r exp

onen

ts

-a–m

= 1

am

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our o

pera

tions

us

ing

num

bers

in e

xpon

entia

l for

m, u

sing

the

law

s of

exp

onen

ts

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

num

bers

in

expo

nent

ial f

orm

, inc

ludi

ng s

cien

tific

nota

tion



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.3

inte

gers

Cou

ntin

g, o

rder

ing

and

com

parin

g in

tege

rs

• C

ount

forw

ards

and

bac

kwar

ds in

inte

gers

for a

ny

inte

rval

• R

ecog

nize

, ord

er a

nd c

ompa

re in

tege

rs

Cal

cula

tions

with

inte

gers

• A

dd a

nd s

ubtra

ct w

ith in

tege

rs

Prop

ertie

s of

inte

gers

• R

ecog

nise

and

use

com

mut

ativ

e an

d as

soci

ativ

e pr

oper

ties

of a

dditi

on a

nd m

ultip

licat

ion

for

inte

gers

solv

ing

prob

lem

s

Sol

ve p

robl

ems

in c

onte

xts

invo

lvin

g ad

ditio

n an

d su

btra

ctio

n w

ith in

tege

rs

Cou

ntin

g, o

rder

ing

and

com

parin

g in

tege

rs

• R

evis

e:

-co

untin

g fo

rwar

ds a

nd b

ackw

ards

in in

tege

rs

for a

ny in

terv

al

-re

cogn

izin

g, o

rder

ing

and

com

parin

g in

tege

rs

Cal

cula

tions

with

inte

gers

• R

evis

e ad

ditio

n an

d su

btra

ctio

n w

ith in

tege

rs

• m

ultip

ly a

nd d

ivid

e w

ith in

tege

rs

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our o

pera

tions

w

ith in

tege

rs

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our o

pera

tions

w

ith n

umbe

rs th

at in

volv

e th

e sq

uare

s, c

ubes

, sq

uare

root

s an

d cu

be ro

ots

of in

tege

rs

Prop

ertie

s of

inte

gers

• R

ecog

nise

and

use

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e pr

oper

ties

of a

dditi

on a

nd

mul

tiplic

atio

n fo

r int

eger

s

• R

ecog

nize

and

use

add

itive

and

mul

tiplic

ativ

e in

vers

es fo

r int

eger

s

solv

ing

prob

lem

s

Sol

ve p

robl

ems

in c

onte

xts

invo

lvin

g m

ultip

le

oper

atio

ns w

ith in

tege

rs

Cal

cula

tions

with

inte

gers

• R

evis

e:

-pe

rform

cal

cula

tions

invo

lvin

g al

l fou

r op

erat

ions

with

inte

gers

-pe

rform

cal

cula

tions

invo

lvin

g al

l fou

r op

erat

ions

with

num

bers

that

invo

lve

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

in

tege

rs

Prop

ertie

s of

inte

gers

• R

evis

e:

-C

omm

utat

ive,

ass

ocia

tive

and

dist

ribut

ive

prop

ertie

s of

add

ition

and

mul

tiplic

atio

n fo

r in

tege

rs

-ad

ditiv

e an

d m

ultip

licat

ive

inve

rses

for i

nteg

ers

solv

ing

prob

lem

s

Sol

ve p

robl

ems

in c

onte

xts

invo

lvin

g m

ultip

le

oper

atio

ns w

ith in

tege

rs


17CAPS

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.4

Com

mon

fr

actio

ns

ord

erin

g, c

ompa

ring

and

sim

plify

ing

frac

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

6

-co

mpa

re a

nd o

rder

com

mon

frac

tions

, inc

ludi

ng

spec

ifica

lly te

nths

and

hun

dred

ths

• E

xten

d to

thou

sand

ths

Cal

cula

tions

with

frac

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-ad

ditio

n an

d su

btra

ctio

n of

com

mon

frac

tions

, in

clud

ing

mix

ed n

umbe

rs, l

imite

d to

frac

tions

w

ith th

e sa

me

deno

min

ator

or w

here

one

de

nom

inat

or is

a m

ultip

le o

f ano

ther

-fin

ding

frac

tions

of w

hole

num

bers

• E

xten

d ad

ditio

n an

d su

btra

ctio

n to

frac

tions

w

here

one

den

omin

ator

is n

ot a

mul

tiple

of t

he

othe

r

• m

ultip

licat

ion

of c

omm

on fr

actio

ns, i

nclu

ding

m

ixed

num

bers

, not

lim

ited

to fr

actio

ns w

here

one

de

nom

inat

or is

a m

ultip

le o

f ano

ther

Cal

cula

tion

tech

niqu

es

• C

onve

rt m

ixed

num

bers

to c

omm

on fr

actio

ns in

or

der t

o pe

rform

cal

cula

tions

with

them

• U

se k

now

ledg

e of

mul

tiple

s an

d fa

ctor

s to

writ

e fra

ctio

ns in

the

sim

ples

t for

m b

efor

e or

afte

r ca

lcul

atio

ns

• U

se k

now

ledg

e of

equ

ival

ent f

ract

ions

to a

dd a

nd

subt

ract

com

mon

frac

tions

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

com

mon

fra

ctio

ns a

nd m

ixed

num

bers

, inc

ludi

ng g

roup

ing,

sh

arin

g an

d fin

ding

frac

tions

of w

hole

num

bers

Cal

cula

tions

with

frac

tions

• R

evis

e:

-ad

ditio

n an

d su

btra

ctio

n of

com

mon

frac

tions

, in

clud

ing

mix

ed n

umbe

rs

-fin

ding

frac

tions

of w

hole

num

bers

-m

ultip

licat

ion

of c

omm

on fr

actio

ns, i

nclu

ding

m

ixed

num

bers

• D

ivid

e w

hole

num

bers

and

com

mon

frac

tions

by

com

mon

frac

tions

• C

alcu

late

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

com

mon

frac

tions

Cal

cula

tion

tech

niqu

es

• R

evis

e:

-co

nver

t mix

ed n

umbe

rs to

com

mon

frac

tions

in

orde

r to

perfo

rm c

alcu

latio

ns w

ith th

em

-us

e kn

owle

dge

of m

ultip

les

and

fact

ors

to w

rite

fract

ions

in th

e si

mpl

est f

orm

bef

ore

or a

fter

calc

ulat

ions

-us

e kn

owle

dge

of e

quiv

alen

t fra

ctio

ns to

add

an

d su

btra

ct c

omm

on fr

actio

ns

• U

se k

now

ledg

e of

reci

proc

al re

latio

nshi

ps to

di

vide

com

mon

frac

tions

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

com

mon

fra

ctio

ns a

nd m

ixed

num

bers

, inc

ludi

ng g

roup

ing,

sh

arin

g an

d fin

ding

frac

tions

of w

hole

num

bers

Cal

cula

tions

with

frac

tions

• A

ll fo

ur o

pera

tions

with

com

mon

frac

tions

and

m

ixed

num

bers

• A

ll fo

ur o

pera

tions

, with

num

bers

that

invo

lve

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

co

mm

on fr

actio

ns

Cal

cula

tion

tech

niqu

es

• R

evis

e:

-co

nver

t mix

ed n

umbe

rs to

com

mon

frac

tions

in

orde

r to

perfo

rm c

alcu

latio

ns w

ith th

em

-us

e kn

owle

dge

of m

ultip

les

and

fact

ors

to w

rite

fract

ions

in th

e si

mpl

est f

orm

bef

ore

or a

fter

calc

ulat

ions

-us

e kn

owle

dge

of e

quiv

alen

t fra

ctio

ns to

add

an

d su

btra

ct c

omm

on fr

actio

ns

-us

e kn

owle

dge

of re

cipr

ocal

rela

tions

hips

to

divi

de c

omm

on fr

actio

ns

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

com

mon

fra

ctio

ns, m

ixed

num

bers

and

per

cent

ages



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.4

Com

mon

Fr

actio

ns

Perc

enta

ges

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-Fi

ndin

g pe

rcen

tage

s of

who

le n

umbe

rs

• C

alcu

late

the

perc

enta

ge o

f par

t of a

who

le

• C

alcu

late

per

cent

age

incr

ease

or d

ecre

ase

of

who

le n

umbe

rs

• S

olve

pro

blem

s in

con

text

s in

volv

ing

perc

enta

ges

equi

vale

nt fo

rms

Rev

ise

the

follo

win

g do

ne in

Gra

de 6

:

• re

cogn

ize

and

use

equi

vale

nt fo

rms

of c

omm

on

fract

ions

with

1-d

igit

or 2

-dig

it de

nom

inat

ors

(frac

tions

whe

re o

ne d

enom

inat

or is

a m

ultip

le o

f th

e ot

her)

• re

cogn

ize

equi

vale

nce

betw

een

com

mon

frac

tion

and

deci

mal

frac

tion

form

s of

the

sam

e nu

mbe

r

• re

cogn

ize

equi

vale

nce

betw

een

com

mon

frac

tion,

de

cim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

Perc

enta

ges

• R

evis

e:

-fin

ding

per

cent

ages

of w

hole

num

bers

-ca

lcul

atin

g th

e pe

rcen

tage

of p

art o

f a w

hole

-ca

lcul

atin

g pe

rcen

tage

incr

ease

or d

ecre

ase

• C

alcu

late

am

ount

s if

give

n pe

rcen

tage

incr

ease

or

dec

reas

e

• S

olve

pro

blem

s in

con

text

s in

volv

ing

perc

enta

ges

equi

vale

nt fo

rms

• R

evis

e eq

uiva

lent

form

s be

twee

n:

-co

mm

on fr

actio

ns (f

ract

ions

whe

re o

ne

deno

min

ator

is a

mul

tiple

of t

he o

ther

)

-co

mm

on fr

actio

n an

d de

cim

al fr

actio

n fo

rms

of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

equi

vale

nt fo

rms

• R

evis

e eq

uiva

lent

form

s be

twee

n:

-co

mm

on fr

actio

ns w

here

one

den

omin

ator

is a

m

ultip

le o

f ano

ther

-co

mm

on fr

actio

n an

d de

cim

al fr

actio

n fo

rms

of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r


19CAPS

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.5

dec

imal

fr

actio

ns

ord

erin

g an

d co

mpa

ring

deci

mal

frac

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-co

unt f

orw

ards

and

bac

kwar

ds in

dec

imal

fra

ctio

ns to

at l

east

two

deci

mal

pla

ces

-co

mpa

re a

nd o

rder

dec

imal

frac

tions

to a

t lea

st

two

deci

mal

pla

ces

-pl

ace

valu

e of

dig

its to

at l

east

two

deci

mal

pl

aces

-ro

undi

ng o

ff de

cim

al fr

actio

ns to

at l

east

1

deci

mal

pla

ce

• E

xten

d al

l of t

he a

bove

to d

ecim

al fr

actio

ns to

at

leas

t thr

ee d

ecim

al p

lace

s an

d ro

undi

ng o

ff to

at

leas

t 2 d

ecim

al p

lace

s

Cal

cula

tions

with

dec

imal

frac

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-ad

ditio

n an

d su

btra

ctio

n of

dec

imal

frac

tions

of

at le

ast t

wo

deci

mal

pla

ces

-m

ultip

licat

ion

of d

ecim

al fr

actio

ns b

y 10

and

10

0

• E

xten

d ad

ditio

n an

d su

btra

ctio

n to

dec

imal

fra

ctio

ns o

f at l

east

thre

e de

cim

al p

lace

s

• m

ultip

ly d

ecim

al fr

actio

ns to

incl

ude:

-de

cim

al fr

actio

ns to

at l

east

3 d

ecim

al p

lace

s by

who

le n

umbe

rs

-de

cim

al fr

actio

ns to

at l

east

2 d

ecim

al p

lace

s by

dec

imal

frac

tions

to a

t lea

st 1

dec

imal

pla

ce

• D

ivid

e de

cim

al fr

actio

ns to

incl

ude

deci

mal

fra

ctio

ns to

at l

east

3 d

ecim

al p

lace

s by

who

le

num

bers

Cal

cula

tion

tech

niqu

es

• U

se k

now

ledg

e of

pla

ce v

alue

to e

stim

ate

the

num

ber o

f dec

imal

pla

ces

in th

e re

sult

befo

re

perfo

rmin

g ca

lcul

atio

ns

• U

se ro

undi

ng o

ff an

d a

calc

ulat

or to

che

ck re

sults

w

here

app

ropr

iate

ord

erin

g an

d co

mpa

ring

deci

mal

frac

tions

• R

evis

e:

-or

derin

g, c

ompa

ring

and

plac

e va

lue

of d

ecim

al

fract

ions

to a

t lea

st 3

dec

imal

pla

ces

-ro

undi

ng o

ff de

cim

al fr

actio

ns to

at l

east

2

deci

mal

pla

ce

Cal

cula

tions

with

dec

imal

frac

tions

• R

evis

e:

-ad

ditio

n, s

ubtra

ctio

n, m

ultip

licat

ion

and

of

deci

mal

frac

tions

to a

t lea

st 3

dec

imal

pla

ces

-di

visi

on o

f dec

imal

frac

tions

by

who

le n

umbe

rs

• E

xten

d m

ultip

licat

ion

to 'm

ultip

licat

ion

by d

ecim

al

fract

ions

' not

lim

ited

to o

ne d

ecim

al p

lace

• E

xten

d di

visi

on to

'div

isio

n of

dec

imal

frac

tions

by

deci

mal

frac

tions

'

• C

alcu

late

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

dec

imal

frac

tions

Cal

cula

tion

tech

niqu

es

• U

se k

now

ledg

e of

pla

ce v

alue

to e

stim

ate

the

num

ber o

f dec

imal

pla

ces

in th

e re

sult

befo

re

perfo

rmin

g ca

lcul

atio

ns

• U

se ro

undi

ng o

ff an

d a

calc

ulat

or to

che

ck re

sults

w

here

app

ropr

iate

Cal

cula

tions

with

dec

imal

frac

tions

• m

ultip

le o

pera

tions

with

dec

imal

frac

tions

, usi

ng a

ca

lcul

ator

whe

re a

ppro

pria

te

• m

ultip

le o

pera

tions

with

or w

ithou

t bra

cket

s, w

ith

num

bers

that

invo

lve

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

dec

imal

frac

tions

Cal

cula

tion

tech

niqu

es

• U

se k

now

ledg

e of

pla

ce v

alue

to e

stim

ate

the

num

ber o

f dec

imal

pla

ces

in th

e re

sult

befo

re

perfo

rmin

g ca

lcul

atio

ns

• U

se ro

undi

ng o

ff an

d a

calc

ulat

or to

che

ck re

sults

w

here

app

ropr

iate



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

1.5

dec

imal

fr

actio

ns

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

invo

lvin

g de

cim

al

fract

ions

equi

vale

nt fo

rms

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-re

cogn

ize

equi

vale

nce

betw

een

com

mon

fra

ctio

n an

d de

cim

al fr

actio

n fo

rms

of th

e sa

me

num

ber

-re

cogn

ize

equi

vale

nce

betw

een

com

mon

fra

ctio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

invo

lvin

g de

cim

al

fract

ions

equi

vale

nt fo

rms

• R

evis

e eq

uiva

lent

form

s be

twee

n:

-co

mm

on fr

actio

n an

d de

cim

al fr

actio

n fo

rms

of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

invo

lvin

g de

cim

al

fract

ions

equi

vale

nt fo

rms

Rev

ise

equi

vale

nt fo

rms

betw

een:

-co

mm

on fr

actio

n an

d de

cim

al fr

actio

n fo

rms

of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r


21CAPS

sPeC

iFiC

atio

n o

F C

on

ten

t (P

Ha

se o

ver

vieW

)

Patt

ern

s, F

un

Cti

on

s a

nd

alG

eBr

a

• P

rogr

essi

on in

Pat

tern

s, F

unct

ions

and

Alg

ebra

is a

chie

ved

prim

arily

by

-in

crea

sing

the

rang

e an

d co

mpl

exity

of:

♦re

latio

nshi

ps b

etw

een

num

bers

in g

iven

pat

tern

s

♦ru

les,

form

ulae

and

equ

atio

ns fo

r whi

ch in

put a

nd o

utpu

t val

ues

can

be fo

und

♦eq

uatio

ns th

at c

an b

e so

lved

-de

velo

ping

mor

e so

phis

ticat

ed s

kills

and

tech

niqu

es fo

r:

♦so

lvin

g eq

uatio

ns

♦ex

pand

ing

and

sim

plify

ing

alge

brai

c ex

pres

sion

s

♦dr

awin

g an

d in

terp

retin

g gr

aphs

-de

velo

ping

the

use

of a

lgeb

raic

lang

uage

and

con

vent

ions

.

• In

Pat

tern

s, F

unct

ions

and

Alg

ebra

, lea

rner

s’ c

once

ptua

l dev

elop

men

t pro

gres

ses

from

:

-an

und

erst

andi

ng o

f num

ber t

o an

und

erst

andi

ng o

f var

iabl

es, w

here

the

varia

bles

are

num

bers

of a

giv

en ty

pe (e

.g. n

atur

al n

umbe

rs, i

nteg

ers,

ratio

nal n

umbe

rs) i

n ge

nera

lized

form

-th

e re

cogn

ition

of p

atte

rns

and

rela

tions

hips

to th

e re

cogn

ition

of f

unct

ions

, whe

re fu

nctio

ns h

ave

uniq

ue o

utpu

ts v

alue

s fo

r spe

cifie

d in

put v

alue

s

-a

view

of M

athe

mat

ics

as m

emor

ized

fact

s an

d se

para

te to

pics

to s

eein

g M

athe

mat

ics

as in

terr

elat

ed c

once

pts

and

idea

s re

pres

ente

d in

a v

arie

ty o

f equ

ival

ent f

orm

s (e

.g. a

num

ber

patte

rn, a

n eq

uatio

n an

d a

grap

h re

pres

entin

g th

e sa

me

rela

tions

hip)

• W

hile

tech

niqu

es fo

r sol

ving

equ

atio

ns a

re d

evel

oped

in P

atte

rns,

Fun

ctio

ns a

nd A

lgeb

ra, l

earn

ers

also

pra

ctis

e so

lvin

g eq

uatio

ns in

mea

sure

men

t and

Spa

ce a

nd S

hape

, whe

n th

ey

appl

y kn

own

form

ulae

to s

olve

pro

blem

s.

toPi

Cs

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

2.1

num

eric

and

ge

omet

ric

patte

rns

inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

and

geo

met

ric

patte

rns

look

ing

for r

elat

ions

hips

bet

wee

n nu

mbe

rs, i

nclu

ding

pat

tern

s:

-re

pres

ente

d in

phy

sica

l or d

iagr

am fo

rm

-no

t lim

ited

to s

eque

nces

invo

lvin

g a

cons

tant

di

ffere

nce

or ra

tio

-of

lear

ner’s

ow

n cr

eatio

n

-re

pres

ente

d in

tabl

es

• D

escr

ibe

and

just

ify th

e ge

nera

l rul

es fo

r ob

serv

ed re

latio

nshi

ps b

etw

een

num

bers

in o

wn

wor

ds

inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

and

geo

met

ric

patte

rns

look

ing

for r

elat

ions

hips

bet

wee

n nu

mbe

rs, i

nclu

ding

pat

tern

s:

-re

pres

ente

d in

phy

sica

l or d

iagr

am fo

rm

-no

t lim

ited

to s

eque

nces

invo

lvin

g a

cons

tant

di

ffere

nce

or ra

tio

-of

lear

ner’s

ow

n cr

eatio

n

-re

pres

ente

d in

tabl

es

-re

pres

ente

d al

gebr

aica

lly

• D

escr

ibe

and

just

ify th

e ge

nera

l rul

es fo

r ob

serv

ed re

latio

nshi

ps b

etw

een

num

bers

in o

wn

wor

ds o

r in

alge

brai

c la

ngua

ge

inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

and

geo

met

ric

patte

rns

look

ing

for r

elat

ions

hips

bet

wee

n nu

mbe

rs, i

nclu

ding

pat

tern

s:

-re

pres

ente

d in

phy

sica

l or d

iagr

am fo

rm

-no

t lim

ited

to s

eque

nces

invo

lvin

g a

cons

tant

di

ffere

nce

or ra

tio

-of

lear

ner’s

ow

n cr

eatio

n

-re

pres

ente

d in

tabl

es

-re

pres

ente

d al

gebr

aica

lly

• D

escr

ibe

and

just

ify th

e ge

nera

l rul

es fo

r ob

serv

ed re

latio

nshi

ps b

etw

een

num

bers

in o

wn

wor

ds o

r in

alge

brai

c la

ngua

ge



toPi

Cs

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

2.2

Func

tions

and

re

latio

nshi

ps

inpu

t and

out

put v

alue

s

• D

eter

min

e in

put v

alue

s, o

utpu

t val

ues

or ru

les

for

patte

rns

and

rela

tions

hips

usi

ng:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

equ

ival

ence

of

diffe

rent

des

crip

tions

of t

he s

ame

rela

tions

hip

or

rule

pre

sent

ed:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

num

ber s

ente

nces

inpu

t and

out

put v

alue

s

• D

eter

min

e in

put v

alue

s, o

utpu

t val

ues

or ru

les

for

patte

rns

and

rela

tions

hips

usi

ng:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

-eq

uatio

ns

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

equ

ival

ence

of

diffe

rent

des

crip

tions

of t

he s

ame

rela

tions

hip

or

rule

pre

sent

ed:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

equ

atio

ns

inpu

t and

out

put v

alue

s

• D

eter

min

e in

put v

alue

s, o

utpu

t val

ues

or ru

les

for

patte

rns

and

rela

tions

hips

usi

ng:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

-eq

uatio

ns

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

equ

ival

ence

of

diffe

rent

des

crip

tions

of t

he s

ame

rela

tions

hip

or

rule

pre

sent

ed:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

equ

atio

ns

-by

gra

phs

on a

Car

tesi

an p

lane


23CAPS

toPi

Cs

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

2.3

alg

ebra

ic

expr

essi

ons

alg

ebra

ic la

ngua

ge

• R

ecog

nize

and

inte

rpre

t rul

es o

r rel

atio

nshi

ps

repr

esen

ted

in s

ymbo

lic fo

rm

• Id

entif

y va

riabl

es a

nd c

onst

ants

in g

iven

form

ulae

an

d/or

equ

atio

ns

alg

ebra

ic la

ngua

ge

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-re

cogn

ize

and

inte

rpre

t rul

es o

r rel

atio

nshi

ps

repr

esen

ted

in s

ymbo

lic fo

rm

-id

entif

y va

riabl

es a

nd c

onst

ants

in g

iven

fo

rmul

ae a

nd/o

r equ

atio

ns

• R

ecog

nize

and

iden

tify

conv

entio

ns fo

r writ

ing

alge

brai

c ex

pres

sion

s

• Id

entif

y an

d cl

assi

fy li

ke a

nd u

nlik

e te

rms

in

alge

brai

c ex

pres

sion

s

• R

ecog

nize

and

iden

tify

coef

ficie

nts

and

expo

nent

s in

alg

ebra

ic e

xpre

ssio

ns

expa

nd a

nd s

impl

ify a

lgeb

raic

exp

ress

ions

Use

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e la

ws

for r

atio

nal n

umbe

rs a

nd la

ws

of e

xpon

ents

to:

• ad

d an

d su

btra

ct li

ke te

rms

in a

lgeb

raic

ex

pres

sion

s

• m

ultip

ly in

tege

rs a

nd m

onom

ials

by:

-m

onom

ials

-bi

nom

ials

-tri

nom

ials

• di

vide

the

follo

win

g by

inte

gers

or m

onom

ials

:

-m

onom

ials

-B

inom

ials

-tri

nom

ials

• si

mpl

ify a

lgeb

raic

exp

ress

ions

invo

lvin

g th

e ab

ove

oper

atio

ns

alg

ebra

ic la

ngua

ge

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-re

cogn

ize

and

iden

tify

conv

entio

ns fo

r writ

ing

alge

brai

c ex

pres

sion

s

-id

entif

y an

d cl

assi

fy li

ke a

nd u

nlik

e te

rms

in

alge

brai

c ex

pres

sion

s

-re

cogn

ize

and

iden

tify

coef

ficie

nts

and

expo

nent

s in

alg

ebra

ic e

xpre

ssio

ns

• R

ecog

nize

and

diff

eren

tiate

bet

wee

n m

onom

ials

, bi

nom

ials

and

trin

omia

ls

expa

nd a

nd s

impl

ify a

lgeb

raic

exp

ress

ions

• R

evis

e th

e fo

llow

ing

done

in G

rade

8, u

sing

the

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e la

ws

for

ratio

nal n

umbe

rs a

nd la

ws

of e

xpon

ents

to:

-ad

d an

d su

btra

ct li

ke te

rms

in a

lgeb

raic

ex

pres

sion

s

-m

ultip

ly in

tege

rs a

nd m

onom

ials

by:

♦m

onom

ials

♦bi

nom

ials

♦tri

nom

ials

-di

vide

the

follo

win

g by

inte

gers

or m

onom

ials

:

♦m

onom

ials

♦bi

nom

ials

♦tri

nom

ials

-si

mpl

ify a

lgeb

raic

exp

ress

ions

invo

lvin

g th

e ab

ove

oper

atio

ns



toPi

Cs

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

2.3

alg

ebra

ic

expr

essi

ons

• D

eter

min

e th

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of s

ingl

e al

gebr

aic

term

s or

like

al

gebr

aic

term

s

• D

eter

min

e th

e nu

mer

ical

val

ue o

f alg

ebra

ic

expr

essi

ons

by s

ubst

itutio

n

-D

eter

min

e th

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of s

ingl

e al

gebr

aic

term

s or

like

al

gebr

aic

term

s

-D

eter

min

e th

e nu

mer

ical

val

ue o

f alg

ebra

ic

expr

essi

ons

by s

ubst

itutio

n

• E

xten

d th

e ab

ove

alge

brai

c m

anip

ulat

ions

to

incl

ude:

-m

ultip

ly in

tege

rs a

nd m

onom

ials

by

poly

nom

ials

-D

ivid

e po

lyno

mia

ls b

y in

tege

rs o

r mon

omia

ls

-Th

e pr

oduc

t of t

wo

bino

mia

ls

-Th

e sq

uare

of a

bin

omia

l

Fact

oriz

e al

gebr

aic

expr

essi

ons

• Fa

ctor

ize

alge

brai

c ex

pres

sion

s th

at in

volv

e:

-co

mm

on fa

ctor

s

-di

ffere

nce

of tw

o sq

uare

s

-tri

nom

ials

of t

he fo

rm:

♦x2 +

bx

+ c

♦ax

2 + b

x +

c, w

here

a is

a c

omm

on fa

ctor

.

• S

impl

ify a

lgeb

raic

exp

ress

ions

that

invo

lve

the

abov

e fa

ctor

isat

ion

proc

esse

s.

• S

impl

ify a

lgeb

raic

frac

tions

usi

ng fa

ctor

isat

ion.


25CAPS

toPi

Cs

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

2.4

alg

ebra

ic

equa

tions

num

ber s

ente

nces

• W

rite

num

ber s

ente

nces

to d

escr

ibe

prob

lem

si

tuat

ions

• A

naly

se a

nd in

terp

ret n

umbe

r sen

tenc

es th

at

desc

ribe

a gi

ven

situ

atio

n

• S

olve

and

com

plet

e nu

mbe

r sen

tenc

es b

y:

-in

spec

tion

-tri

al a

nd im

prov

emen

t

• D

eter

min

e th

e nu

mer

ical

val

ue o

f an

expr

essi

on

by s

ubst

itutio

n.

• Id

entif

y va

riabl

es a

nd c

onst

ants

in g

iven

form

ulae

or

equ

atio

ns

equa

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-se

t up

equa

tions

to d

escr

ibe

prob

lem

situ

atio

ns

-an

alys

e an

d in

terp

ret e

quat

ions

that

des

crib

e a

give

n si

tuat

ion

-so

lve

equa

tions

by

insp

ectio

n

-de

term

ine

the

num

eric

al v

alue

of a

n ex

pres

sion

by

sub

stitu

tion.

-id

entif

y va

riabl

es a

nd c

onst

ants

in g

iven

fo

rmul

ae o

r equ

atio

ns

• U

se s

ubst

itutio

n in

equ

atio

ns to

gen

erat

e ta

bles

of

ord

ered

pai

rs

• E

xten

d so

lvin

g eq

uatio

ns to

incl

ude:

-us

ing

addi

tive

and

mul

tiplic

ativ

e in

vers

es

-us

ing

law

s of

exp

onen

ts

equa

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-se

t up

equa

tions

to d

escr

ibe

prob

lem

situ

atio

ns

-an

alys

e an

d in

terp

ret e

quat

ions

that

des

crib

e a

give

n si

tuat

ion

-so

lve

equa

tions

by

insp

ectio

n

-us

ing

addi

tive

and

mul

tiplic

ativ

e in

vers

es

-us

ing

law

s of

exp

onen

ts

-de

term

ine

the

num

eric

al v

alue

of a

n ex

pres

sion

by

sub

stitu

tion.

-us

e su

bstit

utio

n in

equ

atio

ns to

gen

erat

e ta

bles

of

ord

ered

pai

rs

• E

xten

d so

lvin

g eq

uatio

ns to

incl

ude:

-us

ing

fact

oris

atio

n

-eq

uatio

ns o

f the

form

: a p

rodu

ct o

f fac

tors

= 0



toPi

Cs

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

2.5

Gra

phs

inte

rpre

ting

grap

hs

• A

naly

se a

nd in

terp

ret g

loba

l gra

phs

of p

robl

em

situ

atio

ns, w

ith s

peci

al fo

cus

on th

e fo

llow

ing

trend

s an

d fe

atur

es:

-lin

ear o

r non

-line

ar

-co

nsta

nt, i

ncre

asin

g or

dec

reas

ing

dra

win

g gr

aphs

• D

raw

glo

bal g

raph

s fro

m g

iven

des

crip

tions

of

a pr

oble

m s

ituat

ion,

iden

tifyi

ng fe

atur

es li

sted

ab

ove

inte

rpre

ting

grap

hs

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-an

alys

e an

d in

terp

ret g

loba

l gra

phs

of p

robl

em

situ

atio

ns, w

ith a

spe

cial

focu

s on

the

follo

win

g tre

nds

and

feat

ures

:

♦lin

ear o

r non

-line

ar

♦co

nsta

nt, i

ncre

asin

g or

dec

reas

ing

• E

xten

d th

e fo

cus

on fe

atur

es o

f gra

phs

to in

clud

e:

-m

axim

um o

r min

imum

-di

scre

te o

r con

tinuo

us

dra

win

g gr

aphs

• D

raw

glo

bal g

raph

s fro

m g

iven

des

crip

tions

of

a pr

oble

m s

ituat

ion,

iden

tifyi

ng fe

atur

es li

sted

ab

ove

• U

se ta

bles

or o

rder

ed p

airs

to p

lot p

oint

s an

d dr

aw g

raph

s on

the

Car

tesi

an p

lane

inte

rpre

ting

grap

hs

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-an

alys

e an

d in

terp

ret g

loba

l gra

phs

of p

robl

em

situ

atio

ns, w

ith a

spe

cial

focu

s on

the

follo

win

g tre

nds

and

feat

ures

:

♦lin

ear o

r non

-line

ar

♦co

nsta

nt, i

ncre

asin

g or

dec

reas

ing

♦m

axim

um o

r min

imum

♦di

scre

te o

r con

tinuo

us

• E

xten

d th

e ab

ove

with

spe

cial

focu

s on

the

follo

win

g fe

atur

es o

f lin

ear g

raph

s:

-x-

inte

rcep

t and

y-in

terc

ept

-gr

adie

nt

dra

win

g gr

aphs

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-dr

aw g

loba

l gr

aphs

fro

m g

iven

des

crip

tions

of

a pr

oble

m s

ituat

ion,

ide

ntify

ing

feat

ures

lis

ted

abov

e.

-us

e ta

bles

of

orde

red

pairs

to

plot

poi

nts

and

draw

gra

phs

on th

e C

arte

sian

pla

ne

• E

xten

d th

e ab

ove

with

spe

cial

focu

s on

:

-dr

awin

g lin

ear g

raph

s fro

m g

iven

equ

atio

ns

-de

term

inin

g eq

uatio

ns fr

om g

iven

line

ar g

raph

s


27CAPS

sPeC

iFiC

atio

n o

F C

on

ten

t (P

Ha

se o

ver

vieW

)sP

aC

e a

nd

sH

aPe

(Geo

met

ry)

• P

rogr

essi

on in

geo

met

ry in

the

Sen

ior P

hase

is a

chie

ved

prim

arily

by:

-in

vest

igat

ing

new

pro

perti

es o

f sha

pes

and

obje

cts

-de

velo

ping

from

info

rmal

des

crip

tions

of g

eom

etric

figu

res

to m

ore

form

al d

efini

tions

and

cla

ssifi

catio

n of

sha

pes

and

obje

cts

-so

lvin

g m

ore

com

plex

geo

met

ric p

robl

ems

usin

g kn

own

prop

ertie

s of

geo

met

ric fi

gure

s

-de

velo

ping

from

indu

ctiv

e re

ason

ing

to d

educ

tive

reas

onin

g.

• Th

e ge

omet

ry to

pics

are

muc

h m

ore

inte

r-re

late

d th

an in

the

Inte

rmed

iate

Pha

se, e

spec

ially

thos

e re

latin

g to

con

stru

ctio

ns a

nd g

eom

etry

of 2

D s

hape

s an

d st

raig

ht li

nes,

hen

ce c

are

has

to b

e ta

ken

rega

rdin

g se

quen

cing

of t

opic

s th

roug

h th

e te

rms.

• In

the

Sen

ior P

hase

, tra

nsfo

rmat

ion

geom

etry

dev

elop

s fro

m g

ener

al d

escr

iptio

ns o

f mov

emen

t in

spac

e to

mor

e sp

ecifi

c de

scrip

tions

of m

ovem

ent i

n co

-ord

inat

e pl

anes

. Thi

s la

ys

the

foun

datio

n fo

r ana

lytic

geo

met

ry in

the

FET

phas

e.

• S

olvi

ng p

robl

ems

in g

eom

etry

to fi

nd u

nkno

wn

angl

es o

r len

gths

pro

vide

s a

usef

ul c

onte

xt to

pra

ctis

e so

lvin

g eq

uatio

ns.

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

3.1

Geo

met

ry o

f 2d

sh

apes

Cla

ssify

ing

2d s

hape

s

• D

escr

ibe,

sor

t, na

me

and

com

pare

tria

ngle

s ac

cord

ing

to th

eir s

ides

and

ang

les,

focu

sing

on:

-eq

uila

tera

l tria

ngle

s

-is

osce

les

trian

gles

-rig

ht-a

ngle

d tri

angl

es

• D

escr

ibe,

sor

t, na

me

and

com

pare

qua

drila

tera

ls

in te

rms

of:

-le

ngth

of s

ides

-pa

ralle

l and

per

pend

icul

ar s

ides

-si

ze o

f ang

les

(rig

ht-a

ngle

s or

not

)

• D

escr

ibe

and

nam

e pa

rts o

f a c

ircle

Cla

ssify

ing

2d s

hape

s

• Id

entif

y an

d w

rite

clea

r defi

nitio

ns o

f tria

ngle

s in

te

rms

of th

eir s

ides

and

ang

les,

dis

tingu

ishi

ng

betw

een:

-eq

uila

tera

l tria

ngle

s

-is

osce

les

trian

gles

-rig

ht-a

ngle

d tri

angl

es

• Id

entif

y an

d w

rite

clea

r defi

nitio

ns o

f qua

drila

tera

ls

in te

rms

of th

eir s

ides

and

ang

les,

dis

tingu

ishi

ng

betw

een:

-pa

ralle

logr

am

-re

ctan

gle

-sq

uare

-rh

ombu

s

-tra

pezi

um

-ki

te

Cla

ssify

ing

2d s

hape

s

• R

evis

e pr

oper

ties

and

defin

ition

s of

tria

ngle

s in

te

rms

of th

eir s

ides

and

ang

les,

dis

tingu

ishi

ng

betw

een:

-eq

uila

tera

l tria

ngle

s

-is

osce

les

trian

gles

-rig

ht-a

ngle

d tri

angl

es

• R

evis

e an

d w

rite

clea

r defi

nitio

ns o

f qua

drila

tera

ls

in te

rms

of th

eir s

ides

, ang

les

and

diag

onal

s,

dist

ingu

ishi

ng b

etw

een:

-pa

ralle

logr

am

-re

ctan

gle

-sq

uare

-rh

ombu

s

-tra

pezi

um

-ki

te



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

3.1

Geo

met

ry o

f 2d

sh

apes

sim

ilar a

nd c

ongr

uent

2d

sha

pes

• R

ecog

nize

and

des

crib

e si

mila

r and

con

grue

nt

figur

es b

y co

mpa

ring:

-sh

ape

-si

ze

solv

ing

prob

lem

s

• S

olve

sim

ple

geom

etric

pro

blem

s in

volv

ing

unkn

own

side

s an

d an

gles

in tr

iang

les

and

quad

rilat

eral

s, u

sing

kno

wn

prop

ertie

s.

sim

ilar a

nd c

ongr

uent

2d

sha

pes

• Id

entif

y an

d de

scrib

e th

e pr

oper

ties

of c

ongr

uent

sh

apes

• Id

entif

y an

d de

scrib

e th

e pr

oper

ties

of s

imila

r sh

apes

solv

ing

prob

lem

s

• S

olve

geo

met

ric p

robl

ems

invo

lvin

g un

know

n si

des

and

angl

es in

tria

ngle

s an

d qu

adril

ater

als,

us

ing

know

n pr

oper

ties

and

defin

ition

s.

sim

ilar a

nd c

ongr

uent

tria

ngle

s

• Th

roug

h in

vest

igat

ion,

est

ablis

h th

e m

inim

um

cond

ition

s fo

r con

grue

nt tr

iang

les

• Th

roug

h in

vest

igat

ion,

est

ablis

h th

e m

inim

um

cond

ition

s fo

r sim

ilar t

riang

les

solv

ing

prob

lem

s

• S

olve

geo

met

ric p

robl

ems

invo

lvin

g un

know

n si

des

and

angl

es in

tria

ngle

s an

d qu

adril

ater

als,

us

ing

know

n pr

oper

ties

of tr

iang

les

and

quad

rilat

eral

s, a

s w

ell a

s pr

oper

ties

of c

ongr

uent

an

d si

mila

r tria

ngle

s.

3.2

Geo

met

ry o

f 3d

ob

ject

s

Cla

ssify

ing

3d o

bjec

ts

• D

escr

ibe,

sor

t and

com

pare

pol

yhed

ra in

term

s of

: -sh

ape

and

num

ber o

f fac

es

-nu

mbe

r of v

ertic

es

-nu

mbe

r of e

dges

Bui

ldin

g 3d

mod

els

• R

evis

e us

ing

nets

to c

reat

e m

odel

s of

geo

met

ric

solid

s, in

clud

ing:

-cu

bes

-pr

ism

s

Cla

ssify

ing

3d o

bjec

ts

• D

escr

ibe,

nam

e an

d co

mpa

re th

e 5

Pla

toni

c so

lids

in te

rms

of th

e sh

ape

and

num

ber o

f fac

es,

the

num

ber o

f ver

tices

and

the

num

ber o

f edg

es

Bui

ldin

g 3d

mod

els

• R

evis

e us

ing

nets

to c

reat

e m

odel

s of

geo

met

ric

solid

s, in

clud

ing:

-cu

bes

-pr

ism

s

-py

ram

ids

Cla

ssify

ing

3d o

bjec

ts

• R

evis

e pr

oper

ties

and

defin

ition

s of

the

5 P

lato

nic

solid

s in

term

s of

the

shap

e an

d nu

mbe

r of f

aces

, th

e nu

mbe

r of v

ertic

es a

nd th

e nu

mbe

r of e

dges

• R

ecog

nize

and

des

crib

e th

e pr

oper

ties

of:

-sp

here

s

-cy

linde

rs

Bui

ldin

g 3d

mod

els

• U

se n

ets

to c

reat

e m

odel

s of

geo

met

ric s

olid

s,

incl

udin

g:

-cu

bes

-pr

ism

s

-py

ram

ids

-cy

linde

rs

3.3

Geo

met

ry o

f st

raig

ht li

nes

Define:

• Li

ne s

egm

ent

• R

ay

• S

traig

ht li

ne

• P

aral

lel l

ines

• P

erpe

ndic

ular

line

s

ang

le re

latio

nshi

ps

• R

ecog

nize

and

des

crib

e pa

irs o

f ang

les

form

ed

by:

-pe

rpen

dicu

lar l

ines

-in

ters

ectin

g lin

es

-pa

ralle

l lin

es c

ut b

y a

trans

vers

al

solv

ing

prob

lem

s

• S

olve

geo

met

ric p

robl

ems

usin

g th

e re

latio

nshi

ps

betw

een

pairs

of a

ngle

s de

scrib

ed a

bove

ang

le re

latio

nshi

ps

• R

evis

e an

d w

rite

clea

r des

crip

tions

of t

he

rela

tions

hip

betw

een

angl

es fo

rmed

by:

-pe

rpen

dicu

lar l

ines

-in

ters

ectin

g lin

es

-pa

ralle

l lin

es c

ut b

y a

trans

vers

al

solv

ing

prob

lem

s

• S

olve

geo

met

ric p

robl

ems

usin

g th

e re

latio

nshi

ps

betw

een

pairs

of a

ngle

s de

scrib

ed a

bove


29CAPS

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

3.4

tran

sfor

mat

ion

Geo

met

ry

tran

sfor

mat

ions

• R

ecog

nize

, des

crib

e an

d pe

rform

tran

slat

ions

, re

flect

ions

and

rota

tions

with

geo

met

ric fi

gure

s an

d sh

apes

on

squa

red

pape

r

• Id

entif

y an

d dr

aw li

nes

of s

ymm

etry

in g

eom

etric

fig

ures

enla

rgem

ents

and

redu

ctio

ns

• D

raw

enl

arge

men

ts a

nd re

duct

ions

of g

eom

etric

fig

ures

on

squa

red

pape

r and

com

pare

them

in

term

s of

sha

pe a

nd s

ize

tran

sfor

mat

ions

• R

ecog

nize

, des

crib

e an

d pe

rform

tran

sfor

mat

ions

w

ith p

oint

s on

a c

o-or

dina

te p

lane

, foc

usin

g on

:

-re

flect

ing

a po

int i

n th

e X

-axi

s or

Y-a

xis

-tra

nsla

ting

a po

int w

ithin

and

acr

oss

quad

rant

s

• R

ecog

nize

, des

crib

e an

d pe

rform

tran

sfor

mat

ions

w

ith tr

iang

les

on a

co-

ordi

nate

pla

ne, f

ocus

ing

on

the

co-o

rdin

ates

of t

he v

ertic

es w

hen:

-re

flect

ing

a tri

angl

e in

the

X-a

xis

or Y

-axi

s

-tra

nsla

ting

a tri

angl

e w

ithin

and

acr

oss

quad

rant

s

-ro

tatin

g a

trian

gle

arou

nd th

e or

igin

enla

rgem

ents

and

redu

ctio

ns

• U

se p

ropo

rtion

to d

escr

ibe

the

effe

ct o

f en

larg

emen

t or r

educ

tion

on a

rea

and

perim

eter

of

geo

met

ric fi

gure

s

tran

sfor

mat

ions

• R

ecog

nize

, des

crib

e an

d pe

rform

tran

sfor

mat

ions

w

ith p

oint

s, li

ne s

egm

ents

and

sim

ple

geom

etric

fig

ures

on

a co

-ord

inat

e pl

ane,

focu

sing

on:

-re

flect

ion

in th

e X

-axi

s or

Y-a

xis

-tra

nsla

tion

with

in a

nd a

cros

s qu

adra

nts

-re

flect

ion

in th

e lin

e y

= x

• Id

entif

y w

hat t

he tr

ansf

orm

atio

n of

a p

oint

is, i

f gi

ven

the

co-o

rdin

ates

of i

ts im

age

enla

rgem

ents

and

redu

ctio

ns

• U

se p

ropo

rtion

to d

escr

ibe

the

effe

ct o

f en

larg

emen

t or r

educ

tion

on a

rea

and

perim

eter

of

geo

met

ric fi

gure

s

• In

vest

igat

e th

e co

-ord

inat

es o

f the

ver

tices

of

figur

es th

at h

ave

been

enl

arge

d or

redu

ced

by a

gi

ven

scal

e fa

ctor



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

3.5

Con

stru

ctio

n of

geo

met

ric

figures

mea

surin

g an

gles

• A

ccur

atel

y us

e a

prot

ract

or to

mea

sure

and

cl

assi

fy a

ngle

s:

-<

90o (

acut

e an

gles

)

-R

ight

-ang

les

->

90o (

obtu

se a

ngle

s)

-S

traig

ht a

ngle

s

->

180o (

refle

x an

gles

)

Con

stru

ctio

ns

• A

ccur

atel

y co

nstru

ct g

eom

etric

figu

res

appr

opria

tely

usi

ng a

com

pass

, rul

er a

nd

prot

ract

or, i

nclu

ding

:

-an

gles

, to

one

degr

ee o

f acc

urac

y

-ci

rcle

s

-pa

ralle

l lin

es

-pe

rpen

dicu

lar l

ines

Con

stru

ctio

ns

• A

ccur

atel

y co

nstru

ct g

eom

etric

figu

res

appr

opria

tely

usi

ng a

com

pass

, rul

er a

nd

prot

ract

or, i

nclu

ding

:

-bi

sect

ing

lines

and

ang

les

-pe

rpen

dicu

lar l

ines

at a

giv

en p

oint

or f

rom

a

give

n po

int

-tri

angl

es

-qu

adril

ater

als

• C

onst

ruct

ang

les

of 3

0°, 4

5°, 6

0° a

nd th

eir

mul

tiple

s w

ithou

t usi

ng a

pro

tract

or

Investigatingprop

ertiesofgeometric

figu

res

• B

y co

nstru

ctio

n, in

vest

igat

e th

e an

gles

in a

tri

angl

e, fo

cusi

ng o

n:

-th

e su

m o

f the

inte

rior a

ngle

s of

tria

ngle

s

-th

e si

ze o

f ang

les

in a

n eq

uila

tera

l tria

ngle

-th

e si

des

and

base

ang

les

of a

n is

osce

les

trian

gle

• B

y co

nstru

ctio

n, in

vest

igat

e si

des

and

angl

es in

qu

adril

ater

als,

focu

sing

on:

-th

e su

m o

f the

inte

rior a

ngle

s of

qua

drila

tera

ls

-th

e si

des

and

oppo

site

ang

les

of p

aral

lelo

gram

s

Con

stru

ctio

ns

• A

ccur

atel

y co

nstru

ct g

eom

etric

figu

res

appr

opria

tely

usi

ng a

com

pass

, rul

er a

nd

prot

ract

or, i

nclu

ding

bis

ectin

g an

gles

of a

tria

ngle

• C

onst

ruct

ang

les

of 3

0°, 4

5°, 6

0° a

nd th

eir

mul

tiple

s w

ithou

t usi

ng a

pro

tract

or

Investigatingprop

ertiesofgeometric

figu

res

• B

y co

nstru

ctio

n, in

vest

igat

e th

e an

gles

in a

tri

angl

e, fo

cusi

ng o

n th

e re

latio

nshi

p be

twee

n th

e ex

terio

r ang

le o

f a tr

iang

le a

nd it

s in

terio

r ang

les

• B

y co

nstru

ctio

n, in

vest

igat

e si

des,

ang

les

and

diag

onal

s in

qua

drila

tera

ls, f

ocus

ing

on:

-th

e di

agon

als

of re

ctan

gles

, squ

ares

, pa

ralle

logr

ams,

rhom

bi a

nd k

ites

-ex

plor

ing

the

sum

of t

he in

terio

r ang

les

of

poly

gons

• B

y co

nstru

ctio

n, e

xplo

re th

e m

inim

um c

ondi

tions

fo

r tw

o tri

angl

es to

be

cong

ruen

t


31CAPS

sPeC

iFiC

atio

n o

F C

on

ten

t (P

Ha

se o

ver

vieW

)

mea

sur

emen

t

• P

rogr

essi

on in

Mea

sure

men

t is

achi

eved

by

the

sele

ctio

n of

sha

pes

and

obje

cts

in e

ach

grad

e fo

r whi

ch th

e fo

rmul

ae fo

r find

ing

area

, per

imet

er, s

urfa

ce a

rea

and

volu

me

beco

me

mor

e co

mpl

ex.

• Th

e us

e of

form

ulae

in th

is p

hase

pro

vide

s a

usef

ul c

onte

xt to

pra

ctis

e so

lvin

g eq

uatio

ns.

• Th

e in

trodu

ctio

n of

the

Theo

rem

of P

ytha

gora

s is

a w

ay o

f int

rodu

cing

a fo

rmul

a to

cal

cula

te th

e le

ngth

s of

sid

es in

righ

t-ang

led

trian

gles

. Hen

ce, t

he T

heor

em o

f Pyt

hago

ras

beco

mes

a u

sefu

l too

l whe

n le

arne

rs s

olve

geo

met

ric p

robl

ems

invo

lvin

g rig

ht-a

ngle

d tri

angl

es.

• M

easu

rem

ent d

isap

pear

s as

a s

epar

ate

topi

c in

the

FET

phas

e, a

nd b

ecom

es p

art o

f the

stu

dy o

f Geo

met

ry a

nd T

rigon

omet

ry.

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

4.1

are

a an

d pe

rimet

er o

f 2d

sh

apes

are

a an

d pe

rimet

er

• C

alcu

late

the

perim

eter

of r

egul

ar a

nd ir

regu

lar

poly

gons

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te p

erim

eter

an

d ar

ea o

f:

-sq

uare

s

-re

ctan

gles

-tri

angl

es

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s in

volv

ing

perim

eter

and

are

a of

po

lygo

ns

• C

alcu

late

to a

t lea

st 1

dec

imal

pla

ce

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

SI u

nits

, in

clud

ing:

-m

m2

↔ cm

2

-cm

2 ↔

m2

are

a an

d pe

rimet

er

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te p

erim

eter

an

d ar

ea o

f:

-sq

uare

s

-re

ctan

gles

-tri

angl

es

-ci

rcle

s

• C

alcu

late

the

area

s of

pol

ygon

s, to

at l

east

2

deci

mal

pla

ces,

by

deco

mpo

sing

them

into

re

ctan

gles

and

/or t

riang

les

• U

se a

nd d

escr

ibe

the

rela

tions

hip

betw

een

the

radi

us, d

iam

eter

and

circ

umfe

renc

e of

a c

ircle

in

calc

ulat

ions

• U

se a

nd d

escr

ibe

the

rela

tions

hip

betw

een

the

radi

us a

nd a

rea

of a

circ

le in

cal

cula

tions

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s, w

ith o

r with

out a

cal

cula

tor,

invo

lvin

g pe

rimet

er a

nd a

rea

of p

olyg

ons

and

circ

les

• C

alcu

late

to a

t lea

st 2

dec

imal

pla

ces

• U

se a

nd d

escr

ibe

the

mea

ning

of t

he ir

ratio

nal

num

ber P

i (π)

in c

alcu

latio

ns in

volv

ing

circ

les

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

SI u

nits

, in

clud

ing:

mm

2 ↔

cm2

↔ m

2 ↔

km2

are

a an

d pe

rimet

er

• U

se a

ppro

pria

te fo

rmul

ae a

nd c

onve

rsio

ns

betw

een

SI u

nits

, to

solv

e pr

oble

ms

and

calc

ulat

e pe

rimet

er a

nd a

rea

of:

-po

lygo

ns

-ci

rcle

s

• In

vest

igat

e ho

w d

oubl

ing

any

or a

ll of

the

dim

ensi

ons

of a

2D

figu

re a

ffect

s its

per

imet

er

and

its a

rea



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

4.2

surf

ace

area

and

vo

lum

e of

3d

ob

ject

s

surf

ace

area

and

vol

ume

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te th

e su

rface

ar

ea, v

olum

e an

d ca

paci

ty o

f:

-cu

bes

-re

ctan

gula

r pris

ms

• D

escr

ibe

the

inte

rrel

atio

nshi

p be

twee

n su

rface

ar

ea a

nd v

olum

e of

the

obje

cts

men

tione

d ab

ove

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s in

volv

ing

surfa

ce a

rea,

vol

ume

and

capa

city

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

SI u

nits

, in

clud

ing:

-m

m2

↔ cm

2

-cm

2 ↔

m2

-m

m3

↔ cm

3

-cm

3 ↔

m3

• U

se e

quiv

alen

ce b

etw

een

units

whe

n so

lvin

g pr

oble

ms:

-1

cm3

↔ 1

ml

-1

m3

↔ 1

kl

surf

ace

area

and

vol

ume

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te th

e su

rface

ar

ea, v

olum

e an

d ca

paci

ty o

f:

-cu

bes

-re

ctan

gula

r pris

ms

-tri

angu

lar p

rism

s

• D

escr

ibe

the

inte

rrel

atio

nshi

p be

twee

n su

rface

ar

ea a

nd v

olum

e of

the

obje

cts

men

tione

d ab

ove

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s, w

ith o

r with

out a

cal

cula

tor,

invo

lvin

g su

rface

are

a, v

olum

e an

d ca

paci

ty

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

SI u

nits

, in

clud

ing:

-m

m2

↔ cm

2 ↔

m2

↔ km

2

-m

m3

↔ cm

3 ↔

m3

-m

l (cm

3 ) ↔ l ↔

kl

surf

ace

area

and

vol

ume

• U

se a

ppro

pria

te fo

rmul

ae a

nd c

onve

rsio

ns

betw

een

SI u

nits

to s

olve

pro

blem

s an

d ca

lcul

ate

the

surfa

ce a

rea,

vol

ume

and

capa

city

of:

-cu

bes

-re

ctan

gula

r pris

ms

-tri

angu

lar p

rism

s

-cy

linde

rs

• In

vest

igat

e ho

w d

oubl

ing

any

or a

ll th

e di

men

sion

s of

righ

t pris

ms

and

cylin

ders

affe

cts

thei

r vol

ume

4.3

the

theo

rem

of

Pyth

agor

as

dev

elop

and

use

the

theo

rem

of P

ytha

gora

s

• In

vest

igat

e th

e re

latio

nshi

p be

twee

n th

e le

ngth

s of

the

side

s of

a ri

ght-a

ngle

d tri

angl

e to

dev

elop

th

e Th

eore

m o

f Pyt

hago

ras

• D

eter

min

e w

heth

er a

tria

ngle

is a

righ

t-ang

led

trian

gle

or n

ot if

the

leng

th o

f the

thre

e si

des

of

the

trian

gle

are

know

n

• U

se th

e Th

eore

m o

f Pyt

hago

ras

to c

alcu

late

a

mis

sing

leng

th in

a ri

ght-a

ngle

d tri

angl

e, le

avin

g irr

atio

nal a

nsw

ers

in s

urd

form

solv

e pr

oble

ms

usin

g th

e th

eore

m o

f Py

thag

oras

• U

se th

e Th

eore

m o

f Pyt

hago

ras

to s

olve

pr

oble

ms

invo

lvin

g un

know

n le

ngth

s in

geo

met

ric

figur

es th

at c

onta

in ri

ght-a

ngle

d tri

angl

es


33CAPS

sPeC

iFiC

atio

n o

F C

on

ten

t (P

Ha

se o

ver

vieW

)

dat

a H

an

dli

nG

• P

rogr

essi

on in

Dat

a H

andl

ing

is a

chie

ved

prim

arily

by:

-in

crea

sing

com

plex

ity o

f dat

a se

ts a

nd c

onte

xts

-re

adin

g, in

terp

retin

g an

d dr

awin

g ne

w ty

pes

of d

ata

grap

hs

-be

com

ing

mor

e ef

ficie

nt a

t org

aniz

ing

and

sum

mar

izin

g da

ta

-be

com

ing

mor

e cr

itica

l and

aw

are

of b

ias

and

man

ipul

atio

n in

repr

esen

ting,

ana

lysi

ng a

nd re

porti

ng d

ata

• Le

arne

rs s

houl

d w

ork

thro

ugh

at le

ast 1

dat

a cy

cle

for t

he y

ear –

this

invo

lves

col

lect

ing

and

orga

nizi

ng, r

epre

sent

ing,

ana

lysi

ng, s

umm

ariz

ing,

inte

rpre

ting

and

repo

rting

dat

a. T

he

data

cyc

le p

rovi

des

the

oppo

rtuni

ty fo

r doi

ng p

roje

cts.

• A

ll of

the

abov

e as

pect

s of

dat

a ha

ndlin

g sh

ould

als

o be

dea

lt w

ith a

s di

scre

te a

ctiv

ities

in o

rder

to c

onso

lidat

e co

ncep

ts a

nd p

ract

ise

skill

s. F

or e

xam

ple,

lear

ners

nee

d to

pra

ctis

e su

mm

ariz

ing

data

pre

sent

ed in

diff

eren

t for

ms,

and

sum

mar

ies

shou

ld b

e us

ed w

hen

repo

rting

dat

a.

• D

ata

hand

ling

cont

exts

sho

uld

be s

elec

ted

to b

uild

aw

aren

ess

of s

ocia

l, ec

onom

ic a

nd e

nviro

nmen

tal i

ssue

s.

• Le

arne

rs s

houl

d be

com

e se

nsiti

zed

to b

ias

in th

e co

llect

ion

of d

ata,

as

wel

l as

mis

repr

esen

tatio

n of

dat

a th

roug

h th

e us

e of

diff

eren

t sca

les

and

diffe

rent

mea

sure

s of

cen

tral

tend

ency

.

• Th

e fo

llow

ing

reso

urce

s pr

ovid

e in

tere

stin

g co

ntex

ts fo

r dat

a co

mpa

rison

and

ana

lysi

s th

at c

an b

e us

ed in

this

pha

se:

-C

ensu

s at

Sch

ool –

for s

choo

l bas

ed s

urve

ys

-na

tiona

l sur

veys

from

Sta

tistic

s S

outh

Afri

ca (S

tats

SA

) – fo

r hou

seho

ld a

nd p

opul

atio

n su

rvey

s.

-in

tern

atio

nal s

urve

ys fr

om U

nite

d N

atio

ns (U

N D

ata)

– fo

r int

erna

tiona

l soc

ial,

dem

ogra

phic

and

env

ironm

enta

l sur

veys

. Man

y ot

her w

ebsi

tes

may

be

cons

ulte

d, e

spec

ially

for h

ealth

an

d en

viro

nmen

tal d

ata.

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

5.1

Col

lect

, org

aniz

e an

d su

mm

ariz

e da

ta

Col

lect

dat

a

• P

ose

ques

tions

rela

ting

to s

ocia

l, ec

onom

ic, a

nd

envi

ronm

enta

l iss

ues

in o

wn

envi

ronm

ent

• S

elec

t app

ropr

iate

sou

rces

for t

he c

olle

ctio

n of

da

ta (i

nclu

ding

pee

rs, f

amily

, new

spap

ers,

boo

ks,

mag

azin

es)

• D

istin

guis

h be

twee

n sa

mpl

es a

nd p

opul

atio

ns

and

sugg

est a

ppro

pria

te s

ampl

es fo

r inv

estig

atio

n

• D

esig

n an

d us

e si

mpl

e qu

estio

nnai

res

to a

nsw

er

ques

tions

:

-w

ith y

es/n

o ty

pe re

spon

ses

-w

ith m

ultip

le c

hoic

e re

spon

ses

Col

lect

dat

a

• P

ose

ques

tions

rela

ting

to s

ocia

l, ec

onom

ic, a

nd

envi

ronm

enta

l iss

ues

• S

elec

t app

ropr

iate

sou

rces

for t

he c

olle

ctio

n of

da

ta (i

nclu

ding

pee

rs, f

amily

, new

spap

ers,

boo

ks,

mag

azin

es)

• D

istin

guis

h be

twee

n sa

mpl

es a

nd p

opul

atio

ns,

and

sugg

est a

ppro

pria

te s

ampl

es fo

r inv

estig

atio

n

• D

esig

n an

d us

e si

mpl

e qu

estio

nnai

res

to a

nsw

er

ques

tions

with

mul

tiple

cho

ice

resp

onse

s

Col

lect

dat

a

• P

ose

ques

tions

rela

ting

to s

ocia

l, ec

onom

ic, a

nd

envi

ronm

enta

l iss

ues

• S

elec

t and

just

ify a

ppro

pria

te s

ourc

es fo

r the

co

llect

ion

of d

ata

• D

istin

guis

h be

twee

n sa

mpl

es a

nd p

opul

atio

ns,

and

sugg

est a

ppro

pria

te s

ampl

es fo

r inv

estig

atio

n

• S

elec

t and

just

ify a

ppro

pria

te m

etho

ds fo

r co

llect

ing

data



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

5.1

Col

lect

, org

aniz

e an

d su

mm

ariz

e da

ta

org

aniz

e an

d su

mm

ariz

e da

ta

• O

rgan

ize

(incl

udin

g gr

oupi

ng w

here

app

ropr

iate

) an

d re

cord

dat

a us

ing

-ta

lly m

arks

-ta

bles

-st

em-a

nd-le

af d

ispl

ays

• G

roup

dat

a in

to in

terv

als

• S

umm

ariz

e an

d di

stin

guis

hing

bet

wee

n un

grou

ped

num

eric

al d

ata

by d

eter

min

ing:

-m

ean

-m

edia

n

-m

ode

• Id

entif

y th

e la

rges

t and

sm

alle

st s

core

s in

a d

ata

set a

nd d

eter

min

e th

e di

ffere

nce

betw

een

them

in

orde

r to

dete

rmin

e th

e sp

read

of t

he d

ata

(ran

ge)

org

aniz

e an

d su

mm

ariz

e da

ta

• O

rgan

ize

(incl

udin

g gr

oupi

ng w

here

app

ropr

iate

) an

d re

cord

dat

a us

ing

-ta

lly m

arks

-ta

bles

-st

em-a

nd-le

af d

ispl

ays

• G

roup

dat

a in

to in

terv

als

• S

umm

ariz

e da

ta u

sing

mea

sure

s of

cen

tral

tend

ency

, inc

ludi

ng:

-m

ean

-m

edia

n

-m

ode

• S

umm

ariz

e da

ta u

sing

mea

sure

s of

dis

pers

ion,

in

clud

ing:

-ra

nge

-ex

trem

es

org

aniz

e an

d su

mm

ariz

e da

ta

• O

rgan

ize

num

eric

al d

ata

in d

iffer

ent w

ays

in o

rder

to

sum

mar

ize

by d

eter

min

ing:

-m

easu

res

of c

entra

l ten

denc

y

-m

easu

res

of d

ispe

rsio

n, in

clud

ing

extre

mes

and

ou

tlier

s

• O

rgan

ize

data

acc

ordi

ng to

mor

e th

an o

ne c

riter

ia

5.2

rep

rese

nt d

ata

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy

to d

ispl

ay a

nd in

terp

ret d

ata

(gro

uped

and

un

grou

ped)

incl

udin

g:

-ba

r gra

phs

and

doub

le b

ar g

raph

s

-hi

stog

ram

s w

ith g

iven

inte

rval

s

-pi

e ch

arts

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy to

di

spla

y an

d in

terp

ret d

ata

incl

udin

g:

-ba

r gra

phs

and

doub

le b

ar g

raph

s

-hi

stog

ram

s w

ith g

iven

and

ow

n in

terv

als

-pi

e ch

arts

-br

oken

-line

gra

phs

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy to

di

spla

y an

d in

terp

ret d

ata

incl

udin

g:

-ba

r gra

phs

and

doub

le b

ar g

raph

s

-hi

stog

ram

s w

ith g

iven

and

ow

n in

terv

als

-pi

e ch

arts

-br

oken

-line

gra

phs

-sc

atte

r plo

ts


35CAPS

Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

5.3

inte

rpre

t, an

alys

e, a

nd

repo

rt d

ata

inte

rpre

t dat

a

• C

ritic

ally

read

and

inte

rpre

t dat

a re

pres

ente

d in

:

-w

ords

-ba

r gra

phs

-do

uble

bar

gra

phs

-pi

e ch

arts

-hi

stog

ram

s

ana

lyse

dat

a

• C

ritic

ally

ana

lyse

dat

a by

ans

wer

ing

ques

tions

re

late

d to

:

-da

ta c

ateg

orie

s, in

clud

ing

data

inte

rval

s

-da

ta s

ourc

es a

nd c

onte

xts

-ce

ntra

l ten

denc

ies

(mea

n, m

ode,

med

ian)

-sc

ales

use

d on

gra

phs

rep

ort d

ata

• S

umm

ariz

e da

ta in

sho

rt pa

ragr

aphs

that

incl

ude

-dr

awin

g co

nclu

sion

s ab

out t

he d

ata

-m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

-id

entif

ying

sou

rces

of e

rror

and

bia

s in

the

data

-ch

oosi

ng a

ppro

pria

te s

umm

ary

stat

istic

s fo

r the

da

ta (m

ean,

med

ian,

mod

e)

inte

rpre

t dat

a

• C

ritic

ally

read

and

inte

rpre

t dat

a re

pres

ente

d in

:

-w

ords

-ba

r gra

phs

-do

uble

bar

gra

phs

-pi

e ch

arts

-hi

stog

ram

s

-br

oken

-line

gra

phs

ana

lyse

dat

a

• C

ritic

ally

ana

lyse

dat

a by

ans

wer

ing

ques

tions

re

late

d to

:

-da

ta c

ateg

orie

s, in

clud

ing

data

inte

rval

s

-da

ta s

ourc

es a

nd c

onte

xts

-ce

ntra

l ten

denc

ies

(mea

n, m

ode,

med

ian)

-sc

ales

use

d on

gra

phs

-sa

mpl

es a

nd p

opul

atio

ns

-di

sper

sion

of d

ata

-er

ror a

nd b

ias

in th

e da

ta

rep

ort d

ata

• S

umm

ariz

e da

ta in

sho

rt pa

ragr

aphs

that

incl

ude

-dr

awin

g co

nclu

sion

s ab

out t

he d

ata

-m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

-id

entif

ying

sou

rces

of e

rror

and

bia

s in

the

data

-ch

oosi

ng a

ppro

pria

te s

umm

ary

stat

istic

s fo

r the

da

ta (m

ean,

med

ian,

mod

e, ra

nge)

-th

e ro

le o

f ext

rem

es in

the

data

inte

rpre

t dat

a

• C

ritic

ally

read

and

inte

rpre

t dat

a re

pres

ente

d in

a

varie

ty o

f way

s

• C

ritic

ally

com

pare

two

sets

of d

ata

rela

ted

to th

e sa

me

issu

e

ana

lyse

dat

a

• C

ritic

ally

ana

lyse

dat

a by

ans

wer

ing

ques

tions

re

late

d to

:

-da

ta c

olle

ctio

n m

etho

ds

-su

mm

ary

of d

ata

-so

urce

s of

err

or a

nd b

ias

in th

e da

ta

rep

ort d

ata

• S

umm

ariz

e da

ta in

sho

rt pa

ragr

aphs

that

incl

ude

-dr

awin

g co

nclu

sion

s ab

out t

he d

ata

-m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

-m

akin

g co

mpa

rison

s be

twee

n tw

o se

ts o

f dat

a

-id

entif

ying

sou

rces

of e

rror

and

bia

s in

the

data

-ch

oosi

ng a

ppro

pria

te s

umm

ary

stat

istic

s fo

r the

da

ta (m

ean,

med

ian,

mod

e, ra

nge)

-th

e ro

le o

f ext

rem

es a

nd o

utlie

rs in

the

data



Co

nte

nt

Gr

ad

e 7

Gr

ad

e 8

Gr

ad

e 9

5.4

Prob

abili

tyPr

obab

ility

• P

erfo

rm s

impl

e ex

perim

ents

whe

re th

e po

ssib

le

outc

omes

are

equ

ally

like

ly a

nd:

-lis

t the

pos

sibl

e ou

tcom

es b

ased

on

the

cond

ition

s of

the

activ

ity

-de

term

ine

the

prob

abili

ty o

f eac

h po

ssib

le

outc

ome

usin

g th

e de

finito

n of

pro

babi

lity

Prob

abili

ty

• C

onsi

der a

sim

ple

situ

atio

n (w

ith e

qual

ly li

kely

ou

tcom

es) t

hat c

an b

e de

scrib

ed u

sing

pro

babi

lity

and:

-lis

t all

the

poss

ible

out

com

es

-de

term

ine

the

prob

abili

ty o

f eac

h po

ssib

le

outc

ome

usin

g th

e de

finiti

on o

f pro

babi

lity

-pr

edic

t with

reas

ons

the

rela

tive

frequ

ency

of

the

poss

ible

out

com

es fo

r a s

erie

s of

tria

ls

base

d on

pro

babi

lity

-co

mpa

re re

lativ

e fre

quen

cy w

ith p

roba

bilit

y an

d ex

plai

ns p

ossi

ble

diffe

renc

es

Prob

abili

ty

• C

onsi

der s

ituat

ions

with

equ

ally

pro

babl

e ou

tcom

es, a

nd:

-de

term

ine

prob

abili

ties

for c

ompo

und

even

ts

usin

g tw

o-w

ay ta

bles

and

tree

dia

gram

s

-de

term

ine

the

prob

abili

ties

for o

utco

mes

of

even

ts a

nd p

redi

ct th

eir r

elat

ive

frequ

ency

in

sim

ple

expe

rimen

ts

-co

mpa

re re

lativ

e fre

quen

cy w

ith p

roba

bilit

y an

d ex

plai

ns p

ossi

ble

diffe

renc

es


37CAPS

SECTION 3: CONTENT CLARIFICATION

3.1 introduCtion

• In this chapter, content clarification includes:

- teaching guidelines

- suggested sequencing of topics per term

- suggested 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.

• Teachers may choose to sequence and pace the contents differently from the recommendations in this section. However, cognisance should be taken of the relative weighting and notional hours of the Content Areas for this phase.

3.2 alloCation oF teaCHinG time

Time has been allocated in the following way:

• 10 weeks per term, with 4,5 hours for Mathematics per week (10 x 4 x 4,5 hours = 180 hours per year)

• Between 6 and 12 hours have been allocated for revision and assessment per term

• Therefore, 150 hours of teaching have been distributed across the Content Areas

• The distribution of time per topic, has taken account of the weighting for the Content Area as specified for the Senior Phase in Section 2.

• The weighting of Content Areas represents notional hours; therefore, the recommended distribution of hours may vary slightly across grades.



3.3 ClariFiCation notes WitH teaCHinG Guidelines

The tables below provide the teacher with:

• content areas and topics per grade per term;

• concepts and skills per term;

• clarification notes with teaching guidelines; and

• the duration of time allocated per topic in hours.

time alloCation Per toPiC: Grade 7

term 1 term 2 term 3 term 4

topic time topic time topic time topic time

Whole numbers 9 hours Common fractions 9

hoursNumeric and geometric patterns

6 hours Integers 9

hours

Exponents 9 hours Decimal fractions 9

hoursFunctions and relationships

3 hours

Numeric and geometric patterns

3 hours

Construction of Geometric figures

10 hours

Functions and relationships

3 hours

Algebraic expressions

3 hours


3 hours

Geometry of 2D shapes

10 hours

Area and perimeter of 2D shapes

7 hours

Algebraic equations

3 hours


3 hours

Geometry of straight lines

2 hours

Surface area and Volume of 3D objects

8 hours Graphs 6

hoursAlgebraic equations

4 hours

Transformation geometry 9

hoursCollect, organize and summarize data

4 hours

Geometry of 3D objects

9 hours Represent data 3

hours

Interpret, analyse and report data

3,5 hours

Probability 4,5 hours

revision/assessment

5 hours

revision/assessment

9 hours

revision/assessment

6 hours

revision/assessment 8hours

total: 45 hours total: 45 hours total: 45 hours total: 45 hours


39CAPS

3.3.1Clarifi

catio

nofcon

tentfo

rGrade7

ter

m 1

– G

ra

de

7

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.1

Who

le

num

bers

men

tal c

alcu

latio

ns

Rev

ise

the

follo

win

g do

ne in

Gra

de 6

:

• m

ultip

licat

ion

of w

hole

num

bers

to a

t le

ast 1

2 x

12

• m

ultip

licat

ion

fact

s fo

r:

-U

nits

and

tens

by

mul

tiple

s of

ten

-U

nits

and

tens

by

mul

tiple

s of

100

-U

nits

and

tens

by

mul

tiple

s of

1 0

00

-U

nits

an

d te

ns

by

mul

tiple

s of

10

000

• In

vers

e op

erat

ion

betw

een

mul

tiplic

atio

n an

d di

visi

on

Wha

t is

diffe

rent

to G

rade

6?

• P

rime

fact

ors

of 3

-dig

it nu

mbe

rs

• LC

m a

nd H

CF

• M

ore

com

plex

fina

ncia

l con

text

s fo

r sol

ving

pro

blem

s

In G

rade

7 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r w

hole

num

bers

, dev

elop

ed in

the

Inte

rmed

iate

Pha

se.

men

tal c

alcu

latio

ns

men

tal c

alcu

latio

ns s

houl

d be

use

d to

pra

ctic

e co

ncep

ts a

nd s

kills

dev

elop

ed

thro

ugh

the

mai

n le

sson

, som

etim

es w

ith s

mal

ler n

umbe

r ran

ges.

Lea

rner

s sh

ould

no

t be

aske

d to

do

rand

om c

alcu

latio

ns e

ach

day.

Rat

her,

men

tal c

alcu

latio

ns

shou

ld b

e us

ed a

s an

opp

ortu

nity

to c

onso

lidat

e fo

ur a

spec

ts o

f lea

rner

s’ n

umbe

r kn

owle

dge:

• nu

mbe

r fac

ts (n

umbe

r bon

ds a

nd ti

mes

tabl

es)

• ca

lcul

atio

n te

chni

ques

( do

ublin

g an

d ha

lvin

g, u

sing

mul

tiplic

atio

n to

do

divi

sion

, m

ultip

lyin

g an

d di

vidi

ng b

y 10

, 100

, 1 0

00

• m

ultip

lyin

g by

mul

tiple

s of

10,

100

, 1 0

00,

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

, rou

ndin

g of

f and

com

pens

atin

g et

c)

• nu

mbe

r con

cept

(cou

ntin

g, o

rder

ing

and

com

parin

g, p

lace

val

ue, o

dd a

nd e

ven

num

bers

, mul

tiple

s an

d fa

ctor

s)

• pr

oper

ties

of n

umbe

rs (i

dent

ity e

lem

ents

for a

dditi

on a

nd m

ultip

licat

ion;

• co

mm

utat

ive

and

asso

ciat

ive

prop

erty

for a

dditi

on a

nd m

ultip

licat

ion;

• in

vers

e op

erat

ion

for m

ultip

licat

ion

and

divi

sion

; inv

erse

ope

ratio

n fo

r add

ition

an

d su

btra

ctio

n)

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.1

Who

le

num

bers

ord

erin

g an

d co

mpa

ring

who

le

num

bers

Rev

ise

the

follo

win

g do

ne in

Gra

de 6

:

• O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

9-d

igit

num

bers

• R

ecog

nise

and

repr

esen

t prim

e nu

mbe

rs to

at l

east

100

• R

ound

ing

off n

umbe

rs to

the

near

est

5, 1

0, 1

00 o

r 1 0

00

Prop

ertie

s of

who

le n

umbe

rs

Rev

ise

the

follo

win

g do

ne in

Gra

de 6

:

• R

ecog

nise

and

use

the

com

mut

ativ

e;

asso

ciat

ive;

dis

tribu

tive

prop

ertie

s w

ith w

hole

num

bers

• R

ecog

nise

and

use

0 in

term

s of

its

addi

tive

prop

erty

(ide

ntity

ele

men

t for

ad

ditio

n)

• R

ecog

nise

and

use

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty (i

dent

ity

elem

ent f

or m

ultip

licat

ion)

ord

erin

g an

d co

mpa

ring

num

bers

Lear

ners

sho

uld

be g

iven

a ra

nge

of e

xerc

ises

suc

h as

:

• A

rran

ge g

iven

num

bers

from

the

smal

lest

to th

e bi

gges

t: or

big

gest

to s

mal

lest

• Fi

ll in

mis

sing

num

bers

in

-a

sequ

ence

-on

a n

umbe

r grid

-on

a n

umbe

r lin

e e.

g. w

hich

who

le n

umbe

r is

hal

fway

bet

wee

n 47

1 34

0 an

d 47

1 35

0.

• Fi

ll in

<, =

or >

exam

ples

:

a)

247

889

*

247

898

b)

784

109

*

785

190

Prop

ertie

s of

who

le n

umbe

rs

• R

evis

ing

the

prop

ertie

s of

who

le n

umbe

rs s

houl

d be

the

star

ting

poin

t for

wor

k w

ith w

hole

num

bers

. The

pro

perti

es o

f num

bers

sho

uld

prov

ide

the

mot

ivat

ion

for w

hy a

nd h

ow o

pera

tions

with

num

bers

wor

k.

• W

hen

lear

ners

are

intro

duce

d to

new

num

bers

, suc

h as

inte

gers

for e

xam

ple,

th

ey c

an a

gain

exp

lore

how

the

prop

ertie

s of

num

bers

wor

k fo

r the

new

set

of

num

bers

.

• Le

arne

rs a

lso

have

to a

pply

the

prop

ertie

s of

num

bers

in a

lgeb

ra, w

hen

they

w

ork

with

var

iabl

es in

pla

ce o

f num

bers

.

• Le

arne

rs s

houl

d kn

ow a

nd b

e ab

le to

use

the

follo

win

g pr

oper

ties:

-Th

e co

mm

utat

ive

prop

erty

of a

dditi

on a

nd m

ultip

licat

ion:

♦a

+ b

= b

+ a

♦a

x b

= b

x a

-Th

e as

soci

ativ

e (g

roup

ing)

pro

perty

of a

dditi

on a

nd m

ultip

licat

ion:

♦(a

+ b

) + c

= a

+ (b

+ a

)

♦(a

x b

) x c

= a

x (b

x c

)

-Th

e di

strib

utiv

e pr

oper

ty o

f mul

tiplic

atio

n ov

er a

dditi

on a

nd s

ubtra

ctio

n:

♦a(

b +

c) =

(a x

b) +

(a x

c)

♦a(

b –

c) =

(a x

b) –

(a x

c)


41CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.1

Who

le

num

bers

Cal

cula

tions

with

who

le n

umbe

rs

• R

evis

e th

e fo

llow

ing

done

in G

rade

6,

with

out u

se o

f cal

cula

tors

:

-A

dditi

on a

nd s

ubtra

ctio

n of

who

le

num

bers

to a

t lea

st 6

–di

git n

umbe

rs

-m

ultip

licat

ion

of a

t lea

st w

hole

4-

digi

t by

2-di

git n

umbe

rs

-D

ivis

ion

of a

t lea

st w

hole

4-d

igit

by

2-di

git n

umbe

rs

• P

erfo

rm c

alcu

latio

ns u

sing

all

four

op

erat

ions

on

who

le n

umbe

rs,

estim

atin

g an

d us

ing

calc

ulat

ors

whe

re a

ppro

pria

te

Cal

cula

tion

tech

niqu

es

• U

se a

rang

e of

tech

niqu

es to

per

form

an

d ch

eck

writ

ten

and

men

tal

calc

ulat

ions

of w

hole

num

bers

in

clud

ing:

-es

timat

ion

-ad

ding

, su

btra

ctin

g an

d m

ultip

lyin

g in

col

umns

-A

dditi

on a

nd s

ubtra

ctio

n as

inve

rse

oper

atio

ns

-m

ultip

licat

ion

and

divi

sion

as

inve

rse

oper

atio

ns

-0

is th

e id

entit

y el

emen

t for

add

ition

: t +

0 =

t

-1

is th

e id

entit

y el

emen

t for

mul

tiplic

atio

n: t

x 1

= t

illus

trat

ing

the

prop

ertie

s w

ith w

hole

num

bers

a)

33 +

99

= 99

+ 3

3 =

132

b)

51 +

(19

+ 46

) = (5

1 +

19) +

46

= 11

6

c)

4(12

+ 9

) = (4

x 1

2) +

(4 x

9) =

48

+ 36

= 8

4

d)

(9 x

64)

+ (9

x 3

6) =

9 x

(64

+ 36

) = 9

x 1

00 =

900

e)

If 33

+ 9

9 =

132,

then

132

– 9

9 =

33 a

nd 1

32 –

33

= 99

f) If

20 x

5 =

110

, the

n 11

0 ÷

20 =

5 a

nd 1

10 ÷

5 =

20

Cal

cula

tions

with

who

le n

umbe

rs

• Le

arne

rs s

houl

d do

con

text

free

cal

cula

tions

and

sol

ve p

robl

ems

in c

onte

xts

• Le

arne

rs s

houl

d be

com

e m

ore

confi

dent

in a

nd m

ore

inde

pend

ent a

t m

athe

mat

ics,

if th

ey h

ave

tech

niqu

es

-to

che

ck th

eir s

olut

ions

them

selv

es, e

.g. u

sing

inve

rse

oper

atio

ns; u

sing

ca

lcul

ator

s

-to

judg

e th

e re

ason

able

ness

of t

heir

solu

tions

e.g

. est

imat

e by

roun

ding

off;

es

timat

e by

dou

blin

g or

hal

ving

;

• A

ddin

g, s

ubtra

ctin

g an

d m

ultip

lyin

g in

col

umns

, and

long

div

isio

n, s

houl

d on

ly b

e us

ed to

pra

ctic

e nu

mbe

r fac

ts a

nd c

alcu

latio

n te

chni

ques

, and

hen

ce

shou

ld b

e do

ne w

ith fa

mili

ar a

nd s

mal

ler n

umbe

r ran

ges.

For

big

and

unw

ield

y ca

lcul

atio

ns, l

earn

ers

shou

ld b

e en

cour

aged

to u

se a

cal

cula

tor.

mul

tiple

s an

d fa

ctor

s

• P

ract

ice

with

find

ing

mul

tiple

s an

d fa

ctor

s of

who

le n

umbe

rs a

re e

spec

ially

im

porta

nt w

hen

lear

ners

do

calc

ulat

ions

with

frac

tions

. The

y us

e th

is k

now

ledg

e to

find

the

LCM

whe

n on

e de

nom

inat

or is

a m

ultip

le o

f ano

ther

, and

als

o w

hen

they

sim

plify

frac

tions

or h

ave

to fi

nd e

quiv

alen

t fra

ctio

ns.

• Fa

ctor

isin

g w

hole

num

bers

lays

the

foun

datio

n fo

r fac

toris

atio

n of

alg

ebra

ic

expr

essi

ons.

• U

sing

the

defin

ition

of p

rime

num

bers

, em

phas

ise

that

1 is

not

cla

ssifi

ed a

s a

prim

e nu

mbe

r



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.1

Who

le

num

bers

-lo

ng d

ivis

ion

-ro

undi

ng o

ff an

d co

mpe

nsat

ing

-us

ing

a ca

lcul

ator

mul

tiple

s an

d fa

ctor

s

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-M

ultip

les

of 2

-dig

it an

d 3-

digi

t w

hole

num

bers

-Fa

ctor

s of

2-d

igit

and

3-di

git w

hole

nu

mbe

rs

-P

rime

fact

ors

of n

umbe

rs to

at

leas

t 100

• Li

st p

rime

fact

ors

of n

umbe

rs to

at

leas

t 3-d

igit

who

le n

umbe

rs

• Fi

nd th

e LC

m a

nd H

CF

of n

umbe

rs

to a

t lea

st 3

-dig

it w

hole

num

bers

, by

insp

ectio

n or

fact

oris

atio

n

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng:

-C

ompa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd (r

atio

)

-C

ompa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

-S

harin

g in

a g

iven

ratio

whe

re th

e w

hole

is g

iven

• S

olve

pro

blem

s th

at in

volv

e w

hole

nu

mbe

rs, p

erce

ntag

es a

nd d

ecim

al

fract

ions

in fi

nanc

ial c

onte

xts

such

as

:

-P

rofit

, los

s an

d di

scou

nt

-B

udge

ts

-A

ccou

nts

-Lo

ans

-S

impl

e in

tere

st

exam

ples

a)

The

mul

tiple

s of

6 a

re 6

, 12,

18,

24,

... o

r M6 =

{6; 1

2; 1

8; 2

4;...

}

b)

LCM

of 6

and

18

is 1

8

LCM

of 6

and

7 is

42

c)

The

fact

ors

of 2

4 ar

e 1,

2, 3

, 4, 6

, 12

and

24 b

y in

spec

tion

and,

the

prim

e fa

ctor

s of

24

are

2 an

d 3

d)

The

fact

ors

of 1

40 a

re 1

, 2, 5

, 7, 1

0, 1

4, 2

8, 3

5, 7

0 an

d 14

0

e)

Det

erm

ine

the

HC

F of

120

; 300

and

900

Lear

ners

do

this

by

findi

ng th

e pr

ime

fact

ors

of th

e nu

mbe

rs fi

rst.

120

= 5

x 3

x 23 .

Initi

ally

lear

ners

may

writ

e th

is a

s: 5

x 3

x 2

x 2

x 2

300

= 52 x

3 x

22

900

= 52 x

32 x

23

HC

F =

5 x

3 x

22 = 6

0 (M

ultip

ly th

e co

mm

on p

rime

fact

ors

of th

e th

ree

num

bers

)

solv

ing

prob

lem

s

• S

olvi

ng p

robl

ems

in c

onte

xts

shou

ld ta

ke a

ccou

nt o

f the

num

ber r

ange

s le

arne

rs

are

fam

iliar

with

.

• C

onte

xts

invo

lvin

g ra

tio a

nd ra

te, s

houl

d in

clud

e sp

eed,

dis

tanc

e an

d tim

e pr

oble

ms.

• In

fina

ncia

l con

text

s, le

arne

rs a

re n

ot e

xpec

ted

to u

se fo

rmul

ae fo

r cal

cula

ting

sim

ple

inte

rest

.


43CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.2

expo

nent

sm

enta

l cal

cula

tions

• D

eter

min

e sq

uare

s to

at l

east

122

and

thei

r squ

are

root

s

• D

eter

min

e cu

bes

to a

t lea

st 6

3 and

th

eir c

ube

root

s

Com

parin

g an

d re

pres

entin

g nu

mbe

rs in

exp

onen

tial f

orm

• C

ompa

re a

nd re

pres

ent w

hole

nu

mbe

rs in

exp

onen

tial f

orm

: ab

= a

x a

x a

x...

for b

num

ber o

f fa

ctor

s

Wha

t is

diffe

rent

to G

rade

6?

• A

lthou

gh le

arne

rs m

ay h

ave

enco

unte

red

squa

re n

umbe

rs in

Gra

de 6

, the

y w

ere

not e

xpec

ted

to w

rite

thes

e nu

mbe

rs in

exp

onen

tial f

orm

.

Com

parin

g an

d re

pres

entin

g nu

mbe

rs in

exp

onen

tial f

orm

• Le

arne

rs n

eed

to u

nder

stan

d th

at in

the

expo

nent

ial f

orm

ab ,

the

num

ber i

s re

ad

as ‘a

to th

e po

wer

b’,

whe

re a

is c

alle

d th

e ba

se a

nd b

is c

alle

d th

e ex

pone

nt o

r in

dex.

indi

cate

s th

e nu

mbe

r of f

acto

rs th

at a

re m

ultip

lied.

exam

ple:

a)

a3 =

a x

a x

a;

b)

a5 =

a x

a x

a x

a x

a

• Le

arne

rs c

an re

pres

ent a

ny n

umbe

r in

expo

nent

ial f

orm

, with

out n

eedi

ng to

co

mpu

te th

e va

lue.

exam

ple:

50 x

50

x 50

x 5

0 x

50 x

50

x 50

= 5

07

• m

ake

sure

lear

ners

und

erst

and

that

squ

are

root

s an

d cu

be ro

ots

are

the

inve

rse

oper

atio

ns o

f squ

arin

g an

d cu

bing

num

bers

.

exam

ples

:32 =

9 th

eref

ore

9√

= 3

• M

ake

sure

lear

ners

und

erst

and

that

any

num

ber r

aise

d to

the

pow

er 1

is e

qual

to

the

num

ber.

Exa

mpl

e m

1 = m

• A

t thi

s po

int,

lear

ners

do

not n

eed

to k

now

the

rule

for r

aisi

ng a

num

ber t

o th

e po

wer

0. T

his

will

onl

y be

intro

duce

d in

Gra

de 8

whe

n th

ey u

se o

ther

law

s of

ex

pone

nts

in c

alcu

latio

ns.

• To

avo

id c

omm

on m

isco

ncep

tions

, em

phas

ize

the

follo

win

g w

ith e

xam

ples

:

-12

2 = 1

2 x

12 a

nd n

ot 1

2 x

2

-13 m

eans

1 x

1 x

1 a

nd n

ot 1

x 3

-10

01 = 1

00

-81

√ =

9 b

ecau

se 9

2 = 8

1

-3

27√

= 3

beca

use

33 = 2

7

-Th

e sq

uare

of 9

= 8

1, w

here

as th

e sq

uare

root

of 9

= 3

• Le

arne

rs s

houl

d us

e th

eir k

now

ledg

e of

repr

esen

ting

num

bers

in e

xpon

entia

l fo

rm w

hen

sim

plify

ing

and

expa

ndin

g al

gebr

aic

expr

essi

ons

and

solv

ing

alge

brai

c eq

uatio

ns.

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.2

expo

nent

sC

alcu

latio

ns u

sing

num

bers

in

expo

nent

ial f

orm

• R

ecog

nize

and

use

the

appr

opria

te

law

s of

ope

ratio

ns w

ith n

umbe

rs

invo

lvin

g ex

pone

nts

and

squa

re a

nd

cube

root

s

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all

four

ope

ratio

ns u

sing

num

bers

in

exp

onen

tial f

orm

, lim

ited

to

expo

nent

s up

to 5

, and

squ

are

and

cube

root

s

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

num

bers

in e

xpon

entia

l for

m

Cal

cula

tions

usi

ng n

umbe

rs in

exp

onen

tial f

orm

• K

now

ing

the

rule

s of

ope

ratio

ns fo

r cal

cula

tions

invo

lvin

g ex

pone

nts,

is

impo

rtant

.

exam

ple:

a)

(7 –

4)3 =

33 a

nd

no

t 73 –

43

b)

16 +

9

√ =

25

√, a

nd

no

t 16

√ +

9

√


45CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

shap

e an

d sp

ace

(Geo

met

ry)

3.5

Con

stru

ctio

n of

geo

met

ric

figures

mea

surin

g an

gles

• A

ccur

atel

y us

e a

prot

ract

or to

m

easu

re a

nd c

lass

ify a

ngle

s:

-<

90o (

acut

e an

gles

)

-R

ight

-ang

les

->

90o (

obtu

se a

ngle

s)

-S

traig

ht a

ngle

s

->

180o (

refle

x an

gles

)

Con

stru

ctio

ns

• A

ccur

atel

y co

nstru

ct g

eom

etric

fig

ures

app

ropr

iate

ly u

sing

com

pass

, ru

ler a

nd p

rotra

ctor

, inc

ludi

ng:

-an

gles

, to

one

degr

ee o

f acc

urac

y

-ci

rcle

s

-pa

ralle

l lin

es

-pe

rpen

dicu

lar l

ines

Wha

t is

diffe

rent

to G

rade

6?

• m

easu

re a

ngle

s w

ith a

pro

tract

or

• G

eom

etric

con

stru

ctio

ns u

sing

a c

ompa

ss, r

uler

and

pro

tract

or

mea

surin

g an

gles

• Le

arne

rs h

ave

to b

e sh

own

how

to p

lace

the

prot

ract

or o

n th

e ar

m o

f the

ang

le

to b

e m

easu

red.

• Le

arne

rs a

lso

have

to le

arn

how

to re

ad th

e si

ze o

f ang

les

on a

pro

tract

or.

Con

stru

ctio

ns

• C

onst

ruct

ions

pro

vide

a u

sefu

l con

text

to e

xplo

re o

r con

solid

ate

know

ledg

e of

an

gles

and

sha

pes.

• Le

arne

rs h

ave

to b

e sh

own

how

to u

se a

com

pass

to d

raw

circ

les,

alth

ough

they

m

ight

hav

e do

ne th

is in

Gra

de 6

.

• Le

arne

rs s

houl

d be

aw

are

that

the

cent

re o

f the

circ

le is

at t

he fi

xed

poin

t of t

he

com

pass

and

the

radi

us o

f the

circ

le is

dep

ende

nt o

n ho

w w

ide

the

com

pass

is

open

ed u

p.

• m

ake

sure

lear

ners

und

erst

and

that

arc

s ar

e pa

rts o

f the

circ

les

of a

par

ticul

ar

radi

us.

• In

itial

ly, le

arne

rs h

ave

to b

e gi

ven

care

ful i

nstru

ctio

ns a

bout

how

to d

o th

e co

nstru

ctio

ns o

f the

var

ious

sha

pes

• O

nce

they

are

com

forta

ble

with

the

appa

ratu

s an

d ca

n do

the

cons

truct

ions

, th

ey c

an p

ract

ise

by d

raw

ing

patte

rns,

for e

xam

ple

of c

ircle

s or

par

alle

l lin

es.

10 h

ours



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

shap

e an

d sp

ace

(Geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

Cla

ssify

ing

2d s

hape

s

• D

escr

ibe,

sor

t, na

me

and

com

pare

tri

angl

es a

ccor

ding

to th

eir s

ides

and

an

gles

, foc

usin

g on

:

-eq

uila

tera

l tria

ngle

s

-is

osce

les

trian

gles

-rig

ht-a

ngle

d tri

angl

es

• D

escr

ibe,

sor

t, na

me

and

com

pare

qu

adril

ater

als

in te

rms

of:

-le

ngth

of s

ides

-pa

ralle

l and

per

pend

icul

ar s

ides

-si

ze o

f ang

les

(rig

ht-a

ngle

s or

not

)

• D

escr

ibe

and

nam

e pa

rts o

f a c

ircle

Wha

t is

diffe

rent

to G

rade

6?

• D

istin

guis

hing

and

nam

ing

trian

gles

in te

rms

of th

eir s

ides

and

ang

les

• D

istin

guis

hing

qua

drila

tera

ls in

term

s of

par

alle

l and

per

pend

icul

ar s

ides

• D

istin

guis

hing

sim

ilar a

nd c

ongr

uent

figu

res

• U

sing

kno

wn

prop

ertie

s of

sha

pes

to s

olve

geo

met

ric p

robl

ems

tria

ngle

s

• Le

arne

rs s

houl

d be

abl

e to

dis

tingu

ish

betw

een

an e

quila

tera

l tria

ngle

(all

the

side

s ar

e eq

ual),

an

isos

cele

s tri

angl

e (a

t lea

st tw

o eq

ual s

ides

) and

a ri

ght-

angl

ed tr

iang

le (o

ne ri

ght-a

ngle

).

Qua

drila

tera

ls

• Le

arne

rs s

houl

d be

abl

e to

sor

t and

gro

up q

uadr

ilate

rals

in th

e fo

llow

ing

way

s:

-al

l sid

es e

qual

(squ

are

and

rhom

bus)

-op

posi

te s

ides

equ

al (r

ecta

ngle

, par

alle

logr

am, s

quar

e, rh

ombu

s)

-at

leas

t one

pai

r of a

djac

ent s

ides

equ

al (s

quar

e, rh

ombu

s, k

ite)

-al

l fou

r ang

les

right

ang

les

(squ

are,

rect

angl

e)

-pe

rpen

dicu

lar s

ides

(squ

are,

rect

angl

e)

-tw

o pa

irs o

f opp

osite

sid

es p

aral

lel (

rect

angl

e, s

quar

e, p

aral

lelo

gram

)

-on

ly o

ne p

air o

f opp

osite

sid

es p

aral

lel (

trape

zium

)

Circ

les

• P

arts

of a

circ

le le

arne

rs s

houl

d kn

ow:

-ra

dius

-ci

rcum

fere

nce

-di

amet

er

-ch

ord

-se

gmen

ts

-se

ctor

s

10 h

ours


47CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

shap

e an

d sp

ace

(Geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

sim

ilar a

nd c

ongr

uent

2d

sha

pes

• R

ecog

nize

and

des

crib

e si

mila

r and

co

ngru

ent fi

gure

s by

com

parin

g:

-sh

ape

-si

ze

solv

ing

prob

lem

s•

Sol

ve s

impl

e ge

omet

ric p

robl

ems

invo

lvin

g un

know

n si

des

and

angl

es

in tr

iang

les

and

quad

rilat

eral

s, u

sing

kn

own

prop

ertie

s.

sim

ilarit

y an

d co

ngru

ency

• S

imila

rity

and

cong

ruen

cy c

an b

e ex

plor

ed w

ith a

ny 2

D fi

gure

s.

• Le

arne

rs s

houl

d re

cogn

ize

that

two

or m

ore

figur

es a

re c

ongr

uent

if th

ey a

re

equa

l in

all r

espe

cts

i.e. a

ngle

s an

d si

des

are

equa

l.

• Le

arne

rs s

houl

d re

cogn

ize

that

two

or m

ore

figur

es a

re s

imila

r if t

hey

have

th

e sa

me

shap

e, b

ut d

iffer

in s

ize

i.e. a

ngle

s ar

e th

e sa

me,

but

sid

es a

re

prop

ortio

nally

long

er o

r sho

rter.

Sim

ilar fi

gure

s ar

e fu

rther

exp

lore

d w

hen

doin

g en

larg

emen

ts a

nd re

duct

ions

. Ref

er to

“Cla

rifica

tion

Not

es” u

nder

3.4

Tr

ansf

orm

atio

n G

eom

etry

.

solv

ing

prob

lem

s•

At t

his

stag

e le

arne

rs c

an s

olve

sim

ple

geom

etric

pro

blem

s to

find

unk

now

n si

des

in e

quila

tera

l and

isos

cele

s tri

angl

es, a

nd u

nkno

wn

side

s an

d an

gles

in

quad

rilat

eral

s.

• Le

arne

rs s

houl

d gi

ve re

ason

s fo

r the

ir so

lutio

ns.

exam

ples

a)

If ∆A

BC

is a

n eq

uila

tera

l tria

ngle

, and

sid

e A

B is

3 c

m, w

hat i

s th

e le

ngth

of

BC

? H

ere

lear

ners

sho

uld

answ

er: B

C =

3 c

m, b

ecau

se th

e si

des

of a

n eq

uila

tera

l tria

ngle

are

equ

al.

b)

If A

BC

D is

a k

ite, a

nd A

B =

2,5

cm

and

BC

= 4

,5 c

m, w

hat i

s th

e le

ngth

of

AD

and

DC

? Le

arne

rs s

houl

d us

e th

e pr

oper

ty fo

r kite

s, th

at a

djac

ent p

airs

of

sid

es a

re e

qual

, to

find

the

unkn

own

side

s.

3.3

Geo

met

ry o

f st

raig

ht li

nes

Define:

• Li

ne s

egm

ent

• R

ay

• S

traig

ht li

ne

• P

aral

lel l

ines

• P

erpe

ndic

ular

line

s

• Li

ne s

egm

ent i

s a

set o

f poi

nts

with

a d

efini

te s

tarti

ng-p

oint

and

an

end-

poin

t.

• R

ay is

a s

et o

f poi

nts

with

a d

efini

te s

tarti

ng-p

oint

and

no

defin

ite e

nd-p

oint

.

• Li

ne is

a s

et o

f poi

nts

with

no

defin

ite s

tarti

ng-p

oint

and

end

-poi

nt.

• If

two

lines

on

the

sam

e pl

ane

are

a co

nsta

nt d

ista

nce

apar

t, th

en th

e lin

es a

re

para

llel.

exam

ple:

>E

F

>G

H

Th

is is

writ

ten

as E

F║G

H

• If

verti

cal l

ine

AO

mee

ts o

r int

erse

cts

with

hor

izon

tal l

ine

BC

at r

ight

ang

le, t

hen

AO

is p

erpe

ndic

ular

to B

C.

exam

ple: A O

CB

This

is w

ritte

n as

AO

┴B

C

2 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• ca

lcul

atin

g an

d so

lvin

g pr

oble

ms

usin

g w

hole

num

bers

• w

orki

ng w

ith n

umbe

rs in

exp

onen

tial f

orm

• co

nstru

ctin

g ge

omet

ric o

bjec

ts

• ge

omet

ry o

f 2D

sha

pes

5 ho

urs


49CAPS

Gr

ad

e 7

– te

rm

2

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.4

Com

mon

fr

actio

ns

ord

erin

g, c

ompa

ring

and

sim

plify

ing

frac

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-C

ompa

re a

nd o

rder

com

mon

fra

ctio

ns, i

nclu

ding

spe

cific

ally

te

nths

and

hun

dred

ths

• E

xten

d to

thou

sand

ths

Cal

cula

tions

usi

ng fr

actio

ns

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-ad

ditio

n an

d su

btra

ctio

n of

co

mm

on fr

actio

ns, i

nclu

ding

mix

ed

num

bers

, lim

ited

to fr

actio

ns w

ith

the

sam

e de

nom

inat

or o

r whe

re

one

deno

min

ator

is a

mul

tiple

of

anot

her

-fin

ding

frac

tions

of w

hole

num

bers

• E

xten

d ad

ditio

n an

d su

btra

ctio

n to

fra

ctio

ns w

here

one

den

omin

ator

is

not a

mul

tiple

of t

he o

ther

• m

ultip

licat

ion

of c

omm

on fr

actio

ns,

incl

udin

g m

ixed

num

bers

, not

lim

ited

to fr

actio

ns w

here

one

den

omin

ator

is

a m

ultip

le o

f ano

ther

Cal

cula

tion

tech

niqu

es

• C

onve

rt m

ixed

num

bers

to c

omm

on

fract

ions

in o

rder

to p

erfo

rm

calc

ulat

ions

with

them

• U

se k

now

ledg

e of

mul

tiple

s an

d fa

ctor

s to

writ

e fra

ctio

ns in

the

sim

ples

t for

m b

efor

e or

afte

r ca

lcul

atio

ns

• U

se k

now

ledg

e of

equ

ival

ent

fract

ions

to a

dd a

nd s

ubtra

ct c

omm

on

fract

ions

Wha

t is

diffe

rent

to G

rade

6?

• C

ompa

re a

nd o

rder

thou

sand

ths

• m

ultip

licat

ion

of c

omm

on fr

actio

ns

• P

erce

ntag

e of

par

t of a

who

le

• P

erce

ntag

e in

crea

se o

r dec

reas

e

In G

rade

7 le

arne

rs a

lso

cons

olid

ate

num

ber k

now

ledg

e an

d ca

lcul

atio

n te

ch-

niqu

es fo

r com

mon

frac

tions

, dev

elop

ed in

the

Inte

rmed

iate

Pha

se.

Cal

cula

tions

with

frac

tions

• Le

arne

rs s

houl

d do

con

text

free

cal

cula

tions

and

sol

ve p

robl

ems

in c

onte

xts.

• It

is n

ot e

xpec

ted

that

lear

ners

kno

w ru

les

for s

impl

ifyin

g fra

ctio

ns o

r for

co

nver

ting

betw

een

mix

ed n

umbe

rs a

nd fr

actio

n fo

rms.

Lea

rner

s sh

ould

kno

w

from

wor

king

with

equ

ival

ence

, whe

n a

fract

ion

is e

qual

to o

r gre

ater

than

1.

• LC

ms

have

to b

e fo

und

whe

n ad

ding

and

sub

tract

ing

fract

ions

of d

iffer

ent

deno

min

ator

s. H

ere

lear

ners

use

kno

wle

dge

of c

omm

on m

ultip

les

to fi

nd th

e LC

M i.

e. w

hat n

umbe

r can

bot

h de

nom

inat

ors

be d

ivid

ed in

to.

• To

sim

plify

frac

tions

, lea

rner

s us

e kn

owle

dge

of c

omm

on fa

ctor

s i.e

. wha

t can

di

vide

equ

ally

into

the

num

erat

or a

nd d

enom

inat

or o

f a fr

actio

n. E

mph

asiz

e th

at

whe

n si

mpl

ifyin

g, th

e fra

ctio

ns m

ust r

emai

n eq

uiva

lent

.

exam

ple

3 4 x 2 5 =

4 20 =

3 10

or 3 4 x 2 5 =

3 10

• Le

arne

rs s

houl

d re

cogn

ize

that

find

ing

a ‘fr

actio

n of

a w

hole

num

ber’

or ‘fi

ndin

g a

fract

ion

of a

frac

tion’

mea

ns m

ultip

lyin

g th

e fra

ctio

n an

d th

e w

hole

num

ber o

r th

e fra

ctio

n w

ith th

e fra

ctio

n.

• W

hen

lear

ners

find

frac

tions

of w

hole

num

bers

, the

exa

mpl

es c

an b

e ch

osen

to

resu

lt ei

ther

in w

hole

num

bers

or f

ract

ions

or b

oth.

• Le

arne

rs s

houl

d al

so u

se th

e co

nven

tion

of w

ritin

g th

e w

hole

num

ber a

s a

fract

ion

over

whe

n m

ultip

lyin

g.

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.4

Com

mon

fr

actio

ns

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

com

mon

frac

tions

and

mix

ed

num

bers

, inc

ludi

ng g

roup

ing,

sha

ring

and

findi

ng fr

actio

ns o

f who

le

num

bers

Perc

enta

ges

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-pe

rcen

tage

s of

who

le n

umbe

rs

• C

alcu

late

the

perc

enta

ge o

f par

t of a

w

hole

• C

alcu

late

per

cent

age

incr

ease

or

decr

ease

of w

hole

num

bers

• S

olve

pro

blem

s in

con

text

s in

volv

ing

perc

enta

ges

equi

vale

nt fo

rms

Rev

ise

the

follo

win

g do

ne in

Gra

de 6

:

• R

ecog

nize

and

use

equ

ival

ent f

orm

s of

com

mon

frac

tions

with

1-d

igit

or

2-di

git d

enom

inat

ors

(frac

tions

whe

re

one

deno

min

ator

is a

mul

tiple

of t

he

othe

r)

• R

ecog

nize

equ

ival

ence

bet

wee

n co

mm

on fr

actio

n an

d de

cim

al fr

actio

n fo

rms

of th

e sa

me

num

ber

• R

ecog

nize

equ

ival

ence

bet

wee

n co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

exam

ples

a) C

alcu

late

4 5 of 2

0A

nsw

er:

4 5 of 2

0 =

4 5 x 20 1

= 4 1 x

4 1 = 1

6 o

r

4 5 of 2

0 =

4 5 x 20 1

= 80 5

= 1

6b)

Cal

cula

te

2 3 of 5 6

Ans

wer

2 3 of 5 6 =

2 3 x 5 6 =

1 3 x 5 3 =

5 9 o

r

2 3 of 5 6 =

2 3 x 5 6 =

1 3 x 10 18

= 5 9

Cal

cula

tion

usin

g pe

rcen

tage

s

• Le

arne

rs s

houl

d do

con

text

free

cal

cula

tions

and

sol

ve p

robl

ems

in c

onte

xts.

• W

hen

doin

g ca

lcul

atio

ns u

sing

per

cent

ages

, lea

rner

s ha

ve to

use

the

equi

vale

nt

com

mon

frac

tion

form

, whi

ch is

a fr

actio

n w

ith d

enom

inat

or 1

00.

Lear

ners

sho

uld

beco

me

fam

iliar

with

the

equi

vale

nt fr

actio

n an

d de

cim

al fo

rms

of c

omm

on p

erce

ntag

es li

ke

a)

25%

or 1 4 o

r 0,2

5;

b)

50%

or 1 2 o

r 0,5

;

c)

60%

or 3 5 o

r 0,6

.

• To

cal

cula

te p

erce

ntag

e of

par

t of a

who

le, o

r per

cent

age

incr

ease

or d

ecre

ase,

le

arne

rs h

ave

to le

arn

the

stra

tegy

of m

ultip

lyin

g by

100 1. I

t is

usef

ul fo

r lea

rner

s to

lear

n to

use

cal

cula

tors

for s

ome

of th

ese

calc

ulat

ions

whe

re th

e fra

ctio

ns a

re

not e

asily

sim

plifi

ed.

• W

hen

usin

g ca

lcul

ator

s, le

arne

rs c

an a

lso

use

the

equi

vale

nt d

ecim

al fr

actio

n fo

rm fo

r per

cent

ages

to d

o th

e ca

lcul

atio

ns.

exam

ples

:

a)

Cal

cula

te 6

0% o

f R10

5

Am

ount

= 3 5 X

R10

5 =

R63

b)

Wha

t per

cent

age

is 4

0c o

f R3,

20?

Per

cent

age

= 40

32

0 x 10

0 1 =

100 8 =

12,

5%

9 ho

urs


51CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

c)

Cal

cula

te th

e pe

rcen

tage

incr

ease

if th

e pr

ice

of a

bus

tick

et o

f R60

is

incr

ease

d to

R84

.

Am

ount

incr

ease

d =

R24

.

Ther

efor

e pe

rcen

tage

incr

ease

= 24

60

x 10

0 1 =

40%

d)

Cal

cula

te th

e pe

rcen

tage

dec

reas

e if

the

pric

e of

pet

rol g

oes

dow

n fro

m 2

0 ce

nts

a lit

re to

18

cent

s a

litre

.

Am

ount

dec

reas

ed =

2 c

ents

. The

refo

re p

erce

ntag

e de

crea

se

= 2 20

x 10

0 1 =

10%

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.5

dec

imal

fr

actio

ns

ord

erin

g an

d co

mpa

ring

deci

mal

fr

actio

ns

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-co

unt f

orw

ards

and

bac

kwar

ds in

de

cim

al fr

actio

ns to

at l

east

two

deci

mal

pla

ces

-co

mpa

re a

nd o

rder

dec

imal

fra

ctio

ns to

at l

east

two

deci

mal

pl

aces

-pl

ace

valu

e of

dig

its to

at l

east

two

deci

mal

pla

ces

-ro

undi

ng o

ff de

cim

al fr

actio

ns to

at

leas

t 1 d

ecim

al p

lace

• E

xten

d al

l of t

he a

bove

to d

ecim

al

fract

ions

of a

t lea

st th

ree

deci

mal

pl

aces

and

roun

ding

off

to a

t lea

st 2

de

cim

al p

lace

s

Wha

t is

diffe

rent

to G

rade

6?

• D

ecim

al fr

actio

ns to

at l

east

dec

imal

pla

ces

• R

ound

ing

off t

o at

leas

t dec

imal

pla

ces

• m

ultip

ly a

nd d

ivid

e de

cim

al fr

actio

ns b

y w

hole

num

bers

• m

ultip

ly d

ecim

al fr

actio

ns b

y de

cim

al fr

actio

ns

In G

rade

7 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r de

cim

al fr

actio

ns, d

evel

oped

in th

e In

term

edia

te P

hase

.

ord

erin

g, c

ount

ing

and

com

parin

g de

cim

al fr

actio

ns

• C

ount

ing

shou

ld n

ot o

nly

be th

ough

t of a

s ve

rbal

cou

ntin

g. L

earn

ers

can

coun

t in

dec

imal

inte

rval

s us

ing:

-st

ruct

ured

, sem

i-stru

ctur

ed o

r em

pty

num

ber l

ines

-ch

ain

diag

ram

s fo

r cou

ntin

g

• Le

arne

rs s

houl

d be

giv

en a

rang

e of

exe

rcis

es s

uch

as:

-ar

rang

e gi

ven

num

bers

from

the

smal

lest

to th

e bi

gges

t: or

big

gest

to s

mal

lest

-fil

l in

mis

sing

num

bers

in

♦a

sequ

ence

♦on

a n

umbe

r grid

♦on

a n

umbe

r lin

e

♦fil

l in

<, =

or >

exa

mpl

e: 0

,4 *

0.0

4

• C

ount

ing

exer

cise

s in

cha

in d

iagr

ams

can

be c

heck

ed u

sing

cal

cula

tors

and

le

arne

rs c

an e

xpla

in a

ny d

iffer

ence

s be

twee

n th

eir a

nsw

ers

and

thos

e sh

own

by

the

calc

ulat

or.

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.5

dec

imal

fr

actio

ns

Cal

cula

tions

usi

ng d

ecim

al fr

actio

ns

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-ad

ditio

n an

d su

btra

ctio

n of

dec

imal

fra

ctio

ns o

f at l

east

two

deci

mal

pl

aces

-m

ultip

licat

ion

of d

ecim

al fr

actio

ns

by 1

0 an

d 10

0

• E

xten

d ad

ditio

n an

d su

btra

ctio

n to

de

cim

al fr

actio

ns o

f at l

east

thre

e de

cim

al p

lace

s

• m

ultip

ly d

ecim

al fr

actio

ns to

incl

ude:

-de

cim

al fr

actio

ns to

at l

east

3

deci

mal

pla

ces

by w

hole

num

bers

-de

cim

al fr

actio

ns to

at l

east

2

deci

mal

pla

ces

by d

ecim

al fr

actio

ns

to a

t lea

st 1

dec

imal

pla

ce

• D

ivid

e de

cim

al fr

actio

ns to

incl

ude

deci

mal

frac

tions

to a

t lea

st 3

dec

imal

pl

aces

by

who

le n

umbe

rs

Cal

cula

tion

tech

niqu

es

• U

se k

now

ledg

e of

pla

ce v

alue

to

estim

ate

the

num

ber o

f dec

imal

pl

aces

in th

e re

sult

befo

re p

erfo

rmin

g ca

lcul

atio

ns

• U

se ro

undi

ng o

ff an

d a

calc

ulat

or to

ch

eck

resu

lts w

here

app

ropr

iate

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

invo

lvin

g de

cim

al fr

actio

ns

Cal

cula

ting

with

dec

imal

frac

tions

• Le

arne

rs s

houl

d do

con

text

free

cal

cula

tions

and

sol

ve p

robl

ems

in c

onte

xts.

• Le

arne

rs s

houl

d es

timat

e th

eir a

nsw

ers

befo

re c

alcu

latin

g, e

spec

ially

w

ith m

ultip

licat

ion

by d

ecim

al fr

actio

ns. T

hey

shou

ld b

e ab

le to

judg

e th

e re

ason

able

ness

of a

nsw

ers

rela

ting

to h

ow m

any

deci

mal

pla

ces

and

also

che

ck

thei

r ow

n an

swer

s.

• m

ultip

licat

ion

by d

ecim

al fr

actio

ns s

houl

d st

art w

ith fa

mili

ar n

umbe

rs th

at

lear

ners

can

cal

cula

te b

y in

spec

tion,

so

that

lear

ners

get

a s

ense

of h

ow

deci

mal

pla

ces

are

affe

cted

by

mul

tiplic

atio

n.

exam

ples

:

a)

3 x

2 =

6

0,3

x 2

= 0,

6

0,3

x 0,

2 =

0,06

0,3

x 0,

02 =

0,0

06

0,03

x 0

,002

= 0

,000

6 et

c

b)

15 x

3 =

45

1,5

x 3

= 4,

5

0,15

x 3

= 0

,45

0,15

x 0

,3 =

0,0

45

0,01

5 x

0,3

= 0,

0045

etc

equi

vale

nce

betw

een

com

mon

frac

tions

and

dec

imal

frac

tions

• Le

arne

rs a

re n

ot e

xpec

ted

to b

e ab

le to

con

vert

any

com

mon

frac

tion

into

its

deci

mal

form

, mer

ely

to s

ee th

e re

latio

nshi

p be

twee

n te

nths

, hun

dred

ths

and

thou

sand

ths

in th

eir d

ecim

al fo

rms.

• Le

arne

rs s

houl

d st

art b

y re

writ

ing

and

conv

ertin

g te

nths

, hun

dred

ths

and

thou

sand

ths

in c

omm

on fr

actio

n fo

rm to

dec

imal

frac

tions

. Whe

re d

enom

inat

ors

of o

ther

frac

tions

are

fact

ors

of 1

0 e.

g. 2

,5 o

r fac

tors

of 1

00 e

.g 2

, 4, 2

0,

25 le

arne

rs c

an c

onve

rt th

ese

to h

undr

edth

s us

ing

wha

t the

y kn

ow a

bout

eq

uiva

lenc

e.

• It

is u

sefu

l to

use

calc

ulat

ors

to h

elp

lear

ners

con

vert

betw

een

com

mon

fra

ctio

ns a

nd d

ecim

al fr

actio

ns (h

ere

lear

ners

will

use

wha

t the

y kn

ow a

bout

the

rela

tions

hip

betw

een

fract

ions

and

div

isio

n).


53CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.5

dec

imal

fr

actio

ns

equi

vale

nt fo

rms

• R

evis

e th

e fo

llow

ing

done

in G

rade

6:

-re

cogn

ize

equi

vale

nce

betw

een

com

mon

frac

tion

and

deci

mal

fra

ctio

n fo

rms

of th

e sa

me

num

ber

-R

ecog

nize

equ

ival

ence

bet

wee

n co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

-D

ivid

ing

who

le n

umbe

rs b

y 10

, 100

, 1 0

00, e

tc. c

an h

elp

to b

uild

lear

ners

’ un

ders

tand

ing

of p

lace

val

ue w

ith d

ecim

als.

Thi

s is

als

o us

eful

to d

o on

the

calc

ulat

or –

lear

ners

can

dis

cuss

the

patte

rns

they

see

whe

n di

vidi

ng.

-S

imila

rly c

alcu

lato

rs c

an b

e us

eful

tool

s fo

r lea

rner

s to

lear

n ab

out p

atte

rns

whe

n m

ultip

lyin

g de

cim

als

by 1

0, 1

00 o

r,1 0

00 e

tc.

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.2

Func

tions

an

d re

latio

nshi

ps

inpu

t and

out

put v

alue

s

• D

eter

min

e in

put v

alue

s, o

utpu

t va

lues

or r

ules

for p

atte

rns

and

rela

tions

hips

usi

ng:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

eq

uiva

lenc

e of

diff

eren

t des

crip

tions

of

the

sam

e re

latio

nshi

p or

rule

pr

esen

ted:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

num

ber s

ente

nces

Wha

t is

diffe

rent

to th

e in

term

edia

te P

hase

?

• Fi

ndin

g in

put o

r out

put v

alue

s us

ing

give

n fo

rmul

ae

• Th

e ru

les

and

num

ber p

atte

rns

for w

hich

lear

ners

hav

e to

find

inpu

t or o

utpu

t va

lues

are

ext

ende

d to

incl

ude

patte

rns

with

inte

gers

, squ

are

num

bers

and

cu

bic

num

bers

Find

ing

inpu

t and

out

put v

alue

s in

flow

dia

gram

s, ta

bles

and

form

ulae

sho

uld

be

done

mor

e th

an ju

st o

nce

a ye

ar. I

t can

be

done

afte

r num

ber w

ork,

to p

ract

ise

prop

ertie

s an

d op

erat

ions

with

num

bers

and

afte

r mea

sure

men

t or g

eom

etry

to

prac

tise

solv

ing

prob

lem

s us

ing

form

ulae

.

in t

erm

2 th

e fo

cus

of F

unct

ions

and

rel

atio

nshi

ps is

on

prac

tisin

g op

erat

ions

with

who

le n

umbe

rs a

s w

ell a

s co

mm

on fr

actio

ns o

r dec

imal

fr

actio

ns a

s in

put v

alue

s, o

r inc

ludi

ng c

omm

on fr

actio

ns a

nd d

ecim

al

fractio

nsinth

erulesforfi

ndingou

tputvalues.InTerm3th

efocusof

Func

tions

and

rel

atio

nshi

ps is

on

usin

g fo

rmul

ae a

nd in

ter

m 4

, the

focu

s is

on

pra

ctis

ing

addi

tion

and

subt

ract

ion

of in

tege

rs.

• In

this

pha

se, i

t is

usef

ul to

beg

in to

spe

cify

whe

ther

the

inpu

t val

ues

are

natu

ral

num

bers

, or i

nteg

ers

or ra

tiona

l num

bers

. Hen

ce, t

o fin

d ou

tput

val

ues,

lear

ners

sh

ould

be

give

n th

e ru

le/fo

rmul

a as

wel

l as

the

inpu

t val

ues.

Flow

dia

gram

s ar

e re

pres

enta

tions

of f

unct

iona

l rel

atio

nshi

ps. H

ence

, whe

n us

ing

flow

dia

gram

s, th

e co

rres

pond

ence

bet

wee

n in

put a

nd o

utpu

t val

ues

shou

ld b

e cl

ear i

n its

repr

esen

tatio

nal f

orm

i.e.

the

first

inpu

t pro

duce

s th

e fir

st o

utpu

t, th

e se

cond

inpu

t pro

duce

s th

e se

cond

out

put,

etc.

3 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.2

Func

tions

an

dre

latio

nshi

ps

exam

ples

a)

Use

the

give

n ru

le to

cal

cula

te th

e va

lues

of t

for e

ach

valu

e of

p, w

here

p is

a

natu

ral n

umbe

r.

t = p

x 3

+ 1

In th

is k

ind

of fl

ow d

iagr

am, l

earn

ers

can

also

be

aske

d to

find

the

valu

e of

p

for a

giv

en v

alue

t.

b)

Find

the

rule

for c

alcu

latin

g th

e ou

tput

val

ue fo

r eve

ry g

iven

inpu

t val

ue in

the

flow

dia

gram

bel

ow.

8 24 40 56 72

1

In fl

ow d

iagr

ams

such

as

thes

e, m

ore

than

one

rule

mig

ht b

e po

ssib

le to

de

scrib

e th

e re

latio

nshi

p be

twee

n in

put a

nd o

utpu

t val

ues.

The

rule

s ar

e ac

cept

able

if th

ey m

atch

the

give

n in

put v

alue

s to

the

corr

espo

ndin

g ou

tput

va

lues

.

c)

If th

e ru

le fo

r find

ing

y in

the

tabl

e be

low

is: y

= 3

x –

1, fi

nd y

for t

he g

iven

x

valu

es:

x0

12

510

5010

0

y


55CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.2

Func

tions

an

dre

latio

nshi

ps

d)

Des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

the

top

row

and

bot

tom

row

in

the

tabl

e. T

hen

writ

e do

wn

the

valu

e of

m a

nd n

.

x1

23

412

n

y5

67

8m

34

In ta

bles

suc

h as

thes

e, m

ore

than

one

rule

mig

ht b

e po

ssib

le to

des

crib

e th

e re

latio

nshi

p be

twee

n x

and

y va

lues

. The

rule

s ar

e ac

cept

able

if th

ey m

atch

the

give

n in

put v

alue

s to

the

corr

espo

ndin

g ou

tput

val

ues.

For

exa

mpl

e, th

e ru

le

y =

x +

4 de

scrib

es th

e re

latio

nshi

p be

twee

n th

e gi

ven

x an

d y

valu

es in

the

tabl

e. T

o fin

d m

and

n, y

ou h

ave

to s

ubst

itute

the

corr

espo

ndin

g va

lues

for x

or

y in

to th

is ru

le a

nd s

olve

the

equa

tion

by in

spec

tion.

mea

sure

men

t4.

1

are

a an

d pe

rimet

er o

f 2d

sha

pes

are

a an

d pe

rimet

er

• C

alcu

late

the

perim

eter

of r

egul

ar

and

irreg

ular

pol

ygon

s

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te

perim

eter

and

are

a of

:

-sq

uare

s

-re

ctan

gles

-tri

angl

es

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s in

volv

ing

perim

eter

an

d ar

ea o

f pol

ygon

s

• C

alcu

late

to a

t lea

st 1

dec

imal

pla

ce

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

S

I uni

ts, i

nclu

ding

:

-m

m2 ↔

cm

2

-cm

2 ↔ m

2

Wha

t is

diffe

rent

to G

rade

6?

• In

Gra

de 6

lear

ners

did

not

hav

e to

use

form

ulae

to c

alcu

late

are

a an

d pe

rimet

er.

•Fo

rmul

ae le

arne

rs s

houl

d kn

ow a

nd u

se a

re:

-pe

rimet

er o

f a s

quar

e =

4s

-pe

rimet

er o

f a re

ctan

gle

= 2(

l + b

) or 2

l + 2

b

-ar

ea o

f a s

quar

e =

l2

-ar

ea o

f a re

ctan

gle

= l x

b

-ar

ea o

f a tr

iang

le =

1 2 (b

x h

)

solv

ing

equa

tions

usi

ng fo

rmul

ae

• Th

e us

e of

form

ulae

pro

vide

s a

cont

ext t

o pr

actis

e so

lvin

g eq

uatio

ns b

y in

spec

tion.

exam

ple

1.

If th

e pe

rimet

er o

f a s

quar

e is

32

cm w

hat i

s th

e le

ngth

of e

ach

side

? Le

arne

rs s

houl

d w

rite

this

as:

4s =

32

and

solv

e by

insp

ectio

n by

ask

ing:

4 ti

mes

wha

t will

be

32?

2.

If th

e ar

ea o

f a re

ctan

gle

is 2

00 c

m2 ,

and

its le

ngth

is 5

0 cm

wha

t is

its w

idth

? Le

arne

rs s

houl

d w

rite

this

as:

50

x b

= 20

and

sol

ve b

y in

spec

tion

by a

skin

g:

50 ti

mes

wha

t will

be

200?

7 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

mea

sure

men

t4.

1 a

rea

and

perim

eter

of

2d s

hape

s

exam

ples

of c

alcu

latio

ns fo

r are

a an

d pe

rimet

erC

alcu

late

:a)

P

erim

eter

of a

rect

angl

e w

hich

is 2

4 cm

long

and

18

cm w

ide.

b)

Per

imet

er o

f a re

gula

r oct

agon

if th

e le

ngth

of e

ach

side

is 1

7 cm

.

c)

Are

a of

∆A

BC

if B

C =

12

cm a

nd it

s he

ight

AT

= 9

cm

d)

Per

imet

er o

f a s

quar

e if

its a

rea

is 2

25 c

m2

For a

reas

of t

riang

les:

• m

ake

sure

lear

ners

kno

w th

at th

e he

ight

of a

tria

ngle

is a

line

seg

men

t dra

wn

from

any

ver

tex

perp

endi

cula

r to

the

oppo

site

sid

e.ex

ampl

e: A

D is

the

heig

ht o

nto

base

BC

of ∆

AB

C.

A D

B C

A D B

C

• P

oint

out

that

eve

ry tr

iang

le h

as 3

bas

es, e

ach

with

a re

late

d he

ight

or a

ltitu

de.

•Fo

r con

vers

ions

, not

e: -

if 1

cm =

10

mm

then

1 c

m2 =

100

mm

2

-if

1 m

= 1

00 c

m th

en 1

m2 =

10

000

cm2

exam

ples

of s

olvi

ng p

robl

ems

invo

lvin

g pe

rimet

er a

nd a

rea.

a)

C

alcu

late

the

area

of t

he s

hade

d pa

rt in

the

diag

ram

if A

BC

D is

a re

ctan

gle,

AB

= 1

8,6

cm, D

C =

2TC

and

BC

= 8

cm

b)

The

area

of t

he fl

oor o

f the

din

ing

room

is 1

8,4

cm2 .

How

man

y sq

uare

tile

s w

ith s

ides

of 2

0 cm

are

nee

ded

to ti

le th

e flo

or?

c)

The

leng

th o

f the

sid

e of

a s

quar

e is

dou

bled

. Will

the

area

of t

he e

nlar

ged

squa

re b

e do

uble

or f

our t

imes

that

of t

he o

rigin

al s

quar

e?


57CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

mea

sure

men

t4.

1 a

rea

and

perim

eter

of

2d s

hape

s

exam

ples

of c

alcu

latio

ns fo

r are

a an

d pe

rimet

erC

alcu

late

:a)

P

erim

eter

of a

rect

angl

e w

hich

is 2

4 cm

long

and

18

cm w

ide.

b)

Per

imet

er o

f a re

gula

r oct

agon

if th

e le

ngth

of e

ach

side

is 1

7 cm

.

c)

Are

a of

∆A

BC

if B

C =

12

cm a

nd it

s he

ight

AT

= 9

cm

d)

Per

imet

er o

f a s

quar

e if

its a

rea

is 2

25 c

m2

For a

reas

of t

riang

les:

• m

ake

sure

lear

ners

kno

w th

at th

e he

ight

of a

tria

ngle

is a

line

seg

men

t dra

wn

from

any

ver

tex

perp

endi

cula

r to

the

oppo

site

sid

e.ex

ampl

e: A

D is

the

heig

ht o

nto

base

BC

of ∆

AB

C.

A D

B C

A D B

C

• P

oint

out

that

eve

ry tr

iang

le h

as 3

bas

es, e

ach

with

a re

late

d he

ight

or a

ltitu

de.

•Fo

r con

vers

ions

, not

e: -

if 1

cm =

10

mm

then

1 c

m2 =

100

mm

2

-if

1 m

= 1

00 c

m th

en 1

m2 =

10

000

cm2

exam

ples

of s

olvi

ng p

robl

ems

invo

lvin

g pe

rimet

er a

nd a

rea.

a)

C

alcu

late

the

area

of t

he s

hade

d pa

rt in

the

diag

ram

if A

BC

D is

a re

ctan

gle,

AB

= 1

8,6

cm, D

C =

2TC

and

BC

= 8

cm

b)

The

area

of t

he fl

oor o

f the

din

ing

room

is 1

8,4

cm2 .

How

man

y sq

uare

tile

s w

ith s

ides

of 2

0 cm

are

nee

ded

to ti

le th

e flo

or?

c)

The

leng

th o

f the

sid

e of

a s

quar

e is

dou

bled

. Will

the

area

of t

he e

nlar

ged

squa

re b

e do

uble

or f

our t

imes

that

of t

he o

rigin

al s

quar

e?

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

mea

sure

men

t4.

2

surf

ace

area

an

d vo

lum

e of

3d

obj

ects

surf

ace

area

and

vol

ume

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te

the

surfa

ce a

rea,

vol

ume

and

capa

city

of:

-cu

bes

-re

ctan

gula

r pris

ms

• D

escr

ibe

the

inte

rrel

atio

nshi

p be

twee

n su

rface

are

a an

d vo

lum

e of

th

e ob

ject

s m

entio

ned

abov

e

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s in

volv

ing

surfa

ce

area

, vol

ume

and

capa

city

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

S

I uni

ts, i

nclu

ding

:

-m

m2 ↔

cm

2

-cm

2 ↔ m

2

-m

m3 ↔

cm

3

-cm

3 ↔ m

3

• U

se e

quiv

alen

ce b

etw

een

units

whe

n so

lvin

g pr

oble

ms:

-1c

m3 ↔

1 m

l

-1

m3 ↔

1 k

l

Wha

t is

diffe

rent

to G

rade

6?

• In

Gra

de 6

lear

ners

did

not

hav

e to

use

form

ulae

to c

alcu

late

sur

face

are

a an

d vo

lum

e.

•Fo

rmul

ae le

arne

rs s

houl

d kn

ow a

nd u

se:

-th

e vo

lum

e of

a p

rism

= th

e ar

ea o

f the

bas

e x

the

heig

ht

-th

e su

rface

are

a of

a p

rism

= th

e su

m o

f the

are

a of

all

its fa

ces

-th

e vo

lum

e of

a c

ube

= l3

-th

e vo

lum

e of

a re

ctan

gula

r pris

m =

l x

b x

h

•Fo

r con

vers

ions

, not

e:

-if

1 cm

= 1

0 cm

then

1 c

m3 =

1 0

00 m

m3 a

nd

-if

1 m

= 1

0 cm

then

1 m

3 = 1

000

000

mm

3 or

1 0

00 0

00 o

r 106

cm3 .

-an

obj

ect w

ith a

vol

ume

of 1

cm

3 w

ill d

ispl

ace

exac

tly 1

ml o

f wat

er; a

nd

-an

obj

ect w

ith a

vol

ume

of 1

m3

will

dis

plac

e ex

actly

1 k

l of w

ater

.

• E

mph

asiz

e th

at th

e am

ount

of s

pace

insi

de a

pris

m is

cal

led

its c

apac

ity; a

nd

the

amou

nt o

f spa

ce o

ccup

ied

by a

pris

m is

cal

led

its v

olum

e.

• In

vest

igat

e th

e ne

ts o

f cub

es a

nd re

ctan

gula

r pris

ms

in o

rder

to d

educ

e fo

rmul

ae fo

r cal

cula

ting

thei

r sur

face

are

as.

8 ho

urs

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

be a

sses

sed

on:

• ca

lcul

atin

g an

d so

lvin

g pr

oble

ms

with

com

mon

frac

tions

and

dec

imal

frac

tions

• us

ing

form

ulae

to fi

nd a

rea

and

perim

eter

of 2

D s

hape

s

• us

ing

form

ulae

to fi

nd v

olum

e an

d su

rface

are

a of

3D

obj

ects

9 ho

urs



Gr

ad

e 7

– te

rm

3

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

n

umer

ic a

nd

geom

etric

pa

ttern

s

inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

an

d ge

omet

ric p

atte

rns

look

ing

for

rela

tions

hips

bet

wee

n nu

mbe

rs,

incl

udin

g pa

ttern

s:

-re

pres

ente

d in

phy

sica

l or

diag

ram

fo

rm

-no

t lim

ited

to s

eque

nces

invo

lvin

g a

cons

tant

diff

eren

ce o

r rat

io

-of

lear

ner’s

ow

n cr

eatio

n

-re

pres

ente

d in

tabl

es

• D

escr

ibe

and

just

ify th

e ge

nera

l rul

es

for o

bser

ved

rela

tions

hips

bet

wee

n nu

mbe

rs in

ow

n w

ords

Wha

t is

diffe

rent

to th

e in

term

edia

te P

hase

?

• In

the

Sen

ior P

hase

the

emph

asis

is le

ss o

n m

erel

y ex

tend

ing

a pa

ttern

, and

m

ore

on d

escr

ibin

g a

gene

ral r

ule

for t

he p

atte

rn o

r seq

uenc

e an

d be

ing

able

to

pred

ict u

nkno

wn

term

s in

a s

eque

nce

usin

g a

gene

ral r

ule.

• In

vest

igat

ing

num

ber p

atte

rns

is a

n op

portu

nity

to g

ener

aliz

e –

to g

ive

gene

ral

alge

brai

c de

scrip

tions

of t

he re

latio

nshi

p be

twee

n te

rms

and

its p

ositi

on in

a

sequ

ence

and

to ju

stify

sol

utio

ns.

• Th

e ra

nge

of n

umbe

r pat

tern

s ar

e ex

tend

ed to

incl

ude

patte

rns

inte

gers

, squ

are

num

bers

and

cub

ic n

umbe

rs

• A

s le

arne

rs b

ecom

e us

ed to

des

crib

ing

patte

rns

in th

eir o

wn

wor

ds, t

heir

desc

riptio

ns s

houl

d be

com

e m

ore

prec

ise

and

effic

ient

with

the

use

of a

lgeb

raic

la

ngua

ge to

des

crib

e ge

nera

l rul

es o

f pat

tern

s.

• It

is u

sefu

l als

o to

intro

duce

the

lang

uage

of ‘

term

in a

seq

uenc

e’ in

ord

er to

di

stin

guis

h th

e te

rm fr

om th

e po

sitio

n of

a te

rm in

a s

eque

nce

num

eric

and

geo

met

ric p

atte

rns

are

done

aga

in in

ter

m 4

, whe

re p

atte

rns

can

incl

ude

inte

gers

. in

this

term

pat

tern

s sh

ould

be

rest

ricte

d to

usi

ng

who

le n

umbe

rs, n

umbe

rs in

exp

onen

tial f

orm

, com

mon

frac

tions

and

de

cim

al fr

actio

ns.

Kin

ds o

f num

eric

pat

tern

s

• P

rovi

de a

seq

uenc

e of

num

bers

, lea

rner

s ha

ve to

iden

tify

a pa

ttern

or

rela

tions

hip

betw

een

cons

ecut

ive

term

s in

ord

er to

ext

end

the

patte

rn.

exam

ples

Giv

e a

rule

to d

escr

ibe

the

rela

tions

hip

betw

een

the

num

bers

in th

e se

quen

ces

belo

w. U

se th

is ru

le to

giv

e th

e ne

xt th

ree

num

bers

in th

e se

quen

ce:

a)

3; 7

; 11;

15;

... ..

.

b)

120;

115

; 110

; 105

;... .

..

Her

e le

arne

rs c

ould

iden

tify

the

cons

tant

diff

eren

ce b

etw

een

cons

ecut

ive

term

s in

ord

er to

ext

end

the

patte

rn. T

hese

pat

tern

s ca

n be

des

crib

ed in

lear

ners

’ ow

n w

ords

as

(a) ‘

addi

ng 4

’ or ‘

coun

ting

in 4

s’ o

r ‘ad

d 4s

to th

e pr

evio

us

num

ber i

n th

e pa

ttern

’ (b)

‘sub

tract

ing

5’ o

r ‘co

untin

g do

wn

in 5

s’, o

r ‘su

btra

ct 5

fro

m th

e pr

evio

us n

umbe

r in

the

patte

rn’.

c)

2; 4

; 8; 1

6;...

...

Her

e le

arne

rs c

ould

iden

tify

the

cons

tant

ratio

bet

wee

n co

nsec

utiv

e te

rms.

Th

is p

atte

rn c

an b

e de

scrib

ed in

lear

ners

’ ow

n w

ords

as

‘mul

tiply

the

prev

ious

nu

mbe

r by

2’.

6 ho

urs


59CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

n

umer

ic a

nd

geom

etric

pa

ttern

s

d)

1; 2

; 4; 7

; 11;

16;

... ..

.

This

pat

tern

has

nei

ther

a c

onst

ant d

iffer

ence

nor

con

stan

t rat

io. T

his

patte

rn

can

be d

escr

ibed

in le

arne

rs’ o

wn

wor

ds a

s ‘in

crea

se th

e di

ffere

nce

betw

een

cons

ecut

ive

term

s by

1 e

ach

time’

or ‘

add

1 m

ore

than

was

add

ed to

get

the

prev

ious

term

’. U

sing

this

rule

, the

nex

t 3 te

rms

will

be

22, 2

9, 3

7.

• P

rovi

de a

seq

uenc

e of

num

bers

, lea

rner

s ha

ve to

iden

tify

a pa

ttern

or

rela

tions

hip

betw

een

the

term

and

its

posi

tion

in th

e se

quen

ce. T

his

enab

les

lear

ners

to p

redi

ct a

term

in a

seq

uenc

e ba

sed

on th

e po

sitio

n of

that

te

rm in

the

sequ

ence

. It i

s us

eful

for l

earn

ers

to re

pres

ent t

hese

seq

uenc

es in

ta

bles

so

that

they

can

con

side

r the

pos

ition

of t

he te

rm.

exam

ples

:

a)

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

this

se

quen

ce:1

; 4; 9

; 16;

... ..

. Use

the

rule

to fi

nd th

e 10

th te

rm in

this

seq

uenc

e.

Firs

tly, l

earn

ers

have

to u

nder

stan

d th

at th

e ‘1

0th

term

’ ref

ers

to p

ositi

on 1

0 in

th

e nu

mbe

r se

quen

ce. T

hey

have

to fi

nd a

rul

e in

ord

er to

det

erm

ine

the

10th

te

rm, r

athe

r tha

n co

ntin

uing

the

sequ

ence

to th

e te

nth

term

.

This

seq

uenc

e ca

n be

repr

esen

ted

in th

e fo

llow

ing

tabl

e:

Pos

ition

in s

eque

nce

12

34

10

Term

14

916

?

Lear

ners

sho

uld

reco

gniz

e th

at e

ach

term

in th

e bo

ttom

row

is o

btai

ned

by

squa

ring

the

posi

tion

num

ber i

n th

e to

p ro

w. T

hus

the

10th

term

will

be

'10

squa

red'

or 1

02 , w

hich

is 1

00. U

sing

the

sam

e ru

le, l

earn

ers

can

also

be

aske

d w

hat t

erm

num

ber o

r pos

ition

will

625

be?

If th

e te

rm is

obt

aine

d by

squ

arin

g th

e po

sitio

n nu

mbe

r of t

he te

rm, t

hen

the

posi

tion

num

ber c

an b

e ob

tain

ed

by fi

ndin

g th

e sq

uare

root

of t

he te

rm. H

ence

, 625

will

be

the

25th

term

in th

e se

quen

ce s

ince

62

5 √

= 2

5



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

b)

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

this

se

quen

ce: 4

; 7; 1

0; 1

3; …

... U

se th

e ru

le to

find

the

20th

term

in th

e se

quen

ce.

If le

arne

rs c

onsi

der o

nly

the

rela

tions

hip

betw

een

cons

ecut

ive

term

s, th

en th

ey

can

cont

inue

the

patte

rn (‘

add

3 to

pre

viou

s nu

mbe

r’) to

the

20th

term

to fi

nd

the

answ

er. H

owev

er, i

f the

y lo

ok fo

r a re

latio

nshi

p or

rule

bet

wee

n th

e te

rm

and

the

posi

tion

of th

e te

rm, t

hey

can

pred

ict t

he a

nsw

er w

ithou

t con

tinui

ng th

e pa

ttern

. Usi

ng n

umbe

r sen

tenc

es c

an b

e us

eful

to fi

nd th

e ru

le:

1st t

erm

: 4 =

3 (1

) + 1

2nd

term

: 7 =

3 (2

) + 1

3rd

term

: 10

= 3

(3) +

1

4th

term

: 13

= 3

(4) +

1

The

num

ber i

n th

e br

acke

ts c

orre

spon

ds to

the

posi

tion

of th

e te

rm. H

ence

, the

20

th te

rm w

ill b

e: 3

(20)

+ 1

= 6

1Th

e ru

le in

lear

ners

’ ow

n w

ords

can

be

writ

ten

as ‘3

x th

e po

sitio

n of

the

te

rm +

1

• Th

ese

type

s of

num

eric

pat

tern

s de

velo

p an

und

erst

andi

ng o

f fun

ctio

nal

rela

tions

hips

, in

whi

ch y

ou h

ave

a de

pend

ent v

aria

ble

(pos

ition

of t

he te

rm) a

nd

inde

pend

ent v

aria

ble

(the

term

itse

lf), a

nd w

here

you

hav

e a

uniq

ue o

utpu

t for

an

y gi

ven

inpu

t val

ue.

Kin

ds o

f geo

met

ric p

atte

rns

• G

eom

etric

pat

tern

s ar

e nu

mbe

r pat

tern

s re

pres

ente

d di

agra

mm

atic

ally.

The

di

agra

mm

atic

repr

esen

tatio

n re

veal

s th

e st

ruct

ure

of th

e nu

mbe

r pat

tern

.•

Hen

ce, r

epre

sent

ing

the

num

ber p

atte

rns

in ta

bles

, mak

es it

eas

ier f

or le

arne

rs

to d

escr

ibe

the

gene

ral r

ule

for t

he p

atte

rn.

exam

ple

Con

side

r thi

s pa

ttern

for b

uild

ing

hexa

gons

with

mat

chst

icks

. How

man

y m

atch

stic

ks w

ill b

e us

ed to

bui

ld th

e 10

th h

exag

on?

The

rule

for t

he p

atte

rn is

con

tain

ed in

the

stru

ctur

e (c

onst

ruct

ion)

of t

he

succ

essi

ve h

exag

onal

sha

pes:

(1)

add

1 on

mat

chst

ick

per s

ide

(2)

ther

e ar

e 6

side

s, s

o (3

) ad

d on

6 m

atch

stic

ks p

er h

exag

on a

s yo

u pr

ocee

d fro

m a

giv

en h

exag

on to

th

e ne

xt o

ne.


61CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

n

umer

ic a

nd

geom

etric

pa

ttern

s

For t

he 2

nd h

exag

on, y

ou h

ave

2 x

6 m

atch

es; f

or th

e 3r

d he

xago

n yo

u ha

ve 3

x 6

m

atch

es; U

sing

this

pat

tern

for b

uild

ing

hexa

gons

, the

10t

h he

xago

n w

ill h

ave

10

x 6

mat

ches

.

Lear

ners

can

als

o us

e a

tabl

e to

reco

rd th

e nu

mbe

r of m

atch

es u

sed

for e

ach

hexa

gon.

Thi

s w

ay th

ey c

an lo

ok a

t the

num

ber p

atte

rn re

late

d to

the

num

ber o

f m

atch

es u

sed

for e

ach

new

hex

agon

.

Posi

tion

of h

exag

on in

pat

tern

12

34

56

10

num

ber o

f mat

ches

612

18

des

crib

ing

patte

rns

• It

does

not

mat

ter i

f lea

rner

s ar

e al

read

y fa

mili

ar w

ith a

par

ticul

ar p

atte

rn. T

heir

desc

riptio

ns o

f the

sam

e pa

ttern

can

be

diffe

rent

whe

n th

ey e

ncou

nter

it a

t di

ffere

nt s

tage

s of

thei

r mat

hem

atic

al d

evel

opm

ent.

exam

ple

The

rule

for t

he s

eque

nce:

4; 7

; 10;

13

can

be d

escr

ibed

in th

e fo

llow

ing

way

s:

a)

add

thre

e to

the

prev

ious

term

b)

3 tim

es th

e po

sitio

n of

the

term

+ 1

or 3

x th

e po

sitio

n of

the

term

+ 1

c)

3(n)

+ 1

, whe

re n

is th

e po

sitio

n of

the

term

d)

3(n)

+ 1

, whe

re n

is a

Nat

ural

num

ber.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.2

Func

tions

an

d re

latio

nshi

ps

inpu

t and

out

put v

alue

s

• D

eter

min

e in

put v

alue

s, o

utpu

t va

lues

or r

ules

for p

atte

rns

and

rela

tions

hips

usi

ng:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

eq

uiva

lenc

e of

diff

eren

t des

crip

tions

of

the

sam

e re

latio

nshi

p or

rule

pr

esen

ted:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

num

ber s

ente

nces

Func

tions

and

rela

tions

hips

wer

e al

so d

one

in t

erm

2, a

nd w

ill b

e do

ne a

gain

inTerm4,focusingon

integers.Inthisterm,thefocusis

onfin

ding

outpu

tva

lues

for g

iven

form

ulae

and

inpu

t val

ues.

see

addi

tiona

l not

es a

nd e

xam

ples

in t

erm

2.

Not

e, w

hen

lear

ners

find

inpu

t or o

utpu

t val

ues

for g

iven

rule

s or

form

ulae

, the

y ar

e ac

tual

ly fi

ndin

g th

e nu

mer

ical

val

ue o

f alg

ebra

ic e

xpre

ssio

ns u

sing

sub

stitu

-tio

n.

exam

ples

Use

the

form

ula

for t

he a

rea

of a

rect

angl

e: A

= l

x b

to c

alcu

late

the

follo

win

g:

a)

The

area

, if t

he le

ngth

is 4

cm

and

the

wid

th is

2 c

m

b)

The

leng

th, i

f the

are

a is

20

cm2 a

nd th

e w

idth

is 4

cm

c)

The

wid

th, i

f the

are

a is

24

cm2 a

nd th

e le

ngth

is 8

cm

Lear

ners

can

writ

e th

ese

as n

umbe

r sen

tenc

es, a

nd s

olve

by

insp

ectio

n.

3 ho

urs


63CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.3

alg

ebra

ic

expr

essi

ons

alg

ebra

ic la

ngua

ge

• R

ecog

nize

and

inte

rpre

t rul

es o

r re

latio

nshi

ps re

pres

ente

d in

sym

bolic

fo

rm

• Id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae a

nd e

quat

ions

Wha

t is

diffe

rent

to th

e in

term

edia

te P

hase

?

This

is a

n in

trodu

ctio

n to

form

al a

lgeb

raic

lang

uage

and

is n

ew in

the

Sen

ior

Pha

se. T

he u

se o

f sym

bolic

lang

uage

hel

ps to

dev

elop

an

unde

rsta

ndin

g of

va

riabl

es.

alg

ebra

ic e

xpre

ssio

ns a

re d

one

agai

n in

ter

m 4

, whe

re ru

les

and

rela

tions

hips

can

incl

ude

inte

gers

.

Lear

ners

hav

e op

portu

nitie

s to

writ

e an

d in

terp

ret a

lgeb

raic

exp

ress

ions

whe

n th

ey

writ

e ge

nera

l rul

es to

des

crib

e re

latio

nshi

ps b

etw

een

num

bers

in n

umbe

r pat

tern

s,

and

whe

n th

ey fi

nd in

put a

nd o

utpu

t val

ues

for g

iven

rule

s in

flow

dia

gram

s, ta

bles

an

d fo

rmul

ae.

exam

ples

a)

Wha

t doe

s th

e ru

le 2

x n

–1

mea

n fo

r the

follo

win

g nu

mbe

r seq

uenc

e: 1

; 3; 5

; 7;

9;..

..

Her

e le

arne

rs s

houl

d re

cogn

ize

that

2 x

n –

1 re

pres

ents

the

gene

ral t

erm

in th

is

sequ

ence

, whe

re n

repr

esen

ts th

e po

sitio

n of

the

term

in th

e se

quen

ce. T

hus

it is

the

rule

that

can

be

used

to fi

nd a

ny te

rm in

the

give

n se

quen

ce.

b)

The

rela

tions

hip

betw

een

a bo

y’s

age

(x y

rs o

ld) a

nd h

is m

othe

r’s a

ge is

giv

en

as 2

5 +

x. H

ow c

an th

is re

latio

nshi

p be

use

d to

find

the

mot

her’s

age

whe

n th

e bo

y is

11

year

s ol

d?

Her

e le

arne

rs s

houl

d re

cogn

ize

that

to fi

nd th

e m

othe

r’s a

ge, t

hey

shou

ld

subs

titut

e th

e bo

y’s

give

n ag

e in

to th

e ru

le 2

5 +

x. T

hey

shou

ld a

lso

reco

gniz

e th

at th

e gi

ven

rule

mea

ns th

e m

othe

r is

25 y

ears

old

er th

an th

e bo

y.

See

furth

er e

xam

ples

giv

en fo

r fun

ctio

ns a

nd re

latio

nshi

ps.

3 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.4

alg

ebra

ic

equa

tions

num

ber s

ente

nces

• W

rite

num

ber s

ente

nces

to d

escr

ibe

prob

lem

situ

atio

ns

• A

naly

se a

nd in

terp

ret n

umbe

r se

nten

ces

that

des

crib

e a

give

n si

tuat

ion

• S

olve

and

com

plet

e nu

mbe

r se

nten

ces

by:

-in

spec

tion

-tri

al a

nd im

prov

emen

t

• Id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae o

r equ

atio

ns

• D

eter

min

e th

e nu

mer

ical

val

ue o

f an

expr

essi

on b

y su

bstit

utio

n.

Wha

t is

diffe

rent

to th

e in

term

edia

te P

hase

? Th

e nu

mbe

r se

nten

ces

that

lear

ners

can

sol

ve a

re e

xten

ded

to in

clud

e nu

mbe

r se

nten

ces

with

inte

gers

, squ

are

num

bers

and

cub

ic n

umbe

rs.

Num

ber s

ente

nces

are

use

d he

re a

s a

mor

e fa

mili

ar te

rm fo

r Gra

de 7

lear

ners

than

eq

uatio

ns.

How

ever

, th

e te

rm e

quat

ion

will

be

used

ins

tead

of

num

ber

sent

ence

s in

lat

er

grad

es.

alg

ebra

ic e

quat

ions

are

don

e ag

ain

in t

erm

4, w

here

num

ber

sent

ence

s ca

n in

clud

e in

tege

rs.

Lear

ners

hav

e op

portu

nitie

s to

writ

e, s

olve

and

com

plet

e nu

mbe

r sen

tenc

es w

hen

they

writ

e ge

nera

l ru

les

to d

escr

ibe

rela

tions

hips

bet

wee

n nu

mbe

rs i

n nu

mbe

r pa

ttern

s, a

nd w

hen

they

find

inpu

t and

out

put v

alue

s fo

r giv

en ru

les

in fl

ow d

iagr

ams,

ta

bles

and

form

ulae

.

Rat

her

than

use

form

al a

lgeb

raic

pro

cess

es, l

earn

ers

solv

e nu

mbe

r se

nten

ces

by

insp

ectio

n or

det

erm

ine

the

num

eric

al v

alue

of e

xpre

ssio

ns b

y su

bstit

utio

n.

In th

is p

hase

, it i

s us

eful

whe

n so

lvin

g eq

uatio

ns to

beg

in to

spe

cify

whe

ther

x is

a

natu

ral n

umbe

r, in

tege

r or

rat

iona

l num

ber.

This

bui

lds

lear

ners

’ aw

aren

ess

of th

e do

mai

n of

x.

exam

ples

a)

Sol

ve x

if x

+ 4

= 7

, whe

re x

is a

nat

ural

num

ber.

(Wha

t mus

t be

adde

d to

4 to

gi

ve 7

?)

b)

Sol

ve x

if x

+ 4

= –

7, w

here

x is

an

inte

ger.

(Wha

t mus

t be

adde

d to

4 to

giv

e –7

?)

c)

Sol

ve x

if 2

x =

30, w

here

x is

a n

atur

al n

umbe

r. (W

hat m

ust b

e m

ultip

lied

by 2

to

giv

e 30

?)

d)

Writ

e a

num

ber s

ente

nce

to fi

nd th

e ar

ea o

f a re

ctan

gle

with

leng

th 4

,5 c

m a

nd

brea

dth

2 cm

.

e)

If y

= x2 +

1, c

alcu

late

y w

hen

x =

3

3 ho

urs


65CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.4

alg

ebra

ic

equa

tions

num

ber s

ente

nces

• W

rite

num

ber s

ente

nces

to d

escr

ibe

prob

lem

situ

atio

ns

• A

naly

se a

nd in

terp

ret n

umbe

r se

nten

ces

that

des

crib

e a

give

n si

tuat

ion

• S

olve

and

com

plet

e nu

mbe

r se

nten

ces

by:

-in

spec

tion

-tri

al a

nd im

prov

emen

t

• Id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae o

r equ

atio

ns

• D

eter

min

e th

e nu

mer

ical

val

ue o

f an

expr

essi

on b

y su

bstit

utio

n.

Wha

t is

diffe

rent

to th

e in

term

edia

te P

hase

? Th

e nu

mbe

r se

nten

ces

that

lear

ners

can

sol

ve a

re e

xten

ded

to in

clud

e nu

mbe

r se

nten

ces

with

inte

gers

, squ

are

num

bers

and

cub

ic n

umbe

rs.

Num

ber s

ente

nces

are

use

d he

re a

s a

mor

e fa

mili

ar te

rm fo

r Gra

de 7

lear

ners

than

eq

uatio

ns.

How

ever

, th

e te

rm e

quat

ion

will

be

used

ins

tead

of

num

ber

sent

ence

s in

lat

er

grad

es.

alg

ebra

ic e

quat

ions

are

don

e ag

ain

in t

erm

4, w

here

num

ber

sent

ence

s ca

n in

clud

e in

tege

rs.

Lear

ners

hav

e op

portu

nitie

s to

writ

e, s

olve

and

com

plet

e nu

mbe

r sen

tenc

es w

hen

they

writ

e ge

nera

l ru

les

to d

escr

ibe

rela

tions

hips

bet

wee

n nu

mbe

rs i

n nu

mbe

r pa

ttern

s, a

nd w

hen

they

find

inpu

t and

out

put v

alue

s fo

r giv

en ru

les

in fl

ow d

iagr

ams,

ta

bles

and

form

ulae

.

Rat

her

than

use

form

al a

lgeb

raic

pro

cess

es, l

earn

ers

solv

e nu

mbe

r se

nten

ces

by

insp

ectio

n or

det

erm

ine

the

num

eric

al v

alue

of e

xpre

ssio

ns b

y su

bstit

utio

n.

In th

is p

hase

, it i

s us

eful

whe

n so

lvin

g eq

uatio

ns to

beg

in to

spe

cify

whe

ther

x is

a

natu

ral n

umbe

r, in

tege

r or

rat

iona

l num

ber.

This

bui

lds

lear

ners

’ aw

aren

ess

of th

e do

mai

n of

x.

exam

ples

a)

Sol

ve x

if x

+ 4

= 7

, whe

re x

is a

nat

ural

num

ber.

(Wha

t mus

t be

adde

d to

4 to

gi

ve 7

?)

b)

Sol

ve x

if x

+ 4

= –

7, w

here

x is

an

inte

ger.

(Wha

t mus

t be

adde

d to

4 to

giv

e –7

?)

c)

Sol

ve x

if 2

x =

30, w

here

x is

a n

atur

al n

umbe

r. (W

hat m

ust b

e m

ultip

lied

by 2

to

giv

e 30

?)

d)

Writ

e a

num

ber s

ente

nce

to fi

nd th

e ar

ea o

f a re

ctan

gle

with

leng

th 4

,5 c

m a

nd

brea

dth

2 cm

.

e)

If y

= x2 +

1, c

alcu

late

y w

hen

x =

3

3 ho

urs

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.5

Gra

phs

inte

rpre

ting

grap

hs

• A

naly

se a

nd in

terp

ret g

loba

l gra

phs

of p

robl

em s

ituat

ions

, with

spe

cial

fo

cus

on th

e fo

llow

ing

trend

s an

d fe

atur

es:

-lin

ear o

r non

-line

ar

-co

nsta

nt, i

ncre

asin

g or

dec

reas

ing

dra

win

g gr

aphs

• D

raw

glo

bal g

raph

s fro

m g

iven

de

scrip

tions

of a

pro

blem

situ

atio

n,

iden

tifyi

ng fe

atur

es li

sted

abo

ve

Wha

t is

diffe

rent

to th

e in

term

edia

te P

hase

?

In th

e In

term

edia

te P

hase

lear

ners

enc

ount

ered

gra

phs

in th

e fo

rm o

f dat

a ba

r gr

aphs

and

pie

cha

rts. T

his

mea

ns th

ey d

o ha

ve s

ome

expe

rienc

e re

adin

g an

d in

terp

retin

g gr

aphs

. How

ever

, in

the

Sen

ior P

hase

, the

y ar

e in

trodu

ced

to li

ne

grap

hs th

at s

how

func

tiona

l rel

atio

nshi

ps d

escr

ibed

in te

rms

of d

epen

dent

and

in

depe

nden

t var

iabl

es.

In G

rade

7, t

he fo

cus

is o

n dr

awin

g, a

naly

sing

and

inte

rpre

ting

glob

al g

raph

s on

ly.

That

is, l

earn

ers

do n

ot h

ave

to p

lot p

oint

s to

dra

w g

raph

s an

d th

ey fo

cus

on th

e fe

atur

es o

f the

glo

bal r

elat

ions

hip

show

n in

the

grap

h.

exam

ples

of c

onte

xts

for g

loba

l gra

phs

incl

ude:

• th

e re

latio

nshi

p be

twee

n tim

e an

d di

stan

ce tr

avel

led

• th

e re

latio

nshi

p be

twee

n te

mpe

ratu

re a

nd ti

me

over

whi

ch it

is m

easu

red

• th

e re

latio

nshi

p be

twee

n ra

infa

ll an

d tim

e ov

er w

hich

it is

mea

sure

d, e

tc.

6 ho

urs

spac

e an

d sh

ape

(geo

met

ry)

3.4

tran

sfor

ma-

tion

G

eom

etry

tran

sfor

mat

ions

• R

ecog

nize

, des

crib

e an

d pe

rform

tra

nsla

tions

, refl

ectio

ns a

nd ro

tatio

ns

with

geo

met

ric fi

gure

s an

d sh

apes

on

squa

red

pape

r

• Id

entif

y an

d dr

aw li

nes

of s

ymm

etry

in

geo

met

ric fi

gure

s

enla

rgem

ents

and

redu

ctio

ns

• D

raw

enl

arge

men

ts a

nd re

duct

ions

of

geom

etric

figu

res

on s

quar

ed p

aper

an

d co

mpa

re th

em in

term

s of

sha

pe

and

size

Wha

t is

diffe

rent

to G

rade

6

Lear

ners

in G

rade

7 h

ave

to d

o tra

nsfo

rmat

ions

on

squa

red

pape

r.

Focu

s of

tran

sfor

mat

ions

• U

sing

squ

ared

pap

er fo

r tra

nsfo

rmat

ions

allo

ws

lear

ners

to m

ore

accu

rate

ly

perfo

rm tr

ansf

orm

atio

ns a

nd to

com

pare

the

shap

e an

d si

ze o

f figu

res.

• Le

arne

rs s

houl

d re

cogn

ize

that

tran

slat

ions

, refl

ectio

ns a

nd ro

tatio

ns o

nly

chan

ge th

e po

sitio

n of

the

figur

e, a

nd n

ot it

s sh

ape

or s

ize.

• Th

ey s

houl

d re

cogn

ize

that

the

abov

e tra

nsfo

rmat

ions

pro

duce

con

grue

nt

figur

es.

• Le

arne

rs s

houl

d re

cogn

ize

that

enl

arge

men

ts a

nd re

duct

ions

cha

nge

the

size

of

figur

es b

y in

crea

sing

or d

ecre

asin

g th

e le

ngth

of s

ides

, but

kee

ping

the

angl

es

the

sam

e, p

rodu

cing

sim

ilar r

athe

r tha

n co

ngru

ent fi

gure

s.

• Le

arne

rs s

houl

d al

so b

e ab

le to

wor

k ou

t the

fact

or o

f enl

arge

men

t or r

educ

tion.

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

3.2

Geo

met

ry o

f 3d

obj

ects

Cla

ssify

ing

3d o

bjec

ts

• D

escr

ibe,

sor

t and

com

pare

po

lyhe

dra

in te

rms

of

-sh

ape

and

num

ber o

f fac

es

-nu

mbe

r of v

ertic

es

-nu

mbe

r of e

dges

Bui

ldin

g 3d

mod

el

• R

evis

e us

ing

nets

to c

reat

e m

odel

s of

ge

omet

ric s

olid

s, in

clud

ing:

-cu

bes

-pr

ism

s

Wha

t is

diffe

rent

to G

rade

6?

• M

ost o

f thi

s w

ork

cons

olid

ates

wha

t has

bee

n do

ne in

Gra

de 6

.

Poly

hedr

a

Exa

mpl

es o

f sor

ting

or g

roup

ing

cate

gorie

s:

• cu

bes

(onl

y sq

uare

face

s)

• re

ctan

gula

r pris

ms

(onl

y re

ctan

gula

r fac

es)

• tri

angu

lar p

rism

s (o

nly

trian

gula

r and

rect

angu

lar f

aces

)

• py

ram

ids

(squ

are

and

trian

gula

r fac

es)

• cy

linde

rs (c

ircul

ar a

nd re

ctan

gula

r fac

es).

usi

ng a

nd c

onst

ruct

ing

nets

• U

sing

and

con

stru

ctin

g ne

ts a

re u

sefu

l con

text

s fo

r exp

lorin

g or

con

solid

atin

g pr

oper

ties

of p

olyh

edra

.

• Le

arne

rs s

houl

d re

cogn

ize

the

nets

of d

iffer

ent s

olid

s.

• Le

arne

rs s

houl

d dr

aw s

ketc

hes

of th

e ne

ts u

sing

thei

r kno

wle

dge

of s

hape

and

nu

mbe

r of f

aces

of t

he s

olid

s, b

efor

e dr

awin

g an

d cu

tting

out

the

nets

to b

uild

m

odel

s.

• Th

e co

nstru

ctio

n of

net

s is

bas

ed o

n th

e nu

mbe

r and

sha

pe o

f fac

es o

f the

so

lids,

and

do

not r

equi

re m

easu

ring

of in

tern

al a

ngle

s of

pol

ygon

s.

• Le

arne

rs h

ave

to w

ork

out t

he re

lativ

e po

sitio

n of

the

face

s of

the

nets

, and

us

e tri

al a

nd e

rror

to m

atch

up

the

edge

s an

d ve

rtice

s, in

ord

er to

bui

ld th

e 3D

ob

ject

.

9 ho

urs

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• nu

mer

ic a

nd g

eom

etric

pat

tern

s

• fu

nctio

ns a

nd re

latio

nshi

ps

• al

gebr

aic

expr

essi

ons

• al

gebr

aic

equa

tions

• gr

aphs

• tra

nsfo

rmat

ion

geom

etry

• ge

omet

ry o

f 3D

obj

ects

6 ho

urs


67CAPS

Gr

ad

e 7

– te

rm

4

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.3

inte

gers

Cou

ntin

g, o

rder

ing

and

com

parin

g in

tege

rs

• C

ount

forw

ards

and

bac

kwar

ds in

in

tege

rs fo

r any

inte

rval

• R

ecog

nize

, ord

er a

nd c

ompa

re

inte

gers

Cal

cula

tions

with

inte

gers

• A

dd a

nd s

ubtra

ct w

ith in

tege

rs

Prop

ertie

s of

inte

gers

• R

ecog

nize

and

use

com

mut

ativ

e an

d as

soci

ativ

e pr

oper

ties

of a

dditi

on a

nd

mul

tiplic

atio

n fo

r int

eger

s

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

addi

tion

and

subt

ract

ion

of in

tege

rs

Wha

t is

diffe

rent

to G

rade

6?

Inte

gers

are

new

num

bers

intro

duce

d in

Gra

de 7

.

Cou

ntin

g, o

rder

ing

and

com

parin

g in

tege

rs

• C

ount

ing

shou

ld n

ot o

nly

be th

ough

t of a

s ve

rbal

cou

ntin

g. L

earn

ers

can

coun

t us

ing:

-st

ruct

ured

, sem

i-stru

ctur

ed o

r em

pty

num

ber l

ines

-ch

ain

diag

ram

s fo

r cou

ntin

g

• Le

arne

rs s

houl

d be

giv

en a

rang

e of

exe

rcis

es s

uch

as:

-ar

rang

e gi

ven

num

bers

from

the

smal

lest

to th

e bi

gges

t: or

big

gest

to s

mal

lest

-fil

l in

mis

sing

num

bers

in

♦a

sequ

ence

♦on

a n

umbe

r grid

♦on

a n

umbe

r lin

e

♦fil

l in

<, =

or >

exa

mpl

e: –

425

* –

450

Cal

cula

tions

usi

ng in

tege

rs

• S

tart

calc

ulat

ions

usi

ng in

tege

rs in

sm

all n

umbe

r ran

ges.

• D

evel

op a

n un

ders

tand

ing

that

sub

tract

ing

an in

tege

r is

the

sam

e as

add

ing

its

addi

tive

inve

rse.

•ex

ampl

e:

7 –

4 =

7 +

(– 4

) = 3

or

–7

– 4

= –7

+ (–

4) =

–11

So

too,

7 –

(– 4

) = 7

+ (+

4) =

11

or

–7

– (–

4) =

–7

+ (+

4) =

–3

Her

e th

e us

e of

bra

cket

s ar

ound

the

inte

gers

are

use

ful.

Prop

ertie

s of

inte

gers

• Le

arne

rs s

houl

d in

vest

igat

e th

e pr

oper

ties

for o

pera

tions

usi

ng w

hole

num

bers

on

the

set o

f int

eger

s.

• Th

ese

prop

ertie

s sh

ould

ser

ve a

s m

otiv

atio

n fo

r the

ope

ratio

ns th

ey c

an p

erfo

rm

usin

g in

tege

rs.

• Le

arne

rs s

houl

d se

e th

at th

e co

mm

utat

ive

prop

erty

for a

dditi

on h

olds

for

inte

gers

e.g

8 +

(–3)

= (–

3) +

8 =

5

• Le

arne

rs s

houl

d se

e th

at th

ey c

an s

till u

se s

ubtra

ctio

n to

che

ck a

dditi

on o

r vic

e ve

rsa.

exa

mpl

e: If

8 +

(–3)

= 5

, the

n 5

– 8

= –

3 an

d 5

–(–3

) = 8

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.3

inte

gers

• Le

arne

rs s

houl

d se

e th

at th

e as

soci

ativ

e pr

oper

ty fo

r add

ition

hol

ds fo

r int

eger

s.

exam

ple:

[(–6

) + 4

] + (–

1) =

(–6)

+ [4

+ (–

1)] =

–3

• Le

arne

rs s

houl

d on

ly e

xplo

re th

e di

strib

utiv

e pr

oper

ty o

nce

they

can

mul

tiply

w

ith in

tege

rs

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

an

d ge

omet

ric p

atte

rns

look

ing

for

rela

tions

hips

bet

wee

n nu

mbe

rs,

incl

udin

g pa

ttern

s:

-re

pres

ente

d in

phy

sica

l or

diag

ram

fo

rm

-no

t lim

ited

to s

eque

nces

invo

lvin

g a

cons

tant

diff

eren

ce o

r rat

io

-of

lear

ner’s

ow

n cr

eatio

n

-re

pres

ente

d in

tabl

es

• D

escr

ibe

and

just

ify th

e ge

nera

l rul

es

for o

bser

ved

rela

tions

hips

bet

wee

n nu

mbe

rs in

ow

n w

ords

The

focu

s of

num

eric

pat

tern

s in

this

term

sho

uld

be o

n pr

actis

ing

oper

atio

ns w

ith

inte

gers

.

see

addi

tiona

l not

es in

ter

m 3

.

3 ho

urs

2.2

Func

tions

an

d re

latio

nshi

ps

inpu

t and

out

put v

alue

s•

Det

erm

ine

inpu

t val

ues,

out

put

valu

es o

r rul

es fo

r pat

tern

s an

d re

latio

nshi

ps u

sing

:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

eq

uiva

lenc

e of

diff

eren

t des

crip

tions

of

the

sam

e re

latio

nshi

p or

rule

pr

esen

ted:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

num

ber s

ente

nces

Func

tions

and

rela

tions

hips

in th

is te

rm s

houl

d in

clud

e in

tege

rs a

s in

put o

r out

put

valu

es, a

s w

ell a

s us

ing

inte

gers

in th

e ru

les

for p

atte

rns

and

rela

tions

hips

.

see

addi

tiona

l not

es in

ter

ms

2 &

3.

3 ho

urs


69CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.3

alg

ebra

ic

expr

essi

ons

alg

ebra

ic la

ngua

ge

• R

ecog

nize

and

inte

rpre

t rul

es o

r re

latio

nshi

ps re

pres

ente

d in

sym

bolic

fo

rm

• Id

entif

y va

riabl

es a

nd c

onst

ants

in

form

ulae

and

equ

atio

ns

Alg

ebra

ic e

xpre

ssio

ns s

houl

d in

clud

e in

tege

rs in

the

rule

s or

rela

tions

hips

repr

e-se

nted

in s

ymbo

lic fo

rm.

see

addi

tiona

l not

es in

ter

m 3

.

3 ho

urs

2.4

alg

ebra

ic

equa

tions

num

ber s

ente

nces

• W

rite

num

ber s

ente

nces

to d

escr

ibe

prob

lem

situ

atio

ns

• A

naly

se a

nd in

terp

ret n

umbe

r se

nten

ces

that

des

crib

e a

give

n si

tuat

ion

• S

olve

and

com

plet

e nu

mbe

r se

nten

ces

by:

-in

spec

tion

-tri

al a

nd im

prov

emen

t

• Id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae o

r equ

atio

ns

• D

eter

min

e th

e nu

mer

ical

val

ue o

f an

expr

essi

on b

y su

bstit

utio

n.

Num

ber s

ente

nces

sho

uld

incl

ude

inte

gers

.

see

addi

tiona

l not

es in

ter

m 3

.

4 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

dat

a H

an

dli

nG

5.1

Col

lect

, or

gani

ze a

nd

sum

mar

ize

data

Col

lect

dat

a

• P

ose

ques

tions

rela

ting

to s

ocia

l, ec

onom

ic, a

nd e

nviro

nmen

tal i

ssue

s in

ow

n en

viro

nmen

t

• S

elec

t app

ropr

iate

sou

rces

for

the

colle

ctio

n of

dat

a (in

clud

ing

peer

s, fa

mily

, new

spap

ers,

boo

ks,

mag

azin

es)

• D

istin

guis

h be

twee

n sa

mpl

es a

nd

popu

latio

ns

• D

esig

n an

d us

e si

mpl

e qu

estio

nnai

res

to a

nsw

er q

uest

ions

:

-w

ith y

es/n

o ty

pe re

spon

ses

-w

ith m

ultip

le c

hoic

e re

spon

ses.

org

aniz

e an

d su

mm

ariz

e da

ta

• O

rgan

ize

(incl

udin

g gr

oupi

ng w

here

ap

prop

riate

) and

reco

rd d

ata

usin

g

-ta

lly m

arks

-ta

bles

-st

em-a

nd-le

af d

ispl

ays

Wha

t is

diffe

rent

to G

rade

6?

The

follo

win

g ar

e ne

w in

Gra

de 7

• sa

mpl

es a

nd p

opul

atio

ns

• m

ultip

le c

hoic

e qu

estio

nnai

res

• st

em-a

nd-le

af d

ispl

ays

• gr

oupi

ng d

ata

in in

terv

als

• m

ean

• ra

nge

• hi

stog

ram

s

• sc

ales

on

grap

hs

dat

a se

ts a

nd c

onte

xts

Lear

ners

sho

uld

be e

xpos

ed to

a v

arie

ty o

f con

text

s th

at d

eal w

ith s

ocia

l and

en

viro

nmen

tal i

ssue

s, a

nd s

houl

d w

ork

with

giv

en d

ata

sets

, rep

rese

nted

in

a va

riety

of w

ays,

that

incl

ude

big

num

ber r

ange

s, p

erce

ntag

es a

nd d

ecim

al

fract

ions

. Lea

rner

s sh

ould

then

pra

ctis

e or

gani

zing

and

sum

mar

izin

g th

is d

ata,

an

alys

ing

and

inte

rpre

ting

the

data

, and

writ

ing

a re

port

abou

t the

dat

a.

Com

plet

e a

data

cyc

le

Lear

ners

sho

uld

com

plet

e at

leas

t one

dat

a cy

cle

for t

he y

ear,

star

ting

with

po

sing

thei

r ow

n qu

estio

ns, s

elec

ting

the

sour

ces

and

met

hod

for c

olle

ctin

g da

ta,

reco

rdin

g th

e da

ta, o

rgan

izin

g th

e da

ta, r

epre

sent

ing

the

data

, the

n an

alys

ing,

su

mm

ariz

ing,

inte

rpre

ting

and

repo

rting

the

data

. Cha

lleng

e le

arne

rs to

thin

k ab

out

wha

t kin

ds o

f que

stio

ns a

nd d

ata

need

to b

e co

llect

ed to

be

repr

esen

ted

in a

hi

stog

ram

, pie

cha

rt, o

r bar

gra

ph.

tim

e fo

r co

llect

ing

and

orga

nizi

ng

data

: 4 h

ours


71CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

dat

a H

an

dli

nG

5.1

Col

lect

, or

gani

ze a

nd

sum

mar

ize

data

• G

roup

dat

a in

to in

terv

als

• S

umm

ariz

e an

d di

stin

guis

h be

twee

n un

grou

ped

num

eric

al d

ata

by

dete

rmin

ing:

-m

ean

-m

edia

n

-m

ode

• Id

entif

y th

e la

rges

t and

sm

alle

st

scor

es in

a d

ata

set a

nd d

eter

min

e th

e di

ffere

nce

betw

een

them

in o

rder

to

det

erm

ine

the

spre

ad o

f the

dat

a (r

ange

).

rep

rese

ntin

g da

ta

• D

raw

ing

pie

char

ts to

repr

esen

t dat

a do

not

hav

e to

be

accu

rate

ly d

raw

n w

ith a

co

mpa

ss a

nd p

rotra

ctor

, etc

. Lea

rner

s ca

n us

e an

y ro

und

obje

ct to

dra

w a

circ

le,

then

div

ide

the

circ

le in

to h

alve

s an

d qu

arte

rs a

nd e

ight

hs if

nee

ded,

as

a gu

ide

to e

stim

ate

the

prop

ortio

ns o

f the

circ

le th

at n

eed

to b

e sh

own

to re

pres

ent t

he

data

. Wha

t is

impo

rtant

is th

at th

e va

lues

or p

erce

ntag

es a

ssoc

iate

d w

ith th

e da

ta, a

re s

how

n pr

opor

tiona

lly o

n th

e pi

e ch

art.

• D

raw

ing,

read

ing

and

inte

rpre

ting

pie

char

ts is

a u

sefu

l con

text

to re

-vis

it eq

uiva

lenc

e be

twee

n fra

ctio

ns a

nd p

erce

ntag

es, e

.g. 2

5% o

f the

dat

a is

re

pres

ente

d by

a 1 4

sec

tor o

f the

circ

le.

• It

is a

lso

a co

ntex

t in

whi

ch le

arne

rs c

an fi

nd p

erce

ntag

es o

f who

le n

umbe

rs e

.g.

if 25

% o

f 300

lear

ners

like

rugb

y, h

ow m

any

(act

ual n

umbe

r) le

arne

rs li

ke ru

gby?

• H

isto

gram

s ar

e us

ed to

repr

esen

t gro

uped

dat

a sh

own

in in

terv

als

on th

e ho

rizon

tal a

xis

of th

e gr

aph.

Poi

nt o

ut th

e di

ffere

nces

bet

wee

n hi

stog

ram

s an

d ba

r gra

phs,

in p

artic

ular

bar

gra

phs

that

repr

esen

t dis

cret

e da

ta (e

.g. f

avou

rite

spor

ts) c

ompa

red

to h

isto

gram

s th

at s

how

cat

egor

ies

in c

onse

cutiv

e, n

on-

over

lapp

ing

inte

rval

s, (e

.g. t

est s

core

s ou

t of 1

00 s

how

n in

inte

rval

s of

10)

. The

ba

rs o

n ba

r gra

phs

do n

ot h

ave

to to

uch

each

oth

er, w

hile

in a

his

togr

am th

ey

have

to to

uch

sinc

e th

ey s

how

con

secu

tive

inte

rval

s.

dev

elop

ing

criti

cal a

naly

sis

skill

s

• Le

arne

rs s

houl

d co

mpa

re th

e sa

me

data

repr

esen

ted

in d

iffer

ent w

ays

e.g.

in a

pi

e ch

art o

r a b

ar g

raph

or a

tabl

e, a

nd d

iscu

ss w

hat i

nfor

mat

ion

is s

how

n an

d w

hat i

s hi

dden

; the

y sh

ould

eva

luat

e w

hat f

orm

of r

epre

sent

atio

n w

orks

bes

t for

th

e gi

ven

data

.

• Le

arne

rs s

houl

d co

mpa

re g

raph

s on

the

sam

e to

pic

but w

here

dat

a ha

s be

en

colle

cted

from

diff

eren

t gro

ups

of p

eopl

e, a

t diff

eren

t tim

es, i

n di

ffere

nt p

lace

s or

in d

iffer

ent w

ays.

Her

e le

arne

rs s

houl

d di

scus

s di

ffere

nces

bet

wee

n th

e da

ta

with

an

awar

enes

s of

bia

s re

late

d to

the

impa

ct o

f dat

a so

urce

s an

d m

etho

ds o

f da

ta c

olle

ctio

n on

the

inte

rpre

tatio

n of

the

data

.

• Le

arne

rs s

houl

d co

mpa

re d

iffer

ent w

ays

of s

umm

aris

ing

the

sam

e da

ta s

ets,

de

velo

ping

an

awar

enes

s of

how

dat

a re

porti

ng c

an b

e m

anip

ulat

ed; e

valu

atin

g w

hich

sum

mar

y st

atis

tics

best

repr

esen

t the

dat

a.

• Le

arne

rs s

houl

d co

mpa

re g

raph

s of

the

sam

e da

ta, w

here

the

scal

es o

f th

e gr

aphs

are

diff

eren

t. H

ere

lear

ners

sho

uld

disc

uss

diffe

renc

es w

ith a

n aw

aren

ess

of h

ow re

pres

enta

tion

of d

ata

can

be m

anip

ulat

ed; t

hey

shou

ld

eval

uate

whi

ch fo

rm o

f rep

rese

ntat

ion

wor

ks b

est f

or th

e gi

ven

data

.

• Le

arne

rs s

houl

d w

rite

repo

rts o

n th

e da

ta in

sho

rt pa

ragr

aphs

.

tim

e fo

r re

pres

entin

g da

ta: 3

hou

rs

tim

e fo

r an

alys

ing,

in

terp

retin

g an

d re

port

ing

data

: 3,5

ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.2

rep

rese

ntin

g da

ta

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy to

dis

play

and

inte

rpre

t da

ta (g

roup

ed a

nd u

ngro

uped

) in

clud

ing:

-ba

r gra

phs

and

doub

le b

ar g

raph

s -

hist

ogra

ms

with

giv

en in

terv

als

-pi

e ch

arts

5.3

inte

rpre

t, an

alys

e an

d re

port

dat

a

inte

rpre

t dat

a•

Crit

ical

ly re

ad a

nd in

terp

ret d

ata

repr

esen

ted

in:

-w

ords

-ba

r gra

phs

-do

uble

bar

gra

phs

-pi

e ch

arts

-hi

stog

ram

sa

naly

se d

ata

• C

ritic

ally

ana

lyse

dat

a by

ans

wer

ing

ques

tions

rela

ted

to:

-da

ta c

ateg

orie

s, in

clud

ing

data

in

terv

als

-da

ta s

ourc

es a

nd c

onte

xts

-ce

ntra

l ten

denc

ies

(mea

n, m

ode,

m

edia

n) -

scal

es u

sed

on g

raph

sr

epor

t dat

a•

Sum

mar

ize

data

in s

hort

para

grap

hs

that

incl

ude

-dr

awin

g co

nclu

sion

s ab

out t

he d

ata

-m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

-id

entif

ying

sou

rces

of e

rror

and

bi

as in

the

data

-ch

oosi

ng a

ppro

pria

te s

umm

ary

stat

istic

s fo

r the

dat

a (m

ean,

m

edia

n, m

ode)


73CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.2

rep

rese

ntin

g da

ta

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy to

dis

play

and

inte

rpre

t da

ta (g

roup

ed a

nd u

ngro

uped

) in

clud

ing:

-ba

r gra

phs

and

doub

le b

ar g

raph

s -

hist

ogra

ms

with

giv

en in

terv

als

-pi

e ch

arts

5.3

inte

rpre

t, an

alys

e an

d re

port

dat

a

inte

rpre

t dat

a•

Crit

ical

ly re

ad a

nd in

terp

ret d

ata

repr

esen

ted

in:

-w

ords

-ba

r gra

phs

-do

uble

bar

gra

phs

-pi

e ch

arts

-hi

stog

ram

sa

naly

se d

ata

• C

ritic

ally

ana

lyse

dat

a by

ans

wer

ing

ques

tions

rela

ted

to:

-da

ta c

ateg

orie

s, in

clud

ing

data

in

terv

als

-da

ta s

ourc

es a

nd c

onte

xts

-ce

ntra

l ten

denc

ies

(mea

n, m

ode,

m

edia

n) -

scal

es u

sed

on g

raph

sr

epor

t dat

a•

Sum

mar

ize

data

in s

hort

para

grap

hs

that

incl

ude

-dr

awin

g co

nclu

sion

s ab

out t

he d

ata

-m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

-id

entif

ying

sou

rces

of e

rror

and

bi

as in

the

data

-ch

oosi

ng a

ppro

pria

te s

umm

ary

stat

istic

s fo

r the

dat

a (m

ean,

m

edia

n, m

ode)

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.4

Prob

abili

ty

Prob

abili

ty

Per

form

sim

ple

expe

rimen

ts w

here

the

poss

ible

out

com

es a

re e

qual

ly li

kely

an

d

• lis

t the

pos

sibl

e ou

tcom

es b

ased

on

the

cond

ition

s of

the

activ

ity

• de

term

ine

the

prob

abili

ty o

f eac

h po

ssib

le o

utco

me,

usi

ng th

e de

finiti

on

of p

roba

bilit

y

Prob

abili

ty e

xper

imen

tsIn

the

Inte

rmed

iate

Pha

se le

arne

rs d

id e

xper

imen

ts w

ith c

oins

, dic

e an

d sp

inne

rs.

In th

is g

rade

exp

erim

ents

can

be

done

with

oth

er o

bjec

ts, l

ike,

diff

eren

t col

oure

d bu

ttons

in a

bag

; cho

osin

g sp

ecifi

c ca

rds

from

a d

eck

of c

ards

, etc

.

exam

ple

If yo

u to

ss a

coi

n th

ere

are

two

poss

ible

out

com

es (h

ead

or ta

il). T

he p

roba

bilit

y of

a h

ead

is 1 2

whi

ch is

equ

ival

ent t

o 50

% (s

ince

it is

one

out

of t

wo

poss

ible

ou

tcom

es)

4,5

hour

s

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• ad

ding

and

sub

tract

ing

with

inte

gers

• co

llect

ing,

org

aniz

ing,

repr

esen

ting,

ana

lysi

ng, s

umm

ariz

ing,

inte

rpre

ting

and

repo

rting

dat

a

• pr

obab

ility

tota

l tim

e fo

r re

visi

on/

asse

ssm

ent

for t

he te

rm8

hour

s



time alloCation Per term: Grade 8



Whole numbers 6 hours


9 hours Common fractions 7

hoursFunctions and relationships

6 hours

Integers 9 hours

Algebraic equations

3 hours Decimal fractions 6

hoursAlgebraic equations

3 hours

Exponents 9 hours

Construction of Geometric figures

8 hours

Theorem of Pythagoras

5 hours Graphs 9

hours


4,5 hours


8 hours

Area and perimeter of 2D shapes

5 hours Transformation

geometry6

hours


3 hours

Geometry of straight lines

9 hours

Surface area and volume of 3D objects

5 hours Geometry of 3D

objects7

hours


4,5 hours

Collect, organize and summarize data

4 hours Probability

4,5 hours

Algebraic equations

3 hours Represent data 3

hours


3,5 hours

revision/assessment

6 hours

revision/assessment

8 hours

revision/assessment

6,5 hours

revision/assessment

9,5 hours



75CAPS

3.3.2Clarifi

catio

nofcon

tentfo

rGrade8

Gr

ad

e 8

– te

rm

1

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.1

Who

le

num

bers

men

tal c

alcu

latio

ns

Rev

ise:

• m

ultip

licat

ion

of w

hole

num

bers

to a

t le

ast 1

2 x

12

ord

erin

g an

d co

mpa

ring

who

le

num

bers

Rev

ise

Prim

e nu

mbe

rs to

at l

east

100

Prop

ertie

s of

who

le n

umbe

rs

• R

evis

e:

-Th

e co

mm

utat

ive;

ass

ocia

tive;

di

strib

utiv

e pr

oper

ties

of w

hole

nu

mbe

rs

-0

in te

rms

of it

s ad

ditiv

e pr

oper

ty

(iden

tity

elem

ent f

or a

dditi

on)

-1

in te

rms

of it

s m

ultip

licat

ive

prop

erty

(ide

ntify

ele

men

t for

m

ultip

licat

ion)

-R

ecog

nise

the

divi

sion

pro

perty

of

0, w

here

by a

ny n

umbe

r div

ided

by

0 is

und

efine

d

Wha

t is

diffe

rent

to G

rade

7?

In G

rade

8 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r w

hole

num

bers

, dev

elop

ed in

the

Inte

rmed

iate

Pha

se a

nd G

rade

7

Prop

ertie

s of

who

le n

umbe

rs

• R

evis

ing

the

prop

ertie

s of

who

le n

umbe

rs s

houl

d be

the

star

ting

poin

t for

wor

k w

ith w

hole

num

bers

. The

pro

perti

es o

f num

bers

sho

uld

prov

ide

the

mot

ivat

ion

for

why

and

how

ope

ratio

ns w

ith n

umbe

rs w

ork.

• W

hen

lear

ners

are

intro

duce

d to

new

num

bers

, suc

h as

inte

gers

for e

xam

ple,

th

ey c

an a

gain

exp

lore

how

the

prop

ertie

s of

num

bers

wor

k fo

r the

new

set

of

num

bers

.

• Le

arne

rs a

lso

have

to a

pply

the

prop

ertie

s of

num

bers

in a

lgeb

ra, w

hen

they

w

ork

with

var

iabl

es in

pla

ce o

f num

bers

.

• Le

arne

rs s

houl

d kn

ow a

nd b

e ab

le to

use

the

follo

win

g pr

oper

ties:

-Th

e co

mm

utat

ive

prop

erty

of a

dditi

on a

nd m

ultip

licat

ion:

♦a

+ b

= b

+ a

♦a

x b

= b

x a

-Th

e as

soci

ativ

e (g

roup

ing)

pro

perty

of a

dditi

on a

nd m

ultip

licat

ion:

♦(a

+ b

) + c

= a

+ (b

+ c

)

♦(a

x b

) x c

= a

x (b

x c

)

-Th

e di

strib

utiv

e pr

oper

ty o

f mul

tiplic

atio

n ov

er a

dditi

on a

nd s

ubtra

ctio

n:

♦a(

b +

c) =

(a x

b) +

(a x

c)

♦a(

b –

c) =

(a x

b) –

(a x

c)

-A

dditi

on a

nd s

ubtra

ctio

n as

inve

rse

oper

atio

ns

-m

ultip

licat

ion

and

divi

sion

as

inve

rse

oper

atio

ns

-0

is th

e id

entit

y el

emen

t for

add

ition

: t +

0 =

t

-1

is th

e id

entit

y el

emen

t for

mul

tiplic

atio

n: t

x 1

= t

6 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.1

Who

le

num

bers

Cal

cula

tions

usi

ng w

hole

num

bers

Rev

ise:

• C

alcu

latio

ns u

sing

all

four

ope

ratio

ns

on w

hole

num

bers

, est

imat

ing

and

usin

g ca

lcul

ator

s w

here

app

ropr

iate

Cal

cula

tion

tech

niqu

es

• U

se a

rang

e of

stra

tegi

es to

per

form

an

d ch

eck

writ

ten

and

men

tal

calc

ulat

ions

with

who

le n

umbe

rs

incl

udin

g:

-es

timat

ion

-ad

ding

, sub

tract

ing

and

mul

tiply

ing

in c

olum

ns

-lo

ng d

ivis

ion

-ro

undi

ng o

ff an

d co

mpe

nsat

ing

-us

ing

a ca

lcul

ator

mul

tiple

s an

d fa

ctor

s

Rev

ise:

-P

rime

fact

ors

of n

umbe

rs to

at l

east

3-

digi

t who

le n

umbe

rs

-LC

m a

nd H

CF

of n

umbe

rs to

at

leas

t 3-d

igit

who

le n

umbe

rs, b

y in

spec

tion

or fa

ctor

isat

ion

illus

trat

ing

the

prop

ertie

s w

ith w

hole

num

bers

a)

33 +

99

= 99

+ 3

3 =

132

b)

51 +

(19

+ 46

) = (5

1 +

19) +

46

= 11

6

c)

4(12

+ 9

) = (4

x 1

2) +

(4 x

9) =

48

+ 36

= 8

4

d)

(9 x

64)

+ (9

x 3

6) =

9(6

4 +

36) =

9 x

100

= 9

00

e)

if 33

+ 9

9 =

132,

then

132

– 9

9 =

33 a

d 13

2 –

33 =

99

f) if

20 x

5 =

110

, the

n 11

0 ÷

20 =

5 a

nd 1

10 ÷

5 =

20

Cal

cula

tions

with

who

le n

umbe

rs

• Le

arne

rs s

houl

d co

ntin

ue to

do

cont

ext f

ree

calc

ulat

ions

and

sol

ve p

robl

ems

in

cont

exts

usi

ng w

hole

num

bers

, int

eger

s an

d fra

ctio

ns

• Le

arne

rs s

houl

d be

com

e m

ore

confi

dent

in a

nd m

ore

inde

pend

ent a

t m

athe

mat

ics,

if th

ey h

ave

tech

niqu

es

-to

che

ck th

eir s

olut

ions

them

selv

es, e

.g. u

sing

inve

rse

oper

atio

ns; u

sing

ca

lcul

ator

s

-to

judg

e th

e re

ason

able

ness

of t

heir

solu

tions

e.g

. est

imat

e by

roun

ding

off;

es

timat

e by

dou

blin

g or

hal

ving

;

• A

ddin

g, s

ubtra

ctin

g an

d m

ultip

lyin

g in

col

umns

, and

long

div

isio

n, s

houl

d on

ly b

e us

ed to

pra

ctic

e nu

mbe

r fac

ts a

nd c

alcu

latio

n te

chni

ques

, and

hen

ce s

houl

d be

do

ne w

ith fa

mili

ar a

nd s

mal

ler n

umbe

r ran

ges.

For

big

and

unw

ield

y ca

lcul

atio

ns,

lear

ners

sho

uld

be e

ncou

rage

d to

use

a c

alcu

lato

r.

mul

tiple

s an

d fa

ctor

s

• Le

arne

rs s

houl

d co

ntin

ue p

ract

isin

g fin

ding

mul

tiple

s an

d fa

ctor

s of

who

le

num

bers

. Thi

s is

esp

ecia

lly im

porta

nt w

hen

lear

ners

do

calc

ulat

ions

with

fra

ctio

ns. T

hey

use

this

kno

wle

dge

to fi

nd th

e LC

M fo

r den

omin

ator

s th

at a

re

diffe

rent

from

eac

h ot

her,

and

also

whe

n th

ey s

impl

ify fr

actio

ns o

r hav

e to

find

eq

uiva

lent

frac

tions

.

• Fa

ctor

isin

g w

hole

num

bers

lays

the

foun

datio

n fo

r fac

toris

atio

n of

alg

ebra

ic

expr

essi

ons.

• U

sing

the

defin

ition

of p

rime

num

bers

, em

phas

ise

that

1 is

not

cla

ssifi

ed a

s a

prim

e nu

mbe

r


77CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, o

pera

tions

and

r

elat

ions

hips

1.1

Who

le

num

bers

solv

ing

prob

lem

s•

Sol

ve p

robl

ems

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

:

-C

ompa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd (r

atio

)

-C

ompa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

-S

harin

g in

a g

iven

ratio

whe

re th

e w

hole

is g

iven

-In

crea

sing

or d

ecre

asin

g of

a

num

ber i

n a

give

n ra

tio

• S

olve

pro

blem

s th

at in

volv

e w

hole

nu

mbe

rs, p

erce

ntag

es a

nd d

ecim

al

fract

ions

in fi

nanc

ial c

onte

xts

such

as:

-P

rofit

, los

s, d

isco

unt a

nd V

AT

-B

udge

ts

-A

ccou

nts

-Lo

ans

-S

impl

e in

tere

st

-H

ire P

urch

ase

-E

xcha

nge

rate

s

exam

ples

a)

LCM

of 6

and

18

is 1

8LC

M o

f 6 a

nd 7

is 4

2b)

Th

e fa

ctor

s of

24

are

1, 2

, 3, 4

, 6,1

2 an

d 24

by

insp

ectio

n.A

nd, t

he p

rime

fact

ors

of 2

4 ar

e 2

and

3c)

Th

e fa

ctor

s of

140

are

1, 2

, 5, 7

, 10,

14,

28,

35,

70

and

140

d)

Det

erm

ine

the

HC

F of

120

; 300

and

900

. Le

arne

rs d

o th

is b

y fin

ding

the

prim

e fa

ctor

s of

the

num

bers

firs

t.12

0 =

5 x

3 x

23 Ini

tially

lear

ners

may

writ

e th

is a

s: 5

x 3

x 2

x 2

x 2

300

= 52 x

3 x

22

900

= 52 x

32 x

23

HC

F =

5 x

3 x

22 = 6

0 (M

ultip

ly th

e co

mm

on p

rime

fact

ors

of th

e th

ree

num

bers

)so

lvin

g pr

oble

ms

• S

olvi

ng p

robl

ems

in c

onte

xts

shou

ld ta

ke a

ccou

nt o

f the

num

ber r

ange

s le

arne

rs

are

fam

iliar

with

.

• C

onte

xts

invo

lvin

g ra

tio a

nd ra

te s

houl

d in

clud

e sp

eed,

dis

tanc

e an

d tim

e pr

oble

ms.

• In

fina

ncia

l con

text

s, le

arne

rs a

re n

ot e

xpec

ted

to u

se fo

rmul

ae fo

r cal

cula

ting

sim

ple

inte

rest

.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.3

inte

gers

Cou

ntin

g, o

rder

ing

and

com

parin

g in

tege

rs

• R

evis

e:

-co

untin

g fo

rwar

ds a

nd b

ackw

ards

in

inte

gers

for a

ny in

terv

al

-re

cogn

isin

g, o

rder

ing

and

com

parin

g in

tege

rs

Cal

cula

tions

with

inte

gers

• R

evis

e ad

ditio

n an

d su

btra

ctio

n w

ith

inte

gers

• m

ultip

ly a

nd d

ivid

e w

ith in

tege

rs

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our

oper

atio

ns w

ith in

tege

rs

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our

oper

atio

ns w

ith n

umbe

rs th

at in

volv

e th

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of in

tege

rs

Wha

t is

diffe

rent

to G

rade

7?

• m

ultip

ly a

nd d

ivid

e w

ith in

tege

rs

• A

ll fo

ur o

pera

tions

with

inte

gers

• A

ll fo

ur o

pera

tions

with

squ

ares

, cub

es, s

quar

e an

d cu

be ro

ots

of in

tege

rs

In G

rade

8 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r in

tege

rs, d

evel

oped

in G

rade

7.

Cou

ntin

g, o

rder

ing

and

com

parin

g in

tege

rs

• Le

arne

rs s

houl

d co

ntin

ue p

ract

isin

g co

untin

g, o

rder

ing

and

com

parin

g in

tege

rs.

Cou

ntin

g sh

ould

not

onl

y be

thou

ght o

f as

verb

al c

ount

ing.

Lea

rner

s ca

n co

unt

usin

g:

-st

ruct

ured

, sem

i-stru

ctur

ed o

r em

pty

num

ber l

ines

-ch

ain

diag

ram

s fo

r cou

ntin

g

• Le

arne

rs s

houl

d be

giv

en a

rang

e of

exe

rcis

es

• A

rran

ge g

iven

num

bers

from

the

smal

lest

to th

e bi

gges

t: or

big

gest

to s

mal

lest

• Fi

ll in

mis

sing

num

bers

in

-a

sequ

ence

-on

a n

umbe

r grid

-on

a n

umbe

r lin

e

-fil

l in

<, =

or >

e.g

. – 4

25 *

– 4

50;

Cal

cula

tions

usi

ng in

tege

rs

• S

tart

calc

ulat

ions

with

inte

gers

usi

ng s

mal

l num

ber r

ange

s.

• D

evel

op a

n un

ders

tand

ing

that

sub

tract

ing

an in

tege

r is

the

sam

e as

add

ing

its

addi

tive

inve

rse.

exam

ple:

7 –

4 =

7 +

(– 4

) = 3

OR

–7

– 4

= –7

+ (–

4) =

–11

So

too,

7 –

(– 4

) = 7

+ (+

4) =

11

OR

–7

– (–

4) =

–7

+ (+

4) =

– 3

. Her

e th

e us

e of

br

acke

ts a

roun

d th

e in

tege

rs a

re u

sefu

l.

• A

usef

ul s

trate

gy is

to u

se re

peat

ed a

dditi

on a

nd n

umbe

r pat

tern

s to

sho

w

lear

ners

the

reas

onab

lene

ss o

f rul

es fo

r the

resu

ltant

sig

n fo

r mul

tiplic

atio

n w

ith

inte

gers

.

Tota

l tim

e fo

r In

tege

rs:

9 ho

urs


79CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.3

inte

gers

exam

ple:

a)

Rep

eate

d ad

ditio

n of

(–3)

: (–3

) + (–

3) +

(–3)

= –

9 =

3 x

(–3)

b)

Rep

eate

d ad

ditio

n of

(–2)

: (–2

) + (–

2) +

(–2)

+ (–

2) =

–8

= 4

x (–

2)

c)

Cou

ntin

g do

wn

in in

terv

als

of 3

from

9:

3 x

3 =

9

3 x

2 =

6

3 x

1 =

3

3 x

0 =

0

3 x

–1 =

–3

3 x

–2 =

?

3 x

–3 =

?

Hen

ce th

e ru

le: a

pos

itive

inte

ger x

a n

egat

ive

inte

ger =

a n

egat

ive

inte

ger

d)

Usi

ng th

e ru

le th

at a

pos

itive

inte

ger x

a n

egat

ive

inte

ger =

a n

egat

ive

inte

ger,

esta

blis

hed

from

exa

mpl

es a

bove

, the

follo

win

g pa

ttern

can

be

used

:

–1 x

3 =

–3

–1 x

2 =

–2

–1 x

1 =

–1

–1 x

0 =

0

–1 x

–1

= 1

–1 x

–2

= ?

–1 x

–3

= ?

Hen

ce th

e ru

le: a

neg

ativ

e in

tege

r x a

neg

ativ

e in

tege

r = a

pos

itive

inte

ger

• U

se th

e in

vers

e op

erat

ion

for m

ultip

licat

ion

and

divi

sion

to d

evel

op a

rule

for t

he

resu

ltant

sig

n fo

r div

isio

n w

ith in

tege

rs.

exam

ple:

a)

If 4

x (–

2) =

–8,

then

–8

÷ 4

= –2

and

–8

÷ (–

2) =

4

b)

If (–

1) x

(–3)

= 3

, the

n 3

÷ (–

1) =

–3

and

3 ÷

(–3)

= –

1

Hen

ce th

e ru

les:

div

isio

n of

a p

ositi

ve a

nd n

egat

ive

inte

ger e

qual

s a

nega

tive

inte

ger a

nd d

ivis

ion

of tw

o ne

gativ

e in

tege

rs e

qual

a p

ositi

ve in

tege

r.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.3

inte

gers

Prop

ertie

s of

inte

gers

• R

ecog

nize

and

use

com

mut

ativ

e,

asso

ciat

ive

and

dist

ribut

ive

prop

ertie

s of

add

ition

and

mul

tiplic

atio

n fo

r in

tege

rs

• R

ecog

nize

and

use

add

itive

and

m

ultip

licat

ive

inve

rses

for i

nteg

ers

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

mul

tiple

ope

ratio

ns w

ith in

tege

rs

• Fi

ndin

g th

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of in

tege

rs a

re a

lso

oppo

rtuni

ties

to c

heck

that

lear

ners

kno

w th

e ru

les

for r

esul

tant

sig

ns w

hen

mul

tiply

ing

inte

gers

.

• Th

eref

ore,

mak

e su

re th

at le

arne

rs u

nder

stan

d w

hy y

ou c

anno

t find

the

squa

re

root

of a

neg

ativ

e in

tege

r, an

d th

at th

e sq

uare

of a

neg

ativ

e in

tege

r is

alw

ays

posi

tive.

exam

ple:

a)

(–5)

2 = (–

5) x

(–5)

= 2

5

b)

(–4)

3 = (–

4) x

(–4)

x (–

4) =

–64

c)

3–2

7√

= –

3 be

caus

e –3

x –

3 x–

3 =

–27

Prop

ertie

s of

inte

gers

• Le

arne

rs s

houl

d in

vest

igat

e th

e pr

oper

ties

for o

pera

tions

with

who

le n

umbe

rs o

n th

e se

t of i

nteg

ers.

• Th

ese

prop

ertie

s sh

ould

ser

ve a

s m

otiv

atio

n fo

r the

ope

ratio

ns th

ey c

an p

erfo

rm

with

inte

gers

.

• Le

arne

rs s

houl

d se

e th

at th

e co

mm

utat

ive

prop

erty

for a

dditi

on a

nd m

ultip

licat

ion

hold

s fo

r int

eger

s, e

.g.

8 +

(–3)

= (–

3) +

8 =

5;

8 x

(–3)

= (–

3) x

8 =

–24

• Le

arne

rs s

houl

d se

e th

at th

ey c

an s

till u

se s

ubtra

ctio

n to

che

ck a

dditi

on o

r vic

e ve

rsa,

e.g

. if 8

+ (–

3) =

5, t

hen

5 –

8 =

–3 a

nd 5

–(–

3) =

8

• Le

arne

rs s

houl

d se

e th

at th

e as

soci

ativ

e pr

oper

ty fo

r add

ition

hol

ds fo

r int

eger

s,

e.g.

[(–6

) + 4

] + (–

1) =

(–6)

+ [4

+ (–

1)] =

–3

• Le

arne

rs s

houl

d se

e th

at th

e in

vers

e op

erat

ion

for m

ultip

licat

ion

and

divi

sion

ho

lds

for i

nteg

ers,

e.g

. if 5

x (–

6) =

–30

, the

n –3

0 ÷

5 =

–6 a

nd –

30 ÷

(–6)

= 5

• Le

arne

rs s

houl

d de

velo

p th

e ru

les,

thro

ugh

patte

rnin

g, fo

r res

ulta

nt s

igns

whe

n m

ultip

lyin

g an

d di

vidi

ng in

tege

rs:

(+5)

x (+

5) =

(+25

);

(–5)

x (–

5) =

(+25

);

(–5)

x (+

5) =

(–25

);

(+25

) ÷ (+

5) =

(+5)

;

(–25

) ÷ (–

5) =

(+5)

;

(–25

) ÷ (+

5) =

(–5)

;


81CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.2

expo

nent

sm

enta

l cal

cula

tions

• R

evis

e:

-S

quar

es to

at l

east

122 a

nd th

eir

squa

re ro

ots

-C

ubes

to a

t lea

st 6

3 and

thei

r cub

e ro

ots

Com

parin

g an

d re

pres

entin

g nu

mbe

rs in

exp

onen

tial f

orm

• R

evis

e co

mpa

re a

nd re

pres

ent w

hole

nu

mbe

rs in

exp

onen

tial f

orm

• C

ompa

re a

nd re

pres

ent i

nteg

ers

in

expo

nent

ial f

orm

• C

ompa

re a

nd re

pres

ent n

umbe

rs in

sc

ient

ific

nota

tion,

lim

ited

to p

ositi

ve

expo

nent

s

Wha

t is

diffe

rent

to G

rade

7?

• In

tege

rs a

nd ra

tiona

l num

bers

in e

xpon

entia

l for

m

• S

cien

tific

nota

tion

of n

umbe

rs

Com

parin

g an

d re

pres

entin

g nu

mbe

rs in

exp

onen

tial f

orm

• Le

arne

rs n

eed

to u

nder

stan

d th

at in

the

expo

nent

ial f

orm

ab ,

the

num

ber i

s re

ad

as ‘a

to th

e po

wer

of b

, whe

re a

is c

alle

d th

e ba

se a

nd b

is c

alle

d th

e ex

pone

nt o

r in

dex

and

b in

dica

tes

the

num

ber o

f fac

tors

that

are

mul

tiplie

d.

exam

ple:

a3 =

a x

a x

a;

a5 =

a x

a x

a x

a x

a

Lear

ners

can

repr

esen

t any

num

ber i

n ex

pone

ntia

l for

m, w

ithou

t nee

ding

to

com

pute

the

valu

e. e

xam

ple:

50

x 50

x 5

0 x

50 x

50

x 50

x 5

0 =

507

• m

ake

sure

lear

ners

und

erst

and

that

squ

are

root

s an

d cu

be ro

ots

are

the

inve

rse

oper

atio

ns o

f squ

arin

g an

d cu

bing

num

bers

. exa

mpl

e: 3

2 =

9 th

eref

ore

√9 =

3

• M

ake

sure

lear

ners

und

erst

and

that

any

num

ber r

aise

d to

the

pow

er 1

is e

qual

to

that

par

ticul

ar n

umbe

r

exam

ple:

m1 =

m

• U

sing

pat

tern

s an

d th

eir k

now

ledg

e of

mul

tiplic

atio

n w

ith in

tege

rs, l

earn

ers

shou

ld a

ntic

ipat

e th

e re

sulta

nt s

ign

of a

n in

tege

r rai

sed

to a

n od

d or

eve

n po

wer

e.

g. (–

15)4 w

ill b

e po

sitiv

e, w

hile

(–15

)3 will

be

nega

tive

• To

avo

id c

omm

on m

isco

ncep

tions

, em

phas

ize

the

follo

win

g

-12

2 = 1

2 x

12 a

nd n

ot 1

2 x

2

-13 m

eans

1 x

1 x

1 a

nd n

ot 1

x 3

-10

01 =

100

-81

√ =

9 b

ecau

se 9

2 =

81

-3

27√

= 3

bec

ause

33 =

27

-th

e sq

uare

of 9

= 8

1, w

here

as th

e sq

uare

root

of 9

= 3

• Le

arne

rs s

houl

d us

e th

eir k

now

ledg

e of

repr

esen

ting

num

bers

in e

xpon

entia

l fo

rm w

hen

sim

plify

ing

and

expa

ndin

g al

gebr

aic

expr

essi

ons

and

solv

ing

alge

brai

c eq

uatio

ns.

Tota

l tim

e fo

r ex

pone

nts:

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.2

expo

nent

sC

alcu

latio

ns u

sing

num

bers

in

expo

nent

ial f

orm

• E

stab

lish

gene

ral l

aws

of e

xpon

ents

, lim

ited

to:

-na

tura

l num

ber e

xpon

ents

♦am

x a

n = a

m+n

♦am

÷ a

n = a

m–n

, if m

> n

♦(a

m)n =

am

x n

♦(a

x t)

n = a

n x

tn

♦a0 =

1

• R

ecog

nize

and

use

the

appr

opria

te

law

s of

ope

ratio

ns u

sing

num

bers

in

volv

ing

expo

nent

s an

d sq

uare

and

cu

be ro

ots

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all f

our

oper

atio

ns w

ith n

umbe

rs th

at in

volv

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of in

tege

rs

• C

alcu

late

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

ratio

nal

num

bers

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

num

bers

in e

xpon

entia

l for

m

law

s of

exp

onen

ts

• Th

e la

ws

of e

xpon

ents

sho

uld

be in

trodu

ced

thro

ugh

a ra

nge

of n

umer

ic

exam

ples

firs

t, th

en v

aria

bles

can

be

used

. In

othe

r wor

ds, t

he n

umbe

rs a

re

repl

aced

with

lette

rs, b

ut th

e ru

les

wor

k th

e sa

me.

• Th

e fo

llow

ing

law

s of

exp

onen

ts s

houl

d be

intro

duce

d, w

here

m a

nd n

are

nat

ural

nu

mbe

rs a

nd a

and

t ar

e no

t equ

al to

0:

am x

an =

am

+n

exam

ple

a)

23 x 2

4 = 2

3+4 =

27

b)

x3 x x

4 = x

3+4 =

x7

am ÷

an =

am

–n if

m >

n

exam

ple:

a)

35 ÷ 3

2 = 3

3 = 2

7

b)

x5 ÷ x

3 = x

2

(am)n =

am

n

exam

ple:

a)

(23 )2 =

26 =

64

b)

(x3 )2 =

x6

(a x

t)n =

an x

tn

exam

ple:

(3x2 )3 =

33 . x

6 = 2

7x6

a0 = 1

exam

ples

: (37

)0 = 1

; (4x

2 )0 = 1

;

• m

ake

sure

lear

ners

und

erst

and

thes

e la

ws

read

ing

from

bot

h si

des

of th

e eq

ual

sign

i.e.

if th

e LH

S =

RH

S, t

hen

the

RH

S =

LH

S.

• Th

e la

w a

0 = 1

can

be

deriv

ed b

y us

ing

the

law

of e

xpon

ents

for d

ivis

ion

in a

few

ex

ampl

es. a

4 ÷

a4 =

a x

a x

a x

a a

x a

x a

x a =

1 th

eref

ore

a4–4 = a0 =

1

• Le

arne

rs s

houl

d be

abl

e to

use

the

law

s of

exp

onen

ts in

cal

cula

tions

and

fo

r sol

ving

sim

ple

expo

nent

ial e

quat

ions

as

wel

l as

expa

ndin

g or

sim

plify

ing

alge

brai

c ex

pres

sion

s.


83CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.2

expo

nent

s•

Look

out

for t

he fo

llow

ing

com

mon

mis

conc

eptio

ns w

here

:

-le

arne

rs m

ultip

ly u

nlik

e ba

ses

and

add

the

expo

nent

.

exam

ple:

xm x

yn =

(xy)

m+n

-le

arne

rs m

ultip

ly li

ke b

ases

and

add

the

expo

nent

s

exam

ple:

25 x 2

7 = 4

12 in

stea

d of

the

corr

ect a

nsw

er 2

12.

-le

arne

rs f

orge

t, fo

r ex

ampl

e, t

hat

whe

n sq

uarin

g a

bino

mia

l the

re is

a m

iddl

e te

rm

exam

ple:

(x +

y)m

= x

m +

ym

-le

arne

rs c

onfu

se a

ddin

g th

e ex

pone

nts

and

addi

ng th

e te

rms

exam

ple:

xm +

xn =

xm

+n o

r xm

n

Cal

cula

tions

usi

ng n

umbe

rs in

exp

onen

tial f

orm

• K

now

ing

the

rule

s of

ope

ratio

ns fo

r cal

cula

tions

invo

lvin

g ex

pone

nts

are

impo

rtant

, e.g

.

a)

(7 –

4)3

= 33 a

nd n

ot

73 –

43

b)

16 +

9

√ =

25

√ a

nd n

ot

the

16√

+

9√

• Le

arne

rs c

an a

lso

do s

impl

e ca

lcul

atio

ns w

here

the

num

erat

or a

nd d

enom

inat

or

of a

frac

tion

are

writ

ten

in e

xpon

entia

l for

m, e

.g. 23 22

= 2

x 2

x 2

2 x

2 =

8 4 =

2

• Le

arne

rs c

an a

lso

find

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

dec

imal

an

d co

mm

on fr

actio

ns b

y in

spec

tion.

exam

ples

a)

(0,7

)2 = 0

,49

b)

(0,1

)3 = 0

,001

c)

0,09

√

= 0

,3

d)

(3 4)2 =

32 42 =

9 16

e) √

9 25 =

9

√ 25√

= 3 5



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.2

expo

nent

sSc

ientificno

tatio

n

• W

hen

writ

ing

num

bers

in s

cien

tific

nota

tion,

lear

ners

hav

e to

und

erst

and

the

rela

tions

hip

betw

een

the

num

ber o

f dec

imal

pla

ces

and

the

inde

x of

10

exam

ples

:

a)

25 =

2,5

x 1

01

b)

250

= 2,

5 x

102

c)

2 50

0 =

2,5

x 10

3

• S

cien

tific

nota

tion

limite

d to

pos

itive

exp

onen

ts, i

nclu

des

writ

ing

very

larg

e nu

mbe

rs in

sci

entifi

c no

tatio

n.

exam

ple:

25

mill

ion

= 2,

5 x

107

• Le

arne

rs p

ract

ise

writ

ing

larg

e nu

mbe

rs in

sci

entifi

c no

tatio

n, th

ey w

ill re

aliz

e th

ey h

ave

enco

unte

red

thes

e in

Nat

ural

Sci

ence

. It i

s us

eful

to re

fer t

o th

ese

cont

exts

whe

n ta

lkin

g ab

out s

cien

tific

nota

tion.


85CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

an

d ge

omet

ric p

atte

rns

look

ing

for

rela

tions

hips

bet

wee

n nu

mbe

rs,

incl

udin

g pa

ttern

s:

-re

pres

ente

d in

phy

sica

l or

diag

ram

fo

rm

-no

t lim

ited

to s

eque

nces

invo

lvin

g a

cons

tant

diff

eren

ce o

r rat

io

-of

lear

ner’s

ow

n cr

eatio

n

-re

pres

ente

d in

tabl

es

-re

pres

ente

d al

gebr

aica

lly

• D

escr

ibe

and

just

ify th

e ge

nera

l rul

es

for o

bser

ved

rela

tions

hips

bet

wee

n nu

mbe

rs in

ow

n w

ords

or i

n al

gebr

aic

lang

uage

Wha

t is

diffe

rent

to G

rade

7?

• Th

e ra

nge

of n

umbe

r pat

tern

s ar

e ex

tend

ed to

incl

ude

patte

rns

with

mul

tiplic

atio

n an

d di

visi

on w

ith in

tege

rs, n

umbe

rs in

exp

onen

tial f

orm

• A

s le

arne

rs b

ecom

e us

ed to

des

crib

ing

patte

rns

in th

eir o

wn

wor

ds, t

heir

desc

riptio

ns s

houl

d be

com

e m

ore

prec

ise

and

effic

ient

with

the

use

of a

lgeb

raic

la

ngua

ge to

des

crib

e ge

nera

l rul

es o

f pat

tern

s

• It

is u

sefu

l als

o to

intro

duce

the

lang

uage

of ‘

term

in a

seq

uenc

e’ in

ord

er to

di

stin

guis

h th

e te

rm fr

om th

e po

sitio

n of

a te

rm in

a s

eque

nce

• In

vest

igat

ing

num

ber p

atte

rns

is a

n op

portu

nity

to g

ener

aliz

e –

to g

ive

gene

ral

alge

brai

c de

scrip

tions

of t

he re

latio

nshi

p be

twee

n te

rms

and

thei

r pos

ition

in a

se

quen

ce a

nd to

just

ify s

olut

ions

.

Kin

ds o

f num

eric

pat

tern

s

• G

iven

a s

eque

nce

of n

umbe

rs, l

earn

ers

have

to id

entif

y a

patte

rn o

r rel

atio

nshi

p be

twee

n co

nsec

utiv

e te

rms

in o

rder

to e

xten

d th

e pa

ttern

.

exam

ples

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

the

sequ

ence

s be

low

. Use

this

rule

to p

rovi

de th

e ne

xt th

ree

num

bers

in th

e se

quen

ce:

a)

–3; –

7; –

11; –

15; .

.. ...

Her

e le

arne

rs s

houl

d id

entif

y th

e co

nsta

nt d

iffer

ence

bet

wee

n co

nsec

utiv

e te

rms

in o

rder

to e

xten

d th

e pa

ttern

. Thi

s pa

ttern

can

be

desc

ribed

in le

arne

rs’

own

wor

ds a

s ‘a

ddin

g –4

’ or ‘

coun

ting

in –

4s’ o

r ‘ad

d –4

to th

e pr

evio

us

num

ber i

n th

e pa

ttern

’.

b)

2; –

4; 8

; –16

; 32

... ..

.

Her

e le

arne

rs s

houl

d id

entif

y th

e co

nsta

nt ra

tio b

etw

een

cons

ecut

ive

term

s.

This

pat

tern

can

be

desc

ribed

in le

arne

rs’ o

wn

wor

ds a

s ‘m

ultip

ly th

e pr

evio

us

num

ber b

y –2

’.

c)

1; 2

; 4; 7

; 11;

16

... ..

.

This

pat

tern

has

nei

ther

a c

onst

ant d

iffer

ence

nor

con

stan

t rat

io. T

his

patte

rn

can

be d

escr

ibed

in le

arne

rs’ o

wn

wor

ds a

s ‘in

crea

se th

e di

ffere

nce

betw

een

cons

ecut

ive

term

s by

1 e

ach

time’

or ‘

add

1 m

ore

than

was

add

ed to

get

the

prev

ious

term

’. U

sing

this

rule

, the

nex

t 3 te

rms

will

be

22; 2

9; 3

7.

• G

iven

a s

eque

nce

of n

umbe

rs, l

earn

ers

have

to id

entif

y a

patte

rn o

r rel

atio

nshi

p be

twee

n th

e te

rm a

nd it

s po

sitio

n in

the

sequ

ence

. Thi

s en

able

s le

arne

rs to

pr

edic

t a te

rm in

a s

eque

nce

base

d on

the

posi

tion

of th

at te

rm in

the

sequ

ence

. It

is u

sefu

l for

lear

ners

to re

pres

ent t

hese

seq

uenc

es in

tabl

es s

o th

at th

ey c

an

cons

ider

the

posi

tion

of th

e te

rm.

Tota

l tim

e fo

r N

umer

ic a

nd

geom

etric

pa

ttern

s:

4,5

hour

s



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

exam

ples

a)

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

this

se

quen

ce: 1

; 8; 2

7; 6

4; ..

. .. U

se th

is ru

le to

find

the

10th

term

in th

is s

eque

nce.

Firs

tly,

lear

ners

hav

e to

und

erst

and

that

the

‘10

th t

erm

’ ref

ers

to p

ositi

on i

n th

e nu

mbe

r se

quen

ce.

They

hav

e to

find

a r

ule

in o

rder

to

dete

rmin

e th

e 10

th t

erm

, ra

ther

than

con

tinui

ng th

e se

quen

ce to

the

tent

h te

rm.

exam

ple

b)

This

seq

uenc

e ca

n be

re-p

rese

nted

in th

e fo

llow

ing

tabl

e:

Pos

ition

in s

eque

nce

12

34

10

Term

18

2764

?

Lear

ners

hav

e to

reco

gniz

e th

at e

ach

term

in th

e bo

ttom

row

is o

btai

ned

by c

ubin

g th

e po

sitio

n nu

mbe

r in

the

top

row

. Thu

s th

e 10

th te

rm w

ill b

e 10

cub

ed o

r 103 ,

whi

ch is

. Usi

ng th

e sa

me

rule

, lea

rner

s ca

n al

so b

e as

ked

wha

t ter

m n

umbe

r or

posi

tion

will

512

be?

If th

e te

rm is

obt

aine

d by

cub

ing

the

posi

tion

num

ber o

f the

te

rm, t

hen

the

posi

tion

num

ber c

an b

e ob

tain

ed b

y fin

ding

the

cube

root

of t

he

term

. Hen

ce, 5

12 w

ill b

e th

e 8th

term

in th

e se

quen

ce s

ince

351

2 √

= 8

c)

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

this

se

quen

ce: 4

; 7; 1

0; 1

3; ..

. Use

this

rule

to fi

nd th

e 20

th te

rm in

this

seq

uenc

e.

If le

arne

rs c

onsi

der o

nly

the

rela

tions

hip

betw

een

cons

ecut

ive

term

s, th

en th

ey

have

to c

ontin

ue th

e pa

ttern

(‘ad

d 3

to p

revi

ous

num

ber’)

to th

e 20

th te

rm to

fin

d th

e an

swer

. How

ever

, if t

hey

look

for a

rela

tions

hip

or ru

le b

etw

een

the

term

an

d th

e po

sitio

n of

the

term

, the

y ca

n pr

edic

t the

ans

wer

with

out c

ontin

uing

the

patte

rn. U

sing

num

ber s

ente

nces

can

be

usef

ul to

find

the

rule

:

1st t

erm

: 4 =

3(1

) + 1

2nd

term

: 7 =

3(2

) + 1

3rd

term

: 10

= 3(

3) +

1

4th

term

: 13

= 3(

4) +

1

The

num

ber i

n th

e br

acke

ts c

orre

spon

ds to

the

posi

tion

of th

e te

rm. H

ence

, the

20

th te

rm w

ill b

e: 3

(20)

+ 1

= 6

1

The

rule

in le

arne

rs’ o

wn

wor

ds c

an b

e w

ritte

n as

‘3 ti

mes

the

posi

tion

of th

e te

rm

+1’ o

r 3n

+ 1

whe

re n

is th

e po

sitio

n of

the

term

.

• Th

ese

type

s of

num

eric

pat

tern

s de

velo

p an

und

erst

andi

ng o

f fun

ctio

nal

rela

tions

hips

, in

whi

ch y

ou h

ave

a de

pend

ent v

aria

ble

(pos

ition

of t

he te

rm) a

nd

inde

pend

ent v

aria

ble

(the

term

itse

lf), a

nd w

here

you

hav

e a

uniq

ue o

utpu

t for

an

y gi

ven

inpu

t val

ue.


87CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

Kin

ds o

f geo

met

ric p

atte

rns

• G

eom

etric

pat

tern

s ar

e nu

mbe

r pat

tern

s re

pres

ente

d di

agra

mm

atic

ally.

The

di

agra

mm

atic

repr

esen

tatio

n re

veal

s th

e st

ruct

ure

of th

e nu

mbe

r pat

tern

.

• H

ence

, rep

rese

ntin

g th

e nu

mbe

r pat

tern

s in

tabl

es m

akes

it e

asie

r for

lear

ners

to

desc

ribe

the

gene

ral r

ule

for t

he p

atte

rn.

exam

ple

Con

side

r thi

s pa

ttern

for b

uild

ing

hexa

gons

with

mat

chst

icks

. How

man

y m

atch

stic

ks w

ill b

e us

ed to

bui

ld th

e 10

th h

exag

on?

The

rule

for t

he p

atte

rn is

con

tain

ed in

the

stru

ctur

e (c

onst

ruct

ion)

of t

he

succ

essi

ve h

exag

onal

sha

pes:

(1) a

dd 1

mat

chst

ick

per s

ide

(2) t

here

are

6 s

ides

(3) a

dd 6

mat

chst

icks

per

hex

agon

as

you

proc

eed

from

a g

iven

hex

agon

to th

e ne

xt o

ne.

So,

for t

he 2

nd h

exag

on, y

ou h

ave

2 x

6 m

atch

es; f

or th

e 3r

d he

xago

n yo

u ha

ve

3 x

6 m

atch

es: U

sing

this

pat

tern

for b

uild

ing

hexa

gons

, the

10t

h he

xago

n w

ill h

ave

10 x

6 m

atch

es.

Lear

ners

can

als

o us

e a

tabl

e to

reco

rd th

e nu

mbe

r of m

atch

es u

sed

for e

ach

hexa

gon.

Thi

s w

ay th

ey c

an lo

ok a

t the

num

ber p

atte

rn re

late

d to

the

num

ber o

f m

atch

es u

sed

for e

ach

new

hex

agon

.

Posi

tion

of h

exag

on in

pat

tern

12

34

56

10

num

ber o

f mat

ches

612

18

des

crib

ing

patte

rns

• It

does

not

mat

ter i

f lea

rner

s ar

e al

read

y fa

mili

ar w

ith a

par

ticul

ar p

atte

rn. T

heir

desc

riptio

ns o

f the

sam

e pa

ttern

can

be

diffe

rent

whe

n th

ey e

ncou

nter

it a

t di

ffere

nt s

tage

s of

thei

r mat

hem

atic

al d

evel

opm

ent.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

exam

ple

The

rule

for t

he s

eque

nce:

can

be

desc

ribed

in th

e fo

llow

ing

way

s:a)

ad

d 3

to th

e pr

evio

us te

rm

b)

3 x

the

posi

tion

of th

e te

rm +

1c)

3(

n) +

1, w

here

n is

the

posi

tion

of th

e te

rmd)

3(

n) +

1, w

here

n w

here

is a

nat

ural

num

ber

2.2

Func

tions

an

d re

latio

nshi

ps

inpu

t and

out

put v

alue

s•

Det

erm

ine

inpu

t val

ues,

out

put v

alue

s or

rule

s fo

r pat

tern

s an

d re

latio

nshi

ps

usin

g:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae -

equa

tions

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

eq

uiva

lenc

e of

diff

eren

t des

crip

tions

of

the

sam

e re

latio

nshi

p or

rule

pr

esen

ted:

-ve

rbal

ly -

in fl

ow d

iagr

ams

-in

tabl

es -

by fo

rmul

ae -

by e

quat

ions

Wha

t is

diffe

rent

to G

rade

7?

• Fi

ndin

g in

put o

r out

put v

alue

s us

ing

give

n eq

uatio

ns•

The

rule

s an

d nu

mbe

r pat

tern

s fo

r whi

ch le

arne

rs h

ave

to fi

nd in

put a

nd o

utpu

t va

lues

are

ext

ende

d to

incl

ude

patte

rns

with

mul

tiplic

atio

n an

d di

visi

on o

f in

tege

rs a

nd n

umbe

rs in

exp

onen

tial f

orm

• In

this

pha

se, i

t is

usef

ul to

beg

in to

spe

cify

whe

ther

the

inpu

t val

ues

are

natu

ral

num

bers

, or i

nteg

ers

or ra

tiona

l num

bers

. Thi

s bu

ilds

lear

ners

’ aw

aren

ess

of th

e do

mai

n of

inpu

t val

ues.

Hen

ce, t

o fin

d ou

tput

val

ues,

lear

ners

sho

uld

be g

iven

th

e ru

le/fo

rmul

a as

wel

l as

the

dom

ain

of th

e in

put v

alue

s.Fu

nctio

ns a

nd re

latio

nshi

ps w

ill b

e do

ne a

gain

in t

erm

4. i

n te

rm 1

the

focu

s of

Fun

ctio

ns a

nd re

latio

nshi

ps is

on

prac

tisin

g op

erat

ions

with

inte

gers

, or

includ

ingintegersinth

erulesforfi

ndingou

tputvalues.

Not

e th

at w

hen

lear

ners

find

inpu

t or o

utpu

t val

ues

for g

iven

rule

s, th

ey a

re a

ctua

lly

findi

ng th

e nu

mer

ical

val

ue o

f alg

ebra

ic e

xpre

ssio

ns u

sing

sub

stitu

tion.

Flow

dia

gram

s ar

e re

pres

enta

tions

of f

unct

iona

l rel

atio

nshi

ps. H

ence

, whe

n us

ing

flow

dia

gram

s, th

e co

rres

pond

ence

bet

wee

n in

put a

nd o

utpu

t val

ues

shou

ld b

e cl

ear i

n its

repr

esen

tatio

nal f

orm

i.e.

the

first

inpu

t pro

duce

s th

e fir

st o

utpu

t, th

e se

cond

inpu

t pro

duce

s th

e se

cond

out

put,

etc.

exam

ples

a)

Use

the

give

n ru

le to

cal

cula

te th

e va

lues

of t

for e

ach

valu

e of

p, w

here

p is

a

natu

ral n

umbe

r.

t

p

Rule

0 -3

-5

-7

-9

t =

p x

3 +

1

In th

is k

ind

of fl

ow d

iagr

am, l

earn

ers

can

also

be

aske

d to

find

the

valu

e of

p fo

r a

give

n t v

alue

.

Tim

e fo

r Fu

nctio

ns

and

Rel

a-tio

nshi

ps in

th

is te

rm:

3 ho

urs


89CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

exam

ple

The

rule

for t

he s

eque

nce:

can

be

desc

ribed

in th

e fo

llow

ing

way

s:a)

ad

d 3

to th

e pr

evio

us te

rm

b)

3 x

the

posi

tion

of th

e te

rm +

1c)

3(

n) +

1, w

here

n is

the

posi

tion

of th

e te

rmd)

3(

n) +

1, w

here

n w

here

is a

nat

ural

num

ber

2.2

Func

tions

an

d re

latio

nshi

ps

inpu

t and

out

put v

alue

s•

Det

erm

ine

inpu

t val

ues,

out

put v

alue

s or

rule

s fo

r pat

tern

s an

d re

latio

nshi

ps

usin

g:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae -

equa

tions

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

eq

uiva

lenc

e of

diff

eren

t des

crip

tions

of

the

sam

e re

latio

nshi

p or

rule

pr

esen

ted:

-ve

rbal

ly -

in fl

ow d

iagr

ams

-in

tabl

es -

by fo

rmul

ae -

by e

quat

ions

Wha

t is

diffe

rent

to G

rade

7?

• Fi

ndin

g in

put o

r out

put v

alue

s us

ing

give

n eq

uatio

ns•

The

rule

s an

d nu

mbe

r pat

tern

s fo

r whi

ch le

arne

rs h

ave

to fi

nd in

put a

nd o

utpu

t va

lues

are

ext

ende

d to

incl

ude

patte

rns

with

mul

tiplic

atio

n an

d di

visi

on o

f in

tege

rs a

nd n

umbe

rs in

exp

onen

tial f

orm

• In

this

pha

se, i

t is

usef

ul to

beg

in to

spe

cify

whe

ther

the

inpu

t val

ues

are

natu

ral

num

bers

, or i

nteg

ers

or ra

tiona

l num

bers

. Thi

s bu

ilds

lear

ners

’ aw

aren

ess

of th

e do

mai

n of

inpu

t val

ues.

Hen

ce, t

o fin

d ou

tput

val

ues,

lear

ners

sho

uld

be g

iven

th

e ru

le/fo

rmul

a as

wel

l as

the

dom

ain

of th

e in

put v

alue

s.Fu

nctio

ns a

nd re

latio

nshi

ps w

ill b

e do

ne a

gain

in t

erm

4. i

n te

rm 1

the

focu

s of

Fun

ctio

ns a

nd re

latio

nshi

ps is

on

prac

tisin

g op

erat

ions

with

inte

gers

, or

includ

ingintegersinth

erulesforfi

ndingou

tputvalues.

Not

e th

at w

hen

lear

ners

find

inpu

t or o

utpu

t val

ues

for g

iven

rule

s, th

ey a

re a

ctua

lly

findi

ng th

e nu

mer

ical

val

ue o

f alg

ebra

ic e

xpre

ssio

ns u

sing

sub

stitu

tion.

Flow

dia

gram

s ar

e re

pres

enta

tions

of f

unct

iona

l rel

atio

nshi

ps. H

ence

, whe

n us

ing

flow

dia

gram

s, th

e co

rres

pond

ence

bet

wee

n in

put a

nd o

utpu

t val

ues

shou

ld b

e cl

ear i

n its

repr

esen

tatio

nal f

orm

i.e.

the

first

inpu

t pro

duce

s th

e fir

st o

utpu

t, th

e se

cond

inpu

t pro

duce

s th

e se

cond

out

put,

etc.

exam

ples

a)

Use

the

give

n ru

le to

cal

cula

te th

e va

lues

of t

for e

ach

valu

e of

p, w

here

p is

a

natu

ral n

umbe

r.

t

p

Rule

0 -3

-5

-7

-9

t =

p x

3 +

1

In th

is k

ind

of fl

ow d

iagr

am, l

earn

ers

can

also

be

aske

d to

find

the

valu

e of

p fo

r a

give

n t v

alue

.

Tim

e fo

r Fu

nctio

ns

and

Rel

a-tio

nshi

ps in

th

is te

rm:

3 ho

urs

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.2

Func

tions

an

d re

latio

nshi

ps

b)

Find

the

rule

for c

alcu

latin

g th

e ou

tput

val

ue fo

r eve

ry g

iven

inpu

t val

ue in

the

flow

dia

gram

bel

ow.

t

p

Rule

0 3 5 7 9

-3

-9

-21

-27

-15

In fl

ow d

iagr

ams

such

as

thes

e, m

ore

than

one

rule

mig

ht b

e po

ssib

le to

des

crib

e th

e re

latio

nshi

p be

twee

n in

put a

nd o

utpu

t val

ues.

The

rule

s ar

e ac

cept

able

if th

ey m

atch

th

e gi

ven

inpu

t val

ues

to th

e co

rres

pond

ing

outp

ut v

alue

s.

c)

If th

e ru

le fo

r find

ing

y in

the

tabl

e be

low

is: y

= –

3x –

1, fi

nd y

for t

he g

iven

x

valu

es:

x0

12

510

5010

0

y

b)

Des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

the

top

row

and

bot

tom

row

in

the

tabl

e. T

hen

writ

e do

wn

the

valu

e of

and

x–2

–10

12

12n

y–5

–4–3

–2–1

m34

In ta

bles

suc

h as

thes

e, m

ore

than

one

rule

mig

ht b

e po

ssib

le to

des

crib

e th

e re

latio

nshi

p be

twee

n x

and

y va

lues

. The

rule

s ar

e ac

cept

able

if th

ey m

atch

the

give

n in

put v

alue

s to

the

corr

espo

ndin

g ou

tput

val

ues.

For

exa

mpl

e, th

e ru

le

y =

x –3

des

crib

es th

e re

latio

nshi

p be

twee

n th

e x

valu

es a

nd g

iven

y v

alue

s. T

o fin

d m

and

n, l

earn

ers

have

to s

ubst

itute

the

corr

espo

ndin

g va

lues

for x

or y

in th

e ru

le

and

solv

e th

e eq

uatio

n by

insp

ectio

n.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.3

alg

ebra

icex

pres

sion

s

alg

ebra

ic la

ngua

ge

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-re

cogn

ize

and

inte

rpre

t rul

es

or re

latio

nshi

ps re

pres

ente

d in

sy

mbo

lic fo

rm

-id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae a

nd e

quat

ions

• R

ecog

nize

and

iden

tify

conv

entio

ns

for w

ritin

g al

gebr

aic

expr

essi

ons

• Id

entif

y an

d cl

assi

fy li

ke a

nd u

nlik

e te

rms

in a

lgeb

raic

exp

ress

ions

• R

ecog

nize

and

iden

tify

coef

ficie

nts

and

expo

nent

s in

alg

ebra

ic

expr

essi

ons

expa

nd a

nd s

impl

ify a

lgeb

raic

ex

pres

sion

s

Use

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e la

ws

for r

atio

nal n

umbe

rs

and

law

s of

exp

onen

ts to

:

• A

dd a

nd s

ubtra

ct li

ke te

rms

in

alge

brai

c ex

pres

sion

s

• D

eter

min

e th

e sq

uare

s, c

ubes

, sq

uare

root

s an

d cu

be ro

ots

of s

ingl

e al

gebr

aic

term

s or

like

alg

ebra

ic te

rms

• D

eter

min

e th

e nu

mer

ical

val

ue o

f al

gebr

aic

expr

essi

ons

by s

ubst

itutio

n

Wha

t is

diffe

rent

to G

rade

7?

• In

trodu

ctio

n to

con

vent

ions

of a

lgeb

raic

lang

uage

• M

anip

ulat

ing

alge

brai

c ex

pres

sion

s

alg

ebra

ic e

xpre

ssio

ns a

re d

one

agai

n in

ter

m 2

, whe

re th

e fo

cus

is m

ore

fully

on

man

ipul

atin

g al

gebr

aic

expr

essi

ons.

in th

is te

rm th

e fo

cus

is o

n in

terp

retin

g al

gebr

aic

expr

essi

ons

and

intr

oduc

ing

conv

entio

ns o

f alg

ebra

ic

lang

uage

thro

ugh

addi

ng a

nd s

ubtr

actin

g lik

e te

rms.

Lear

ners

hav

e op

portu

nitie

s to

writ

e an

d in

terp

ret a

lgeb

raic

exp

ress

ions

whe

n th

ey

writ

e ge

nera

l rul

es to

des

crib

e re

latio

nshi

ps b

etw

een

num

bers

in n

umbe

r pat

tern

s,

and

whe

n th

ey fi

nd in

put o

r out

put v

alue

s fo

r giv

en ru

les

in fl

ow d

iagr

ams,

tabl

es,

form

ulae

and

equ

atio

ns.

exam

ples

of i

nter

pret

ing

alge

brai

c ex

pres

sion

s

a)

Wha

t doe

s th

e ru

le 2

n mea

n fo

r the

follo

win

g nu

mbe

r seq

uenc

e:

2; 4

; 8; 1

6; 3

2...

Her

e le

arne

rs s

houl

d re

cogn

ize

that

2n re

pres

ents

the

gene

ral t

erm

in th

is

sequ

ence

, whe

re n

repr

esen

ts th

e po

sitio

n of

the

term

in th

e se

quen

ce. T

hus,

it

is th

e ru

le th

at c

an b

e us

ed to

find

any

term

in th

e gi

ven

sequ

ence

.

b)

The

rela

tions

hip

betw

een

a bo

y’s

age

(x yr

s ol

d) a

nd h

is m

othe

r’s a

ge is

giv

en

as 2

5 +

x. H

ow c

an th

is re

latio

nshi

p be

use

d to

find

the

mot

her’s

age

whe

n th

e bo

y is

11

year

s ol

d? H

ere

lear

ners

sho

uld

reco

gniz

e th

at to

find

the

mot

her’s

ag

e, th

ey m

ust s

ubst

itute

the

boy’

s gi

ven

age

into

the

rule

25

+ x.

The

y sh

ould

al

so re

cogn

ize

that

the

give

n ru

le m

eans

the

mot

her i

s 25

yea

rs o

lder

than

the

boy.

See

furth

er e

xam

ples

giv

en fo

r fun

ctio

ns a

nd re

latio

nshi

ps, a

s w

ell a

s no

tes

in

Term

2.

Tim

e fo

r al

gebr

aic

expr

essi

ons

in th

is te

rm:

4,5

hour

s


91CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.3

alg

ebra

icex

pres

sion

s

alg

ebra

ic la

ngua

ge

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-re

cogn

ize

and

inte

rpre

t rul

es

or re

latio

nshi

ps re

pres

ente

d in

sy

mbo

lic fo

rm

-id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae a

nd e

quat

ions

• R

ecog

nize

and

iden

tify

conv

entio

ns

for w

ritin

g al

gebr

aic

expr

essi

ons

• Id

entif

y an

d cl

assi

fy li

ke a

nd u

nlik

e te

rms

in a

lgeb

raic

exp

ress

ions

• R

ecog

nize

and

iden

tify

coef

ficie

nts

and

expo

nent

s in

alg

ebra

ic

expr

essi

ons

expa

nd a

nd s

impl

ify a

lgeb

raic

ex

pres

sion

s

Use

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e la

ws

for r

atio

nal n

umbe

rs

and

law

s of

exp

onen

ts to

:

• A

dd a

nd s

ubtra

ct li

ke te

rms

in

alge

brai

c ex

pres

sion

s

• D

eter

min

e th

e sq

uare

s, c

ubes

, sq

uare

root

s an

d cu

be ro

ots

of s

ingl

e al

gebr

aic

term

s or

like

alg

ebra

ic te

rms

• D

eter

min

e th

e nu

mer

ical

val

ue o

f al

gebr

aic

expr

essi

ons

by s

ubst

itutio

n

Wha

t is

diffe

rent

to G

rade

7?

• In

trodu

ctio

n to

con

vent

ions

of a

lgeb

raic

lang

uage

• M

anip

ulat

ing

alge

brai

c ex

pres

sion

s

alg

ebra

ic e

xpre

ssio

ns a

re d

one

agai

n in

ter

m 2

, whe

re th

e fo

cus

is m

ore

fully

on

man

ipul

atin

g al

gebr

aic

expr

essi

ons.

in th

is te

rm th

e fo

cus

is o

n in

terp

retin

g al

gebr

aic

expr

essi

ons

and

intr

oduc

ing

conv

entio

ns o

f alg

ebra

ic

lang

uage

thro

ugh

addi

ng a

nd s

ubtr

actin

g lik

e te

rms.

Lear

ners

hav

e op

portu

nitie

s to

writ

e an

d in

terp

ret a

lgeb

raic

exp

ress

ions

whe

n th

ey

writ

e ge

nera

l rul

es to

des

crib

e re

latio

nshi

ps b

etw

een

num

bers

in n

umbe

r pat

tern

s,

and

whe

n th

ey fi

nd in

put o

r out

put v

alue

s fo

r giv

en ru

les

in fl

ow d

iagr

ams,

tabl

es,

form

ulae

and

equ

atio

ns.

exam

ples

of i

nter

pret

ing

alge

brai

c ex

pres

sion

s

a)

Wha

t doe

s th

e ru

le 2

n mea

n fo

r the

follo

win

g nu

mbe

r seq

uenc

e:

2; 4

; 8; 1

6; 3

2...

Her

e le

arne

rs s

houl

d re

cogn

ize

that

2n re

pres

ents

the

gene

ral t

erm

in th

is

sequ

ence

, whe

re n

repr

esen

ts th

e po

sitio

n of

the

term

in th

e se

quen

ce. T

hus,

it

is th

e ru

le th

at c

an b

e us

ed to

find

any

term

in th

e gi

ven

sequ

ence

.

b)

The

rela

tions

hip

betw

een

a bo

y’s

age

(x yr

s ol

d) a

nd h

is m

othe

r’s a

ge is

giv

en

as 2

5 +

x. H

ow c

an th

is re

latio

nshi

p be

use

d to

find

the

mot

her’s

age

whe

n th

e bo

y is

11

year

s ol

d? H

ere

lear

ners

sho

uld

reco

gniz

e th

at to

find

the

mot

her’s

ag

e, th

ey m

ust s

ubst

itute

the

boy’

s gi

ven

age

into

the

rule

25

+ x.

The

y sh

ould

al

so re

cogn

ize

that

the

give

n ru

le m

eans

the

mot

her i

s 25

yea

rs o

lder

than

the

boy.

See

furth

er e

xam

ples

giv

en fo

r fun

ctio

ns a

nd re

latio

nshi

ps, a

s w

ell a

s no

tes

in

Term

2.

Tim

e fo

r al

gebr

aic

expr

essi

ons

in th

is te

rm:

4,5

hour

s

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.4

alg

ebra

iceq

uatio

ns

equa

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-se

t up

equa

tions

to d

escr

ibe

prob

lem

situ

atio

ns

-an

alys

e an

d in

terp

ret e

quat

ions

that

de

scrib

e a

give

n si

tuat

ion

-so

lve

equa

tions

by

insp

ectio

n

-de

term

ine

the

num

eric

al v

alue

of a

n ex

pres

sion

by

subs

titut

ion.

-id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae o

r equ

atio

ns

• E

xten

d so

lvin

g eq

uatio

ns to

incl

ude:

-U

sing

add

itive

and

mul

tiplic

ativ

e in

vers

es

-U

sing

law

s of

exp

onen

ts

• U

se s

ubst

itutio

n in

equ

atio

ns to

ge

nera

te ta

bles

of o

rder

ed p

airs

Wha

t is

diffe

rent

to G

rade

7?

Up

to a

nd in

clud

ing

Gra

de 7

lear

ners

use

the

term

'num

bers

sen

tenc

es'.

From

G

rade

8 th

e te

rm 'e

quat

ions

' is

used

.

Sol

ving

equ

atio

ns u

sing

add

itive

and

mul

tiplic

ativ

e in

vers

es a

s w

ell a

s la

ws

of

expo

nent

s

alg

ebra

ic e

quat

ions

are

don

e ag

ain

in t

erm

2 a

nd t

erm

4. i

n th

is te

rm th

e fo

cus

is o

n so

lvin

g eq

uatio

ns th

at in

volv

e m

ultip

licat

ion

and

divi

sion

of

inte

gers

and

num

bers

in e

xpon

entia

l for

m, b

y in

spec

tion

only

. Le

arne

rs h

ave

oppo

rtuni

ties

to w

rite

and

solv

e eq

uatio

ns w

hen

they

writ

e ge

nera

l ru

les

to d

escr

ibe

rela

tions

hips

bet

wee

n nu

mbe

rs in

num

ber p

atte

rns,

and

whe

n th

ey fi

nd in

put o

r out

put v

alue

s fo

r giv

en ru

les

in fl

ow d

iagr

ams,

tabl

es a

nd

form

ulae

.

See

furth

er n

otes

in T

erm

2.

Tim

e fo

r A

lgeb

raic

eq

uatio

ns in

th

is te

rm:

3 ho

urs

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• ca

lcul

atin

g an

d so

lvin

g pr

oble

ms

with

who

le n

umbe

rs a

nd in

tege

rs

• re

pres

entin

g, a

nd c

alcu

latin

g w

ith, n

umbe

rs in

exp

onen

tial f

orm

• nu

mer

ic a

nd g

eom

etric

pat

tern

s

• fu

nctio

ns a

nd re

latio

nshi

ps

• al

gebr

aic

expr

essi

ons

• al

gebr

aic

equa

tions

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

6 ho

urs



Gr

ad

e 8

– te

rm

2

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.3

alg

ebra

icex

pres

sion

sa

lgeb

raic

lang

uage

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-re

cogn

ize

and

inte

rpre

t rul

es

or re

latio

nshi

ps re

pres

ente

d in

sy

mbo

lic fo

rm

-id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae a

nd e

quat

ions

• R

ecog

nize

and

iden

tify

conv

entio

ns

for w

ritin

g al

gebr

aic

expr

essi

ons

• Id

entif

y an

d cl

assi

fy li

ke a

nd u

nlik

e te

rms

in a

lgeb

raic

exp

ress

ions

• R

ecog

nize

and

iden

tify

coef

ficie

nts

and

expo

nent

s in

alg

ebra

ic

expr

essi

ons

expa

nd a

nd s

impl

ify a

lgeb

raic

ex

pres

sion

s

Use

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e la

ws

for r

atio

nal n

umbe

rs

and

law

s of

exp

onen

ts to

:

• A

dd a

nd s

ubtra

ct li

ke te

rms

in

alge

brai

c ex

pres

sion

s

• m

ultip

ly in

tege

rs a

nd m

onom

ials

by:

-m

onom

ials

-bi

nom

ials

-tri

nom

ials

•r

evis

e th

e fo

llow

ing

done

in t

erm

1

alg

ebra

ic e

xpre

ssio

ns w

ere

also

don

e in

ter

m 1

. in

this

term

the

focu

s is

on

exp

andi

ng a

nd s

impl

ifyin

g al

gebr

aic

expr

essi

ons.

Lear

ners

hav

e op

portu

nitie

s to

writ

e an

d in

terp

ret a

lgeb

raic

exp

ress

ions

whe

n th

ey w

rite

gene

ral r

ules

to d

escr

ibe

rela

tions

hips

bet

wee

n nu

mbe

rs in

num

ber

patte

rns,

and

whe

n th

ey fi

nd in

put o

r out

put v

alue

s fo

r giv

en ru

les

in fl

ow

diag

ram

s, ta

bles

, for

mul

ae a

nd e

quat

ions

.

exam

ples

of i

nter

pret

ing

alge

brai

c ex

pres

sion

s

a)

Wha

t doe

s th

e ru

le 2

n mea

n fo

r the

follo

win

g nu

mbe

r seq

uenc

e:

2; 4

; 8; 1

6; 3

2....

Her

e le

arne

rs s

houl

d re

cogn

ize

that

2n re

pres

ents

the

gene

ral t

erm

in th

is

sequ

ence

, whe

re n

repr

esen

ts th

e po

sitio

n of

the

term

in th

e se

quen

ce.

Thus

, it i

s th

e ru

le th

at c

an b

e us

ed to

find

any

term

in th

e gi

ven

sequ

ence

.

b)

The

rela

tions

hip

betw

een

a bo

y’s

age

(x y

rs o

ld) a

nd h

is m

othe

r’s a

ge is

gi

ven

as 2

5 +

x. H

ow c

an th

is re

latio

nshi

p be

use

d to

find

the

mot

her’s

age

w

hen

the

boy

is 1

1 ye

ars

old?

Her

e le

arne

rs s

houl

d re

cogn

ize

that

to fi

nd th

e m

othe

r’s a

ge, t

hey

have

to

subs

titut

e th

e bo

y’s

give

n ag

e in

to th

e ru

le 2

5 +

x. T

hey

shou

ld a

lso

reco

gniz

e th

at th

e gi

ven

rule

mea

ns th

e m

othe

r is

25 y

ears

old

er th

an th

e bo

y.

See

furth

er e

xam

ples

giv

en fo

r fun

ctio

ns a

nd re

latio

nshi

ps.

•m

anip

ulat

ing

alge

brai

c ex

pres

sion

s

-m

ake

sure

lear

ners

und

erst

and

that

the

rul

e fo

r op

erat

ing

with

inte

gers

and

ra

tiona

l num

bers

, inc

ludi

ng la

ws

of e

xpon

ents

, app

lies

equa

lly w

hen

num

bers

ar

e re

plac

ed w

ith v

aria

bles

. The

var

iabl

es a

re n

umbe

rs o

f a g

iven

type

(e.

g.

who

le n

umbe

rs, i

nteg

ers

or ra

tiona

l num

bers

) in

gene

raliz

ed fo

rm.

-W

hen

mul

tiply

ing

or d

ivid

ing

expr

essi

ons,

mak

e su

re le

arne

rs u

nder

stan

d ho

w

the

dist

ribut

ive

rule

wor

ks.

-Th

e as

soci

ativ

e ru

le a

llow

s fo

r gro

upin

g of

like

term

s w

hen

addi

ng.

Tim

e fo

r al

gebr

aic

expr

essi

ons

in

this

term

:

9 ho

urs


93CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.3

alg

ebra

icex

pres

sion

s

• D

ivid

e th

e fo

llow

ing

by in

tege

rs o

r m

onom

ials

:

-m

onom

ials

-bi

nom

ials

-tri

nom

ials

• S

impl

ify a

lgeb

raic

exp

ress

ions

in

volv

ing

the

abov

e op

erat

ions

• D

eter

min

e th

e sq

uare

s, c

ubes

, sq

uare

root

s an

d cu

be ro

ots

of s

ingl

e al

gebr

aic

term

s or

like

alg

ebra

ic

term

s

• D

eter

min

e th

e nu

mer

ical

val

ue o

f al

gebr

aic

expr

essi

ons

by s

ubst

itutio

n

Look

out

for t

he fo

llow

ing

com

mon

mis

conc

eptio

ns:

•x

+ x

= 2x

and

no

t x2 .

Not

e th

e co

nven

tion

is to

writ

e 2x

rath

er th

an x

2

•x2 +

x2 =

2x2 a

nd

no

t 2x

4

•a

+ b

= a

+ b

an

d n

ot

ab

• (–

2x2 )3 =

–8x

6 an

d n

ot

– 6x

5

• –x

(3x

+ 1)

= –

3x2 –

x a

nd

no

t –

3x2 +

1

•6x

2 + 1

x2

= 6

+ 1 x2 a

nd

no

t 6

+ 1

• If

x =

2 th

en –

3x2 =

–3(

2)2 =

–3

x 4

= –1

2 a

nd

no

t (–

6)2

• If

x =

–2 th

en –

x2 – x

= –

(–2)

2 – 2

= –

4 +

2 =

–2 a

nd

no

t 4

+ 2

= 6

• 25

x2 – 9

x2√

=

16x2

√ =

4x

an

d n

ot

5x –

3x

= 2x

exam

ples

a)

Sim

plify

: 2(5

+ x

– x

2 ) –

x(3x

+ 1

) [m

ultip

ly in

tege

r or m

onom

ial b

y po

lyno

mia

l]

b)

If x

= –2

det

erm

ine

the

num

eric

al v

alue

of 3

x2 – 4

x +

5 [u

sing

sub

stitu

tion]

c)

Sim

plify

: 6x3 +

2x2 +

4

2x2

, x ≠

0 [d

ivid

e tri

nom

ial b

y m

onom

ial;

rem

inde

r tha

t de

nom

inat

or c

anno

t be

0]

d)

Sim

plify

: 8x3 –

(–x2 )(

2x)

–x2

, x ≠0

[cal

cula

tions

invo

lvin

g m

ultip

le o

pera

tions

; rem

ind

lear

ners

that

den

omin

ator

can

not b

e 0]

e)

Det

erm

ine:

36

x4 √

[squ

are

root

of m

onom

ial]

It m

ight

hel

p to

rem

ind

lear

ners

that

thes

e va

riabl

es (o

r x in

this

cas

e) re

pres

ent

num

bers

of a

par

ticul

ar ty

pe –

thes

e m

ay b

e ra

tiona

l, or

inte

gers

, or p

erha

ps

who

le n

umbe

rs; s

uch

a re

min

der a

lso

then

impl

ies

that

all

the

asso

ciat

ed ru

les

or

prop

ertie

s of

thes

e nu

mbe

rs a

pply

her

e. In

the

abov

e ex

ampl

e, if

x is

an

inte

ger,

then

x =

a o

r x =

–a

beca

use

a4 = (–

a)4



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.4

alg

ebra

iceq

uatio

ns

equa

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-S

et u

p eq

uatio

ns to

des

crib

e pr

oble

m s

ituat

ions

-A

naly

se a

nd in

terp

ret e

quat

ions

th

at d

escr

ibe

a gi

ven

situ

atio

n

-S

olve

equ

atio

ns b

y in

spec

tion

-D

eter

min

e th

e nu

mer

ical

val

ue o

f an

exp

ress

ion

by s

ubst

itutio

n.

-Id

entif

y va

riabl

es a

nd c

onst

ants

in

give

n fo

rmul

ae a

nd e

quat

ions

• E

xten

d so

lvin

g eq

uatio

ns to

incl

ude:

-U

sing

add

itive

and

mul

tiplic

ativ

e in

vers

es

-U

sing

law

s of

exp

onen

ts

-U

se s

ubst

itutio

n in

equ

atio

ns to

ge

nera

te ta

bles

of o

rder

ed p

airs

Wha

t is

diffe

rent

to G

rade

7?

• S

olvi

ng e

quat

ions

usi

ng a

dditi

ve a

nd m

ultip

licat

ive

inve

rses

as

wel

l as

law

s of

ex

pone

nts

alg

ebra

ic e

quat

ions

wer

e al

so d

one

in t

erm

1. i

n th

is te

rm th

e fo

cus

is o

n so

lvin

g eq

uatio

ns u

sing

add

itive

and

mul

tiplic

ativ

e in

vers

es a

s w

ell a

s th

e la

ws

of e

xpon

ents

. alg

ebra

ic e

quat

ions

are

don

e ag

ain

in t

erm

4.

Lear

ners

hav

e op

portu

nitie

s to

writ

e an

d so

lve

equa

tions

whe

n th

ey w

rite

gene

ral

rule

s to

des

crib

e re

latio

nshi

ps b

etw

een

num

bers

in n

umbe

r pat

tern

s, a

nd w

hen

they

find

inpu

t or o

utpu

t val

ues

for g

iven

rule

s in

flow

dia

gram

s, ta

bles

and

fo

rmul

ae.

exam

ples

of e

quat

ions

a)

Sol

ve x

if x

+ 6

= –

9

To s

olve

the

equa

tion:

add

–6

to b

oth

side

s of

the

equa

tion

x +

6 –

6 =

– 9

– 6,

ther

efor

e x

= –1

5

b)

Sol

ve x

if –

2x =

8

To s

olve

the

equa

tion;

div

ide

both

sid

es o

f the

equ

atio

n by

–2:

–2x

–2 =

8

–2

x =

–4

c)

Sol

ve x

if –

x =

–5

To s

olve

the

equa

tion;

div

ide

both

sid

es o

f the

equ

atio

n by

–1

x –1 =

–5

–1 ,

ther

efor

e x

= 5

d)

Sol

ve x

if 3

x +

1 =

7

To s

olve

the

equa

tion

requ

ires

two

step

s:

Add

–1

to b

oth

side

s of

the

equa

tion:

3x +

1 –

1 =

7 –

1, t

here

fore

3x

= 6

Then

div

ide

both

sid

es o

f the

equ

atio

n by

33x

3

= 6 3

, the

refo

re x

= 2

e)

Pro

vide

an

equa

tion

to fi

nd th

e ar

ea o

f a re

ctan

gle

with

leng

th 2

x cm

and

wid

th

2x +

1 c

m.

f) If

the

area

of a

rect

angl

e is

(4x2 –

6x)

cm

2 , an

d its

wid

th is

2x

cm, w

hat w

ill b

e its

leng

th in

term

s of

x?

g)

If y

= x3

+ 1,

cal

cula

te y

whe

n x

= 4

h)

Than

di is

6 y

ears

old

er th

an S

ophi

e. In

3 y

ears

tim

e Th

andi

will

be

twic

e as

old

as

Sop

hie.

How

old

is T

hand

i now

?

Tim

e fo

r al

gebr

aic

equa

tions

in

this

term

:

3 ho

urs


95CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.5

Con

stru

ctio

n of

ge

omet

ric

figures

Con

stru

ctio

ns

• A

ccur

atel

y co

nstru

ct g

eom

etric

fig

ures

app

ropr

iate

ly u

sing

a

com

pass

, rul

er a

nd p

rotra

ctor

, in

clud

ing:

-B

isec

ting

lines

and

ang

les

-P

erpe

ndic

ular

line

s at

a g

iven

poi

nt

or fr

om a

giv

en p

oint

-Tr

iang

les

-Q

uadr

ilate

rals

• C

onst

ruct

ang

les

of 3

0°, 4

5° a

nd 6

0°

and

thei

r mul

tiple

s w

ithou

t usi

ng a

pr

otra

ctor

inve

stig

atin

g pr

oper

ties

of g

eom

etric

fig

ures

• B

y co

nstru

ctio

n, in

vest

igat

e th

e an

gles

in a

tria

ngle

, foc

usin

g on

:

-th

e su

m o

f the

inte

rior a

ngle

s of

tri

angl

es

-th

e si

ze o

f ang

les

in a

n eq

uila

tera

l tri

angl

e

-th

e si

des

and

base

ang

les

of a

n is

ocel

es tr

iang

le

• B

y co

nstru

ctio

n, in

vest

igat

e si

des

and

angl

es in

qua

drila

tera

ls, f

ocus

ing

on:

-th

e su

m o

f the

inte

rior a

ngle

s of

qu

adril

ater

als

-th

e si

des

and

oppo

site

ang

les

of

para

llelo

gram

s

Wha

t is

diffe

rent

to G

rade

7?

• A

ll th

e co

nstru

ctio

ns a

re n

ew. I

n G

rade

7 le

arne

rs o

nly

cons

truct

ed a

ngle

s,

perp

endi

cula

r and

par

alle

l lin

es

• U

sing

con

stru

ctio

ns to

exp

lore

pro

perti

es o

f tria

ngle

s an

d qu

adril

ater

als

Con

stru

ctio

ns

• C

onst

ruct

ions

pro

vide

a u

sefu

l con

text

to e

xplo

re o

r con

solid

ate

know

ledg

e of

an

gles

and

sha

pes

• m

ake

sure

lear

ners

are

com

pete

nt a

nd c

omfo

rtabl

e w

ith th

e us

e of

a c

ompa

ss

and

know

how

to m

easu

re a

nd re

ad a

ngle

siz

es o

n a

prot

ract

or

• R

evis

e th

e co

nstru

ctio

ns o

f ang

les

if ne

cess

ary

befo

re p

roce

edin

g w

ith th

e ne

w

cons

truct

ions

• S

tart

with

the

cons

truct

ions

of l

ines

, so

that

lear

ners

can

firs

t exp

lore

ang

le

rela

tions

hips

on

stra

ight

line

s.

• W

hen

cons

truct

ing

trian

gles

lear

ners

sho

uld

draw

on

know

n pr

oper

ties

and

cons

truct

ion

of c

ircle

s.

• C

onst

ruct

ion

of s

peci

al a

ngle

s w

ithou

t pro

tract

ors

are

done

by:

-bi

sect

ing

a rig

ht a

ngle

to g

et 4

5°

-dr

awin

g an

equ

ilate

ral t

riang

le to

get

60°

-bi

sect

ing

the

angl

es o

f an

equi

late

ral t

riang

le to

get

30°

Tota

l Tim

e fo

r co

nstru

ctio

ns

of g

eom

etric

fig

ures

:

8 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

Cla

ssify

ing

2d s

hape

s

• Id

entif

y an

d w

rite

clea

r defi

nitio

ns o

f tri

angl

es in

term

s of

thei

r sid

es a

nd

angl

es, d

istin

gush

ing

betw

een:

-eq

uila

tera

l tria

ngle

s

-is

osce

les

trian

gles

-rig

ht-a

ngle

d tri

angl

es

• Id

entif

y an

d w

rite

clea

r defi

nito

ns o

f qu

adril

ater

als

in te

rms

of th

eir s

ides

an

d an

gles

, dis

tingu

ishi

ng b

etw

een:

-pa

ralle

logr

am

-re

ctan

gle

-sq

uare

-rh

ombu

s

-tra

pezi

um

-ki

te

Wha

t is

diffe

rent

to G

rade

7?

• N

ew p

rope

rties

in te

rms

of a

ngle

s of

tria

ngle

s

• W

rite

clea

r defi

nitio

ns o

f the

pro

perti

es o

f tria

ngle

s an

d qu

adril

ater

als

• U

se d

efini

tions

to s

olve

geo

met

ric p

robl

ems

• In

vest

igat

e co

nditi

ons

for 2

D s

hape

s to

be

cong

ruen

t or

sim

ilar

tria

ngle

s

• C

onst

ruct

ions

ser

ve a

s a

usef

ul c

onte

xt fo

r exp

lorin

g pr

oper

ties

of tr

iang

les.

S

ee n

otes

on

Con

stru

ctio

ns a

bove

.

• P

rope

rties

of t

riang

les

lear

ners

sho

uld

know

:

-th

e su

m o

f the

inte

rior a

ngle

s of

tria

ngle

s =

180°

-an

equ

ilate

ral t

riang

le h

as a

ll si

des

equa

l and

all

inte

rior a

ngle

s =

60°

-an

isos

cele

s tri

angl

e ha

s at

leas

t tw

o eq

ual s

ides

and

its

base

ang

les

are

equa

l

-a

right

-ang

led

trian

gle

has

one

angl

e th

at is

a ri

ght-a

ngle

-th

e si

de o

ppos

ite th

e rig

ht-a

ngle

in a

righ

t-ang

led

trian

gle,

is c

alle

d th

e hy

pote

nuse

-in

a ri

ght-a

ngle

d tri

angl

e, th

e sq

uare

of t

he h

ypot

enus

e is

equ

al to

the

sum

of

the

squa

res

of th

e ot

her t

wo

side

s (T

heor

em o

f Pyt

hago

ras)

.

Qua

drila

tera

ls

• C

onst

ruct

ions

ser

ve a

s a

usef

ul c

onte

xt fo

r exp

lorin

g pr

oper

ties

of tr

iang

les.

S

ee n

otes

on

Con

stru

ctio

ns a

bove

.

• Th

e cl

assi

ficat

ion

of q

uadr

ilate

rals

sho

uld

incl

ude

the

reco

gniti

on th

at:

-re

ctan

gles

and

rhom

bi a

re s

peci

al k

inds

of p

aral

lelo

gram

s

-a

squa

re is

a s

peci

al k

ind

of re

ctan

gle

and

rhom

bus.

• P

rope

rties

of q

uadr

ilate

rals

lear

ners

sho

uld

know

:

-th

e su

m o

f the

inte

rior a

ngle

s of

qua

drila

tera

ls =

360

°

-th

e op

posi

te s

ides

of p

aral

lelo

gram

s ar

e pa

ralle

l and

equ

al

-th

e op

posi

te a

ngle

s of

par

alle

logr

ams

are

equa

l

-th

e op

posi

te a

ngle

s of

a rh

ombu

s ar

e eq

ual

-th

e op

posi

te s

ides

of a

rhom

bus

are

para

llel a

nd e

qual

-th

e si

ze o

f eac

h an

gle

of re

ctan

gles

and

squ

ares

is 9

0°

-a

trape

zium

has

one

pai

r of o

ppos

ite s

ides

par

alle

l

Tota

l tim

e fo

r ge

omet

ry o

f 2D

sha

pes

8 ho

urs.


97CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

Cla

ssify

ing

2d s

hape

s

• Id

entif

y an

d w

rite

clea

r defi

nitio

ns o

f tri

angl

es in

term

s of

thei

r sid

es a

nd

angl

es, d

istin

gush

ing

betw

een:

-eq

uila

tera

l tria

ngle

s

-is

osce

les

trian

gles

-rig

ht-a

ngle

d tri

angl

es

• Id

entif

y an

d w

rite

clea

r defi

nito

ns o

f qu

adril

ater

als

in te

rms

of th

eir s

ides

an

d an

gles

, dis

tingu

ishi

ng b

etw

een:

-pa

ralle

logr

am

-re

ctan

gle

-sq

uare

-rh

ombu

s

-tra

pezi

um

-ki

te

Wha

t is

diffe

rent

to G

rade

7?

• N

ew p

rope

rties

in te

rms

of a

ngle

s of

tria

ngle

s

• W

rite

clea

r defi

nitio

ns o

f the

pro

perti

es o

f tria

ngle

s an

d qu

adril

ater

als

• U

se d

efini

tions

to s

olve

geo

met

ric p

robl

ems

• In

vest

igat

e co

nditi

ons

for 2

D s

hape

s to

be

cong

ruen

t or

sim

ilar

tria

ngle

s

• C

onst

ruct

ions

ser

ve a

s a

usef

ul c

onte

xt fo

r exp

lorin

g pr

oper

ties

of tr

iang

les.

S

ee n

otes

on

Con

stru

ctio

ns a

bove

.

• P

rope

rties

of t

riang

les

lear

ners

sho

uld

know

:

-th

e su

m o

f the

inte

rior a

ngle

s of

tria

ngle

s =

180°

-an

equ

ilate

ral t

riang

le h

as a

ll si

des

equa

l and

all

inte

rior a

ngle

s =

60°

-an

isos

cele

s tri

angl

e ha

s at

leas

t tw

o eq

ual s

ides

and

its

base

ang

les

are

equa

l

-a

right

-ang

led

trian

gle

has

one

angl

e th

at is

a ri

ght-a

ngle

-th

e si

de o

ppos

ite th

e rig

ht-a

ngle

in a

righ

t-ang

led

trian

gle,

is c

alle

d th

e hy

pote

nuse

-in

a ri

ght-a

ngle

d tri

angl

e, th

e sq

uare

of t

he h

ypot

enus

e is

equ

al to

the

sum

of

the

squa

res

of th

e ot

her t

wo

side

s (T

heor

em o

f Pyt

hago

ras)

.

Qua

drila

tera

ls

• C

onst

ruct

ions

ser

ve a

s a

usef

ul c

onte

xt fo

r exp

lorin

g pr

oper

ties

of tr

iang

les.

S

ee n

otes

on

Con

stru

ctio

ns a

bove

.

• Th

e cl

assi

ficat

ion

of q

uadr

ilate

rals

sho

uld

incl

ude

the

reco

gniti

on th

at:

-re

ctan

gles

and

rhom

bi a

re s

peci

al k

inds

of p

aral

lelo

gram

s

-a

squa

re is

a s

peci

al k

ind

of re

ctan

gle

and

rhom

bus.

• P

rope

rties

of q

uadr

ilate

rals

lear

ners

sho

uld

know

:

-th

e su

m o

f the

inte

rior a

ngle

s of

qua

drila

tera

ls =

360

°

-th

e op

posi

te s

ides

of p

aral

lelo

gram

s ar

e pa

ralle

l and

equ

al

-th

e op

posi

te a

ngle

s of

par

alle

logr

ams

are

equa

l

-th

e op

posi

te a

ngle

s of

a rh

ombu

s ar

e eq

ual

-th

e op

posi

te s

ides

of a

rhom

bus

are

para

llel a

nd e

qual

-th

e si

ze o

f eac

h an

gle

of re

ctan

gles

and

squ

ares

is 9

0°

-a

trape

zium

has

one

pai

r of o

ppos

ite s

ides

par

alle

l

Tota

l tim

e fo

r ge

omet

ry o

f 2D

sha

pes

8 ho

urs.

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

sim

ilar a

nd c

ongr

uent

2-d

sha

pes

• Id

entif

y an

d de

scrib

e th

e pr

oper

ties

of c

ongr

uent

sha

pes

• Id

entif

y an

d de

scrib

e th

e pr

oper

ties

of s

imila

r sha

pes

solv

ing

prob

lem

s

• S

olve

geo

met

ric p

robl

ems

invo

lvin

g un

know

n si

des

and

angl

es in

tri

angl

es a

nd q

uadr

ilate

rals

, usi

ng

know

n pr

oper

ties

and

defin

ition

s.

• a

kite

has

two

pairs

of a

djac

ent s

ides

equ

al

sim

ilarit

y an

d co

ngru

ency

• Th

is c

an b

e ex

plor

ed w

ith a

ny 2

-D s

hape

• Le

arne

rs s

houl

d re

cogn

ise

that

two

or m

ore

figur

es a

re c

ongr

uent

if th

ey a

re

equa

l in

all r

espe

cts

i.e c

orre

spon

ding

ang

les

and

side

s ar

e eq

ual

• Le

arne

rs s

houl

d re

cogn

ise

that

two

or m

ore

figur

es a

re s

imila

r if t

hey

have

. co

rres

pond

ing

angl

es e

qual

and

the

ir si

des

are

prop

ortio

nally

long

er o

r sho

rter.

S

ee e

xam

ples

bel

ow.

• N

ote

that

in G

rade

9 le

arne

rs w

ill fo

cus

on th

e sp

ecia

l cas

es o

f sim

ilarit

y an

d co

ngru

ence

in tr

iang

les.

• S

imiil

ar fi

gure

s ar

e fu

rther

exp

lore

d w

hen

doin

g en

larg

emen

ts a

nd re

duct

ions

. R

efer

to “C

larifi

catio

n N

otes

” und

er 3

.3 T

rans

form

atio

n G

eom

etry

.

exam

ples

:

• C

ompa

ring

squa

res

to o

ther

rect

angl

es, l

earn

ers

can

asce

rtain

that

hav

ing

corr

espo

ndin

g an

gles

equ

al d

oes

not n

eces

saril

y im

ply

that

the

side

s w

ill b

e of

pro

porti

onal

leng

th. H

ence

hav

ing

equa

l ang

les

alon

e, is

not

a s

uffic

ient

co

nditi

on fo

r figu

res

to b

e si

mila

r.

• C

ompa

ring

rhom

bii w

ith s

ides

pro

porti

onal

, lea

rner

s ca

n as

certa

in th

at h

avin

g si

des

prop

ortio

nal d

oes

not n

eces

saril

y im

ply

that

the

corr

espo

ndin

g an

gles

will

be

equ

al. S

o on

ly h

avin

g si

des

of p

ropo

rtion

al le

ngth

is n

ot a

suf

ficie

nt c

ondi

tion

for s

imila

rity

solv

ing

prob

lem

s

• Le

arne

rs c

an s

olve

geo

met

ric p

robl

ems

to fi

nd u

nkno

wn

side

s an

d an

gles

in

tria

ngle

s an

d qu

adril

ater

als,

usi

ng k

now

n de

finiti

ons

as w

ell a

s an

gle

rela

tions

hips

on

stra

ight

line

s (s

ee n

otes

on

Geo

met

ry o

f Stra

ight

line

s.)

• Fo

r rig

ht-a

ngle

d tri

angl

es, l

earn

ers

can

also

use

the

Theo

rem

of P

ytha

gora

s to

fin

d un

know

n le

ngth

s.

• Le

arne

rs s

houl

d gi

ve re

ason

s an

d ju

stify

thei

r sol

utio

ns fo

r eve

ry w

ritte

n st

atem

ent.

• N

ote

that

sol

ving

geo

met

ric p

robl

ems

is a

n op

portu

nity

to p

ract

ise

solv

ing

equa

tions

.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

exam

ple:

Â1,

Â2,

Â3 an

d ar

e th

ree

angl

es o

n a

stra

ight

line

. Â2 =

75°,

Â3 =

55°.

W

hat i

s th

e si

ze o

f Â1?

Lear

ners

can

find

Â1 b

y so

lvin

g th

e fo

llow

ing

equa

tion:

Â1 +

75°

+ 55

° =

180°

(bec

ause

the

sum

of a

ngle

s on

a s

traig

ht li

ne =

180

°)Â

1 =

180°

– 1

30°(

add

–55°

and

–75

° to

bot

h si

des

of th

e eq

uatio

n)Â

1 =

50°

3.3

Geo

met

ry o

f st

raig

ht li

nes

ang

le re

latio

nshi

ps•

Rec

ogni

se a

nd d

escr

ibe

pairs

of

angl

es fo

rmed

by:

-pe

rpen

dicu

lar l

ines

-in

ters

ectin

g lin

es -

para

llel l

ines

cut

by

a tra

nsve

rsal

solv

ing

prob

lem

s•

Sol

ve g

eom

etric

pro

blem

s us

ing

the

rela

tions

hips

bet

wee

n pa

irs o

f ang

les

desc

ribed

abo

ve

Wha

t is

diffe

rent

to G

rade

7?

• In

Gra

de 7

lear

ners

onl

y de

fined

line

seg

men

t, ra

y, s

traig

ht li

ne, p

aral

lel l

ines

an

d pe

rpen

dicu

lar l

ines

ang

le re

latio

nshi

ps le

arne

rs s

houl

d kn

ow:

• th

e su

m o

f the

ang

les

on a

stra

ight

line

is 1

80°

• If

lines

are

per

pend

icul

ar, t

hen

adja

cent

sup

plem

enta

ry a

ngle

s ar

e ea

ch e

qual

to

90°.

• If

lines

inte

rsec

t, th

en v

ertic

ally

opp

osite

ang

les

are

equa

l.•

if pa

ralle

l lin

es a

re c

ut b

y a

trans

vers

al, t

hen

corr

espo

ndin

g an

gles

are

equ

al•

if pa

ralle

l lin

es c

ut b

y a

trans

vers

al, t

hen

alte

rnat

e an

gles

are

equ

al•

if pa

ralle

l lin

es c

ut b

y a

trans

vers

al, t

hen

co-in

terio

r ang

les

are

supp

lem

enta

ryTh

e ab

ove

angl

es h

ave

to b

e id

entifi

ed a

nd n

amed

by

lear

ner

solv

ing

prob

lem

s•

Lear

ners

can

sol

ve g

eom

etric

pro

blem

s to

find

unk

now

n an

gles

usi

ng th

e an

gle

rela

tions

hips

abo

ve, a

s w

ell a

s ot

her k

now

n pr

oper

ties

of tr

iang

les

and

quad

rilat

eral

s.

• Le

arne

rs s

houl

d gi

ve re

ason

s an

d ju

stify

thei

r sol

utio

ns fo

r eve

ry w

ritte

n st

atem

ent.

• N

ote

that

sol

ving

geo

met

ric p

robl

ems

is a

n op

portu

nity

to p

ract

ise

solv

ing

equa

tions

. exa

mpl

e:Â

, B a

nd Ĉ

are

thre

e an

gles

on

a st

raig

ht li

ne. Â

= 5

5°, B

= 7

5°. W

hat i

s th

e si

ze

of Ĉ?

Lear

ners

can

find

Ĉb

y so

lvin

g th

e fo

llow

ing

equa

tion:

55

° +

75°

+ Ĉ

= 1

80°

(be

caus

e th

e su

m o

f ang

les

on a

stra

ight

line

= 1

80°)

Ĉ =

180

° –

130°

(

add

–55°

and

–75

° to

bot

h si

des

of th

e eq

uatio

n)Ĉ

= 5

0°

Tota

l tim

e fo

r ge

omet

ry o

f st

raig

ht li

nes:

9 ho

urs


99CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

exam

ple:

Â1,

Â2,

Â3 an

d ar

e th

ree

angl

es o

n a

stra

ight

line

. Â2 =

75°,

Â3 =

55°.

W

hat i

s th

e si

ze o

f Â1?

Lear

ners

can

find

Â1 b

y so

lvin

g th

e fo

llow

ing

equa

tion:

Â1 +

75°

+ 55

° =

180°

(bec

ause

the

sum

of a

ngle

s on

a s

traig

ht li

ne =

180

°)Â

1 =

180°

– 1

30°(

add

–55°

and

–75

° to

bot

h si

des

of th

e eq

uatio

n)Â

1 =

50°

3.3

Geo

met

ry o

f st

raig

ht li

nes

ang

le re

latio

nshi

ps•

Rec

ogni

se a

nd d

escr

ibe

pairs

of

angl

es fo

rmed

by:

-pe

rpen

dicu

lar l

ines

-in

ters

ectin

g lin

es -

para

llel l

ines

cut

by

a tra

nsve

rsal

solv

ing

prob

lem

s•

Sol

ve g

eom

etric

pro

blem

s us

ing

the

rela

tions

hips

bet

wee

n pa

irs o

f ang

les

desc

ribed

abo

ve

Wha

t is

diffe

rent

to G

rade

7?

• In

Gra

de 7

lear

ners

onl

y de

fined

line

seg

men

t, ra

y, s

traig

ht li

ne, p

aral

lel l

ines

an

d pe

rpen

dicu

lar l

ines

ang

le re

latio

nshi

ps le

arne

rs s

houl

d kn

ow:

• th

e su

m o

f the

ang

les

on a

stra

ight

line

is 1

80°

• If

lines

are

per

pend

icul

ar, t

hen

adja

cent

sup

plem

enta

ry a

ngle

s ar

e ea

ch e

qual

to

90°.

• If

lines

inte

rsec

t, th

en v

ertic

ally

opp

osite

ang

les

are

equa

l.•

if pa

ralle

l lin

es a

re c

ut b

y a

trans

vers

al, t

hen

corr

espo

ndin

g an

gles

are

equ

al•

if pa

ralle

l lin

es c

ut b

y a

trans

vers

al, t

hen

alte

rnat

e an

gles

are

equ

al•

if pa

ralle

l lin

es c

ut b

y a

trans

vers

al, t

hen

co-in

terio

r ang

les

are

supp

lem

enta

ryTh

e ab

ove

angl

es h

ave

to b

e id

entifi

ed a

nd n

amed

by

lear

ner

solv

ing

prob

lem

s•

Lear

ners

can

sol

ve g

eom

etric

pro

blem

s to

find

unk

now

n an

gles

usi

ng th

e an

gle

rela

tions

hips

abo

ve, a

s w

ell a

s ot

her k

now

n pr

oper

ties

of tr

iang

les

and

quad

rilat

eral

s.

• Le

arne

rs s

houl

d gi

ve re

ason

s an

d ju

stify

thei

r sol

utio

ns fo

r eve

ry w

ritte

n st

atem

ent.

• N

ote

that

sol

ving

geo

met

ric p

robl

ems

is a

n op

portu

nity

to p

ract

ise

solv

ing

equa

tions

. exa

mpl

e:Â

, B a

nd Ĉ

are

thre

e an

gles

on

a st

raig

ht li

ne. Â

= 5

5°, B

= 7

5°. W

hat i

s th

e si

ze

of Ĉ?

Lear

ners

can

find

Ĉb

y so

lvin

g th

e fo

llow

ing

equa

tion:

55

° +

75°

+ Ĉ

= 1

80°

(be

caus

e th

e su

m o

f ang

les

on a

stra

ight

line

= 1

80°)

Ĉ =

180

° –

130°

(

add

–55°

and

–75

° to

bot

h si

des

of th

e eq

uatio

n)Ĉ

= 5

0°

Tota

l tim

e fo

r ge

omet

ry o

f st

raig

ht li

nes:

9 ho

urs

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• al

gebr

aic

expr

essi

ons

• al

gebr

aic

equa

tions

• co

nstru

ctin

g ge

omet

ric o

bjec

ts

• ge

omet

ry o

f 2D

sha

pes

• ge

omet

ry o

f stra

ight

line

s

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

8 ho

urs



Gr

ad

e 8

– te

rm

3

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, op

erat

ions

and

re

latio

nshi

ps

1.4

Com

mon

fr

actio

ns

Cal

cula

tions

usi

ng fr

actio

ns

• R

evis

e:

-ad

ditio

n an

d su

btra

ctio

n of

co

mm

on fr

actio

ns, i

nclu

ding

mix

ed

num

bers

-fin

ding

frac

tions

of w

hole

num

bers

-m

ultip

licat

ion

of c

omm

on fr

actio

ns,

incl

udin

g m

ixed

num

bers

• D

ivid

e w

hole

num

bers

and

com

mon

fra

ctio

ns b

y co

mm

on fr

actio

ns

• C

alcu

late

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

com

mon

fra

ctio

ns

Cal

cula

tion

tech

niqu

es

• R

evis

e:

-C

onve

rt m

ixed

num

bers

to

com

mon

frac

tions

in o

rder

to

perfo

rm c

alcu

latio

ns w

ith th

em

-U

se k

now

ledg

e of

mul

tiple

s an

d fa

ctor

s to

writ

e fra

ctio

ns in

the

sim

ples

t for

m b

efor

e or

afte

r ca

lcul

atio

ns

-U

se k

now

ledg

e of

equ

ival

ent

fract

ions

to a

dd a

nd s

ubtra

ct

com

mon

frac

tions

• U

se k

now

ledg

e of

reci

proc

al

rela

tions

hips

to d

ivid

e co

mm

on

fract

ions

Wha

t is

diffe

rent

to G

rade

7?

• D

ivid

e by

com

mon

frac

tions

• S

quar

es, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of c

omm

on fr

actio

ns

In G

rade

8 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r co

mm

on fr

actio

ns, d

evel

oped

in G

rade

7.

Cal

cula

tions

usi

ng fr

actio

ns

• Le

arne

rs s

houl

d co

ntin

ue to

do

cont

ext f

ree

calc

ulat

ions

and

sol

ve p

robl

ems

in

cont

exts

.

-B

y G

rade

8 le

arne

rs s

houl

d be

com

forta

ble

conv

ertin

g m

ixed

num

bers

to

com

mon

frac

tions

for c

alcu

latio

ns.

exam

ple

51 2 =

11 2;

61 3 =

19 3

-To

sim

plify

frac

tions

, lea

rner

s us

e kn

owle

dge

of c

omm

on fa

ctor

s i.e

. wha

t can

di

vide

equ

ally

into

the

num

erat

or a

nd d

enom

inat

or o

f a fr

actio

n. E

mph

asiz

e th

at

whe

n si

mpl

ifyin

g, th

e fra

ctio

ns m

ust r

emai

n eq

uiva

lent

.

add

ition

and

sub

trac

tion

• LC

ms

have

to b

e fo

und

whe

n ad

ding

and

sub

tract

ing

fract

ions

with

diff

eren

t de

nom

inat

ors.

Her

e le

arne

rs u

se k

now

ledg

e of

com

mon

mul

tiple

s to

find

the

LCM

i.e.

wha

t num

ber c

an b

oth

deno

min

ator

s be

div

ided

into

.

mul

tiplic

atio

n

• Fo

r mul

tiplic

atio

n of

frac

tions

, lea

rner

s sh

ould

be

enco

urag

ed to

sim

plify

fra

ctio

ns b

y di

vidi

ng n

umer

ator

s an

d de

nom

inat

ors

by c

omm

on fa

ctor

s.

• Le

arne

rs s

houl

d no

te th

e di

ffere

nce

betw

een

addi

ng o

r sub

tract

ing

fract

ions

, an

d m

ultip

lyin

g fra

ctio

ns

exam

ples

3

4 +

2

5 =

15 20 +

8 20 =

23 20 =

13 20

(usi

ng L

CM

and

equ

ival

ent f

ract

ions

)3

4 x

2

5 =

3 10 (d

ivid

e 2

and

4 by

com

mon

fact

or 2

)

• Le

arne

rs s

houl

d re

cogn

ize

that

find

ing

a ‘fr

actio

n of

a w

hole

num

ber’

or ‘fi

ndin

g a

fract

ion

of a

frac

tion’

mea

ns m

ultip

lyin

g th

e fra

ctio

n an

d th

e w

hole

num

ber o

r th

e fra

ctio

n w

ith th

e fra

ctio

n.

Tota

l tim

e fo

r com

mon

fra

ctio

ns:

7 ho

urs


101CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, op

erat

ions

and

re

latio

nshi

ps

1.4

Com

mon

fr

actio

ns

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

com

mon

frac

tions

and

mix

ed

num

bers

, inc

ludi

ng g

roup

ing,

sha

ring

and

findi

ng fr

actio

ns o

f who

le

num

bers

• W

hen

lear

ners

find

frac

tions

of w

hole

num

bers

, the

exa

mpl

es c

an b

e ch

osen

to

resu

lt ei

ther

in w

hole

num

bers

or f

ract

ions

or b

oth.

• Le

arne

rs s

houl

d al

so u

se th

e co

nven

tion

of w

ritin

g th

e w

hole

num

ber a

s a

fract

ion

over

whe

n m

ultip

lyin

g.

exam

ples

Find

4 5 o

f 20

= 4 5

x 20 1

= 80 5

= 1

6

Find

2 3 o

f 5 6

= 2 3

x 5 6

= 10 18

= 5 9

div

isio

n

• Th

e te

chni

que

of ‘i

nver

t and

mul

tiply

’ app

lies

to d

ivis

ion

in g

ener

al a

nd n

ot ju

st

to d

ivis

ion

by fr

actio

ns. H

ence

, a u

sefu

l way

of m

akin

g le

arne

rs c

omfo

rtabl

e w

ith

divi

sion

by

fract

ions

is to

sta

rt w

ith e

xam

ples

of d

ivis

ion

by w

hole

num

bers

.

• Le

arne

rs h

ave

to u

nder

stan

d th

at d

ivid

ing

by a

num

ber i

s th

e sa

me

as

mul

tiply

ing

by th

e re

cipr

ocal

of t

he n

umbe

r i.e

. the

reci

proc

al o

f n is

1 n

exam

ples

:

a)

10 ÷

5 is

the

sam

e as

10

x 1 5

= 2

(mul

tiply

by

the

reci

proc

al o

f 5)

b)

10 ÷

1 5 =

10

x 5

= 50

(mul

tiply

by

the

reci

proc

al o

f 1 5

)

This

can

als

o be

exp

lain

ed b

y us

ing

diag

ram

mod

els

for f

ract

ions

and

ask

ing,

ho

w m

any

times

doe

s 1 5

fit i

nto

10?

We

know

that

5 fi

fths

fit in

to 1

who

le, s

o (5

x

10) fi

fths

will

fit i

nto

10 w

hole

s. H

ence

, 10

÷1 5

= 5

0

c)

20 ÷

4 is

the

sam

e as

20

x 1 4

= 5

(mul

tiply

by

the

reci

proc

al o

f 4)

d)

20 ÷

1 4 =

20

x 4

= 80

(mul

tiply

by

the

reci

proc

al o

f 1 4

)

This

can

als

o be

exp

lain

ed b

y us

ing

diag

ram

mod

els

for f

ract

ions

and

ask

ing,

ho

w m

any

times

doe

s 1 4

fit i

nto

20?

We

know

that

4 q

uarte

rs fi

t int

o 1

who

le,

so (4

x 2

0) q

uarte

rs w

ill fi

t int

o 20

who

les.

Hen

ce, 2

0 ÷

1 4 =

80

e)

Onc

e le

arne

rs h

ave

done

a fe

w o

f the

abo

ve e

xam

ples

, the

y ca

n us

e th

e te

chni

que

of m

ultip

lyin

g by

the

reci

proc

al to

div

ide

fract

ions

by

fract

ions

:3 4

÷ 1 2

= 3 4

x 2 1

= 6 4

= 3 2

= 1

1 2(m

ultip

ly b

y th

e re

cipr

ocal

of

1 2)

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s

• K

now

ing

the

rule

s of

ope

ratio

ns fo

r cal

cula

ting

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

com

mon

frac

tions

is im

porta

nt



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, op

erat

ions

and

re

latio

nshi

ps

1.4

Com

mon

fr

actio

ns

Perc

enta

ges

• R

evis

e: -

findi

ng p

erce

ntag

es o

f who

le

num

bers

-ca

lcul

atin

g th

e pe

rcen

tage

of p

art

of a

who

le

-ca

lcul

atin

g pe

rcen

tage

incr

ease

or

decr

ease

•

Cal

cula

te a

mou

nts

if gi

ven

perc

enta

ge in

crea

se o

r dec

reas

e•

Sol

ve p

robl

ems

in c

onte

xts

invo

lvin

g pe

rcen

tage

s

equi

vale

nt fo

rms

• R

evis

e eq

uiva

lent

form

s be

twee

n:

-co

mm

on fr

actio

ns (f

ract

ions

whe

re

one

deno

min

ator

is a

mul

tiple

of

the

othe

r)

-co

mm

on fr

actio

n an

d de

cim

al

fract

ion

form

s of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

exam

ples

:a)

(3 4

)2 = 32 42

= 9 16

b) √16

25

=

25√

16√

= 4 5

• O

nce

lear

ners

are

com

forta

ble

doin

g al

l the

ope

ratio

ns w

ith fr

actio

ns,

calc

ulat

ions

do

not h

ave

to b

e re

stric

ted

to p

ositi

ve fr

actio

ns.

Cal

cula

tion

usin

g pe

rcen

tage

s•

Lear

ners

sho

uld

cont

inue

to d

o co

ntex

t fre

e ca

lcul

atio

ns a

nd s

olve

pro

blem

s in

co

ntex

ts.

• W

hen

doin

g ca

lcul

atio

ns u

sing

per

cent

ages

, lea

rner

s ha

ve to

use

the

equi

vale

nt

com

mon

frac

tion

form

, whi

ch is

a fr

actio

n w

ith d

enom

inat

or 1

00.

• Le

arne

rs s

houl

d be

com

e fa

mili

ar w

ith th

e eq

uiva

lent

frac

tion

and

deci

mal

form

s of

com

mon

per

cent

ages

eg.

25%

is e

quiv

alen

t to

1 4or

0,2

5; 5

0% is

equ

ival

ent t

o 1 2

or 0

,5; 6

0% is

equ

ival

ent t

o 3 5

or 0

,6.

• To

cal

cula

te p

erce

ntag

e of

par

t of a

who

le, o

r per

cent

age

incr

ease

or d

ecre

ase,

le

arne

rs h

ave

to le

arn

the

stra

tegy

of m

ultip

lyin

g by

1 100 .

It is

use

ful f

or le

arne

rs

to le

arn

to u

se c

alcu

lato

rs fo

r som

e of

thes

e ca

lcul

atio

ns w

here

the

fract

ions

are

no

t eas

ily s

impl

ified

. •

Whe

n us

ing

calc

ulat

ors,

lear

ners

can

use

the

equi

vale

nt d

ecim

al fr

actio

n fo

rm

for p

erce

ntag

es to

do

the

calc

ulat

ions

.ex

ampl

es:

a)

Cal

cula

te o

f 60%

of R

105

Am

ount

= 60

10

0 x R

105

= R

63b)

W

hat p

erce

ntag

e is

40c

of R

3,20

?

Per

cent

age

= 40

32

0 x 10

0 1 =

100 8 =

12,

5%c)

C

alcu

late

the

perc

enta

ge in

crea

se if

the

pric

e of

a b

us ti

cket

of R

60 is

in

crea

sed

to R

84.

Am

ount

incr

ease

d =

R24

. The

refo

re p

erce

ntag

e in

crea

se =

24

60 x

100 1 =

40%

d)

Cal

cula

te th

e pe

rcen

tage

dec

reas

e if

the

pric

e of

pet

rol g

oes

dow

n fro

m 2

0 ce

nts

a lit

re to

18

cent

s a

litre

.

Am

ount

dec

reas

ed =

2 c

ents

. The

refo

re p

erce

ntag

e de

crea

se =

2 20 x

100 1 =

10%

e)

Cal

cula

te h

ow m

uch

a ca

r will

cos

t if i

ts o

rigin

al p

rice

of R

150

000

is re

duce

d by

15%

Cal

cula

tion

invo

lves

find

ing

15%

of R

150

000

and

then

sub

tract

ing

that

am

ount

fro

m th

e or

igin

al p

rice.

i.e.

15

100 x

R15

0 00

0 1

= R

22 5

00H

ence

new

pric

e of

car

= R

150

000

– R

22 5

00 =

R12

7 50

0

Or

85

100 x

R15

0 00

0 1

= R

127

500


103CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.5

dec

imal

fr

actio

ns

ord

erin

g an

d co

mpa

ring

deci

mal

fr

actio

ns

• R

evis

e:

-O

rder

ing,

com

parin

g an

d pl

ace

valu

e of

dec

imal

frac

tions

to a

t le

ast 3

dec

imal

pla

ces

-R

ound

ing

off d

ecim

al fr

actio

ns to

at

leas

t 2 d

ecim

al p

lace

Cal

cula

tions

with

dec

imal

frac

tions

• R

evis

e:

-ad

ditio

n, s

ubtra

ctio

n an

d m

ultip

licat

ion

of d

ecim

al fr

actio

ns

to a

t lea

st 3

dec

imal

pla

ces

-di

visi

on o

f dec

imal

frac

tions

by

who

le n

umbe

rs

• E

xten

d m

ultip

licat

ion

to 'm

ultip

licat

ion

by d

ecim

al fr

actio

ns' n

ot li

mite

d to

on

e de

cim

al p

lace

Wha

t is

diffe

rent

to G

rade

7?

• m

ultip

licat

ion

by d

ecim

al fr

actio

ns n

ot li

mite

d to

one

dec

imal

pla

ce

• D

ivis

ion

of d

ecim

al fr

actio

ns b

y de

cim

al fr

actio

ns

• S

quar

es, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of d

ecim

al fr

actio

ns

In G

rade

8 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r de

cim

al fr

actio

ns, d

evel

oped

in G

rade

7.

ord

erin

g, c

ount

ing

and

com

parin

g de

cim

al fr

actio

ns

• Le

arne

rs s

houl

d co

ntin

ue to

pra

ctis

e co

untin

g, o

rder

ing

and

com

parin

g de

cim

al

fract

ions

. Cou

ntin

g sh

ould

not

onl

y be

thou

ght o

f as

verb

al c

ount

ing.

Lea

rner

s ca

n co

unt i

n de

cim

al in

terv

als

usin

g:

-st

ruct

ured

, sem

i-stru

ctur

ed o

r em

pty

num

ber l

ines

-ch

ain

diag

ram

s fo

r cou

ntin

g

• Le

arne

rs s

houl

d be

giv

en a

rang

e of

exe

rcis

es

• A

rran

ge g

iven

num

bers

from

the

smal

lest

to th

e bi

gges

t: or

big

gest

to s

mal

lest

.

• Fi

ll in

mis

sing

num

bers

in

-a

sequ

ence

-on

a n

umbe

r grid

-on

a n

umbe

r lin

e

-fil

l in

<, =

or >

e.g

. 0,4

* 0

,04

* 0,

004

• C

ount

ing

exer

cise

s in

cha

in d

iagr

ams

can

be c

heck

ed u

sing

cal

cula

tors

and

le

arne

rs c

an e

xpla

in a

ny d

iffer

ence

s be

twee

n th

eir a

nsw

ers

and

thos

e sh

own

by

the

calc

ulat

or.

Cal

cula

ting

usin

g de

cim

al fr

actio

ns

• Le

arne

rs s

houl

d co

ntin

ue to

do

cont

ext f

ree

calc

ulat

ions

and

sol

ve p

robl

ems

in

cont

exts

.

• Le

arne

rs s

houl

d es

timat

e th

eir a

nsw

ers

befo

re c

alcu

latin

g, e

spec

ially

with

m

ultip

licat

ion

and

divi

sion

by

deci

mal

frac

tions

. The

y sh

ould

be

able

to ju

dge

the

reas

onab

lene

ss o

f ans

wer

s in

resp

ect o

f how

man

y de

cim

al p

lace

s an

d al

so

chec

k th

eir o

wn

answ

ers.

• m

ultip

licat

ion

by d

ecim

al fr

actio

ns s

houl

d st

art w

ith fa

mili

ar n

umbe

rs th

at

lear

ners

can

cal

cula

te b

y in

spec

tion,

so

that

lear

ners

get

a s

ense

of h

ow

deci

mal

pla

ces

are

affe

cted

by

mul

tiplic

atio

n.

Tota

l tim

e fo

r dec

imal

fra

ctio

ns:

6 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.5

dec

imal

fr

actio

ns

• E

xten

d di

visi

on to

'div

isio

n of

dec

imal

fra

ctio

ns b

y de

cim

al fr

actio

ns'

• C

alcu

late

the

squa

res,

cub

es, s

quar

e ro

ots

and

cube

root

s of

dec

imal

fra

ctio

ns

Cal

cula

tion

tech

niqu

es

• U

se k

now

ledg

e of

pla

ce v

alue

to

estim

ate

the

num

ber o

f dec

imal

pl

aces

in th

e re

sult

befo

re p

erfo

rmin

g ca

lcul

atio

ns

• U

se ro

undi

ng o

ff an

d a

calc

ulat

or to

ch

eck

resu

lts w

here

app

ropr

iate

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

invo

lvin

g de

cim

al fr

actio

ns

equi

vale

nt fo

rms

• R

evis

e eq

uiva

lent

form

s be

twee

n:

-co

mm

on fr

actio

n an

d de

cim

al

fract

ion

form

s of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

exam

ples

:

a)

3 x

2 =

6

0,3

x 2

= 0,

6

0,3

x 0,

2 =

0,06

0,3

x 0,

02 =

0,0

06

0,03

x 0

,02

= 0,

0006

etc

b)

15 x

3 =

45

1,5

x 3

= 4,

5

0,15

x 3

= 0

,45

0,15

x 0

,3 =

0,0

45

0,01

5 x

0,3

= 0,

0045

etc

• Fo

r div

isio

n by

dec

imal

frac

tions

with

out c

alcu

lato

rs, l

earn

ers

have

to u

se

thei

r kno

wle

dge

of m

ultip

licat

ion

by 1

0 or

mul

tiple

s of

10

to m

ake

the

divi

sor a

w

hole

num

ber.

Hen

ce s

tart

with

fam

iliar

num

bers

that

lear

ners

can

cal

cula

te b

y in

spec

tion,

so

that

lear

ners

get

a s

ense

of h

ow d

ecim

al p

lace

s ar

e af

fect

ed b

y di

visi

on.

exam

ples

:

a)

54 ÷

6 =

9

54 ÷

0,6

= 5

40 ÷

6 =

90

(mul

tiply

bot

h nu

mbe

rs b

y 10

to m

ake

the

deci

mal

frac

tion

a w

hole

num

ber)

54 ÷

0,0

6 =

5 40

0 ÷

6 =

900

(mul

tiply

bot

h nu

mbe

rs b

y 10

0 to

mak

e th

e de

cim

al fr

actio

n a

who

le n

umbe

r)

0,54

÷ 0

,06

= 54

÷ 6

= 9

(mul

tiply

bot

h nu

mbe

rs b

y 10

0 to

mak

e th

e de

cim

al fr

actio

n a

who

le n

umbe

r)

b)

125

÷ 5

= 25

125

÷ 0,

5 =

1 25

0 ÷

5 =

250

(mul

tiply

bot

h nu

mbe

rs b

y 10

to m

ake

the

deci

mal

frac

tion

a w

hole

num

ber)

125

÷ 0,

05 =

12

500

÷ 5

= 2

500

(mul

tiply

bot

h nu

mbe

rs b

y 10

0 to

mak

e th

e de

cim

al fr

actio

n a

who

le n

umbe

r)

1,25

÷ 0

,05

= 12

5 ÷

5 =

25

(mul

tiply

bot

h nu

mbe

rs b

y 10

0 to

mak

e th

e de

cim

al fr

actio

n a

who

le n

umbe

r


105CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.5

dec

imal

fr

actio

ns

• Fo

r big

ger a

nd u

nfam

iliar

dec

imal

frac

tions

, lea

rner

s sh

ould

use

cal

cula

tors

for

mul

tiplic

atio

n an

d di

visi

on, b

ut s

till j

udge

the

reas

onab

lene

ss o

f the

ir so

lutio

ns.

• S

imila

rly, fi

ndin

g sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

for d

ecim

al

fract

ions

sho

uld

star

t with

fam

iliar

num

bers

that

lear

ners

can

cal

cula

te b

y in

spec

tion.

exam

ples

:

a)

42 = 1

6

(0,4

)2 = 0

,4 x

0,4

= 0

,16

(0,0

4)2 =

0,0

4 x

0,04

= 0

,001

6

b)

(0,1

)3 = 0

,1 x

0,1

x 0

,1 =

0,0

01

c)

0,04

√

= 0

,2

• O

nce

lear

ners

are

com

forta

ble

with

all

the

oper

atio

ns u

sing

dec

imal

frac

tions

, ca

lcul

atio

ns s

houl

d no

t be

rest

ricte

d to

pos

itive

dec

imal

frac

tions

.

mea

sure

men

t4.

3

the

theo

rem

of

Pyth

agor

as

dev

elop

an

d us

e th

e th

eore

m

of

Pyth

agor

as

• In

vest

igat

e th

e re

latio

nshi

p be

twee

n th

e le

ngth

s of

the

side

s of

a ri

ght-

angl

ed tr

iang

le to

dev

elop

the

Theo

rem

of P

ytha

gora

s

• D

eter

min

e w

heth

er a

tria

ngle

is a

rig

ht-a

ngle

d tri

angl

e or

not

if th

e le

ngth

of t

he th

ree

side

s of

the

trian

gle

are

know

n

• U

se th

e Th

eore

m o

f Pyt

hago

ras

to

calc

ulat

e a

mis

sing

leng

th in

a ri

ght-

angl

ed tr

iang

le, l

eavi

ng ir

ratio

nal

answ

ers

in s

urd

form

• Th

e th

eore

m o

f Pyt

hago

ras

is n

ew in

Gra

de 8

.

• It

is im

porta

nt th

at le

arne

rs u

nder

stan

d th

at th

e Th

eore

m o

f Pyt

hago

ras

appl

ies

only

to ri

ght-a

ngle

d tri

angl

es.

• Th

e Th

eore

m o

f Pyt

hago

ras

is b

asic

ally

a fo

rmul

a to

cal

cula

te u

nkno

wn

leng

th

of s

ides

in ri

ght-a

ngle

d tri

angl

es.

• In

the

FET

phas

e, th

e Th

eore

m o

f Pyt

hago

ras

is c

ruci

al to

the

furth

er s

tudy

of

Geo

met

ry a

nd T

rigon

omet

ry

exam

ples

of s

olvi

ng p

robl

ems

usin

g th

e th

eore

m o

f Pyt

hago

ras:

• In

∆A

BC

, ∠B

= 9

0o , A

C =

4 c

m, B

C =

2 c

m. C

alcu

late

the

leng

th o

f AB

with

out

usin

g a

calc

ulat

or. L

eave

the

answ

er in

the

sim

ples

t sur

d fo

rm.

Tota

l tim

e fo

r th

e Th

eore

m

of P

ytha

gora

s:

5 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

4.1

are

a an

d pe

rimet

er o

f 2d

sha

pes

are

a an

d pe

rimet

er

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te

perim

eter

and

are

a of

:

-sq

uare

s

-re

ctan

gles

-tri

angl

es

-ci

rcle

s

• C

alcu

late

the

area

s of

pol

ygon

s,

to a

t lea

st 2

dec

imal

pla

ces,

by

deco

mpo

sing

them

into

rect

angl

es

and/

or tr

iang

les

• U

se a

nd d

escr

ibe

the

rela

tions

hip

betw

een

the

radi

us, d

iam

eter

an

d ci

rcum

fere

nce

of a

circ

le in

ca

lcul

atio

ns

• U

se a

nd d

escr

ibe

the

rela

tions

hip

betw

een

the

radi

us a

nd a

rea

of a

ci

rcle

in c

alcu

latio

ns

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s, w

ith o

r with

out a

ca

lcul

ator

, inv

olvi

ng p

erim

eter

and

ar

ea o

f pol

ygon

s an

d ci

rcle

s

• C

alcu

late

to a

t lea

st 2

dec

imal

pla

ces

• U

se a

nd d

escr

ibe

the

mea

ning

of

the

irrat

iona

l num

ber P

i (π)

in

calc

ulat

ions

invo

lvin

g ci

rcle

s

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

S

I uni

ts, i

nclu

ding

:

-m

m2

↔ cm

2 ↔

m2

↔ km

2

Wha

t is

diffe

rent

to G

rade

7?

• A

reas

of p

olyg

ons

by d

ecom

posi

tion

• C

ircum

fere

nce

and

area

of a

circ

le

•Fo

rmul

ae le

arne

rs s

houl

d kn

ow a

nd u

se:

-pe

rimet

er o

f a s

quar

e =

4s

-pe

rimet

er o

f a re

ctan

gle

= 2(

l + b

) or 2

l + 2

b

-ar

ea o

f a s

quar

e =

l2

-ar

ea o

f a re

ctan

gle

= l x

b

-ar

ea o

f a tr

iang

le =

1 2 (b

x h)

-di

amet

er o

f a c

ircle

: d =

2r

-ci

rcum

fere

nce

of c

ircle

: c =

πd

or 2

πr

-ar

ea o

f a c

ircle

: A =

πr2

solv

ing

equa

tions

usi

ng fo

rmul

ae

• Th

e us

e of

form

ulae

pro

vide

s a

cont

ext t

o pr

actis

e so

lvin

g eq

uatio

ns b

y in

spec

tion

or u

sing

add

itive

or m

ultip

licat

ive

inve

rses

.

exam

ples

:

1. I

f the

per

imet

er o

f a s

quar

e is

32

cm, w

hat i

s th

e le

ngth

of e

ach

side

? Le

arne

rs s

houl

d w

rite

this

as:

4s =

32

and

solv

e by

ask

ing:

4 ti

mes

wha

t will

be

32 O

R s

ayin

g s

= 32

4

?

2. If

the

area

of a

rect

angl

e is

200

cm

2 , an

d its

leng

th is

50

cm, w

hat i

s its

wid

th?

Lear

ners

sho

uld

writ

e th

is a

s:

50 x

b =

200

and

sol

ve b

y in

spec

tion

by a

skin

g: 5

0 tim

es w

hat w

ill b

e 20

0 O

R

sayi

ng b

= 20

0 50

?

For a

reas

of t

riang

les:

• m

ake

sure

lear

ners

kno

w th

at th

e he

ight

of a

tria

ngle

is a

line

seg

men

t dra

wn

from

any

ver

tex

perp

endi

cula

r to

the

oppo

site

sid

e.

Tota

l tim

e fo

r are

a an

d pe

rimet

er:

5 ho

urs.


107CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

4.1

are

a an

d pe

rimet

er o

f 2d

sha

pes

are

a an

d pe

rimet

er

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te

perim

eter

and

are

a of

:

-sq

uare

s

-re

ctan

gles

-tri

angl

es

-ci

rcle

s

• C

alcu

late

the

area

s of

pol

ygon

s,

to a

t lea

st 2

dec

imal

pla

ces,

by

deco

mpo

sing

them

into

rect

angl

es

and/

or tr

iang

les

• U

se a

nd d

escr

ibe

the

rela

tions

hip

betw

een

the

radi

us, d

iam

eter

an

d ci

rcum

fere

nce

of a

circ

le in

ca

lcul

atio

ns

• U

se a

nd d

escr

ibe

the

rela

tions

hip

betw

een

the

radi

us a

nd a

rea

of a

ci

rcle

in c

alcu

latio

ns

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s, w

ith o

r with

out a

ca

lcul

ator

, inv

olvi

ng p

erim

eter

and

ar

ea o

f pol

ygon

s an

d ci

rcle

s

• C

alcu

late

to a

t lea

st 2

dec

imal

pla

ces

• U

se a

nd d

escr

ibe

the

mea

ning

of

the

irrat

iona

l num

ber P

i (π)

in

calc

ulat

ions

invo

lvin

g ci

rcle

s

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

S

I uni

ts, i

nclu

ding

:

-m

m2

↔ cm

2 ↔

m2

↔ km

2

Wha

t is

diffe

rent

to G

rade

7?

• A

reas

of p

olyg

ons

by d

ecom

posi

tion

• C

ircum

fere

nce

and

area

of a

circ

le

•Fo

rmul

ae le

arne

rs s

houl

d kn

ow a

nd u

se:

-pe

rimet

er o

f a s

quar

e =

4s

-pe

rimet

er o

f a re

ctan

gle

= 2(

l + b

) or 2

l + 2

b

-ar

ea o

f a s

quar

e =

l2

-ar

ea o

f a re

ctan

gle

= l x

b

-ar

ea o

f a tr

iang

le =

1 2 (b

x h)

-di

amet

er o

f a c

ircle

: d =

2r

-ci

rcum

fere

nce

of c

ircle

: c =

πd

or 2

πr

-ar

ea o

f a c

ircle

: A =

πr2

solv

ing

equa

tions

usi

ng fo

rmul

ae

• Th

e us

e of

form

ulae

pro

vide

s a

cont

ext t

o pr

actis

e so

lvin

g eq

uatio

ns b

y in

spec

tion

or u

sing

add

itive

or m

ultip

licat

ive

inve

rses

.

exam

ples

:

1. I

f the

per

imet

er o

f a s

quar

e is

32

cm, w

hat i

s th

e le

ngth

of e

ach

side

? Le

arne

rs s

houl

d w

rite

this

as:

4s =

32

and

solv

e by

ask

ing:

4 ti

mes

wha

t will

be

32 O

R s

ayin

g s

= 32

4

?

2. If

the

area

of a

rect

angl

e is

200

cm

2 , an

d its

leng

th is

50

cm, w

hat i

s its

wid

th?

Lear

ners

sho

uld

writ

e th

is a

s:

50 x

b =

200

and

sol

ve b

y in

spec

tion

by a

skin

g: 5

0 tim

es w

hat w

ill b

e 20

0 O

R

sayi

ng b

= 20

0 50

?

For a

reas

of t

riang

les:

• m

ake

sure

lear

ners

kno

w th

at th

e he

ight

of a

tria

ngle

is a

line

seg

men

t dra

wn

from

any

ver

tex

perp

endi

cula

r to

the

oppo

site

sid

e.

Tota

l tim

e fo

r are

a an

d pe

rimet

er:

5 ho

urs.

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

4.1

are

a an

d Pe

rimet

er o

f 2d

sha

pes

exam

ple:

AD

is th

e he

ight

ont

o ba

se B

C o

f ∆A

BC

.

A D

B C

A D B

C

• P

oint

out

that

eve

ry tr

iang

le h

as b

ases

, eac

h w

ith a

rela

ted

heig

ht o

r alti

tude

.

•Fo

r con

vers

ions

, not

e:

-If

1cm

= 1

0 m

m th

en 1

cm

2 = 1

00 m

m2

-If

1m =

100

cm

then

1 m

2 = 1

0 00

0 cm

2

Circ

les

• m

ake

sure

lear

ners

can

iden

tify

the

cent

re, r

adiu

s, d

iam

eter

and

circ

umfe

renc

e of

the

circ

le.

• S

pend

tim

e in

vest

igat

ing

the

rela

tions

hip

betw

een

radi

us, c

ircum

fere

nce

and

diam

eter

, so

that

lear

ners

dev

elop

a s

ense

of w

here

the

irrat

iona

l num

ber P

i (π)

is

der

ived

from

.

• D

evel

op a

n un

ders

tand

ing

of π

, mak

ing

sure

lear

ners

und

erst

and

that

:

-π

repr

esen

ts th

e va

lue

of th

e ci

rcum

fere

nce

divi

ded

by th

e di

amet

er, f

or a

ny

circ

le

-π

is a

n irr

atio

nal n

umbe

r an

d is

giv

en a

s 3,

141

592

654

corr

ect t

o 9

deci

mal

pl

aces

on

the

calc

ulat

or

-22

7

or 3

,14

are

appr

oxim

ate

ratio

nal v

alue

s of

π in

eve

ryda

y us

e.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

4.2

surf

ace

area

an

d vo

lum

e of

3d

obj

ects

surf

ace

area

and

vol

ume

• U

se a

ppro

pria

te fo

rmul

ae to

cal

cula

te

the

surfa

ce a

rea,

vol

ume

and

capa

city

of:

-cu

bes

-re

ctan

gula

r pris

ms

-tri

angu

lar p

rism

s

• D

escr

ibe

the

inte

rrel

atio

nshi

p be

twee

n su

rface

are

a an

d vo

lum

e of

th

e ob

ject

s m

entio

ned

abov

e

Cal

cula

tions

and

sol

ving

pro

blem

s

• S

olve

pro

blem

s, w

ith o

r with

out a

ca

lcul

ator

, inv

olvi

ng s

urfa

ce a

rea,

vo

lum

e an

d ca

paci

ty

• U

se a

nd c

onve

rt be

twee

n ap

prop

riate

S

I uni

ts, i

nclu

ding

:

-m

m2

↔ cm

2 ↔

m2

↔ km

2

-m

m3

↔ cm

3 ↔

m3

-m

l (cm

3 ) ↔

l ↔ kl

Wha

t is

diffe

rent

to G

rade

7?

• S

urfa

ce a

rea

and

volu

me

of tr

iang

ular

pris

ms

•Fo

rmul

ae le

arne

rs s

houl

d kn

ow a

nd u

se:

-th

e vo

lum

e of

a p

rism

= th

e ar

ea o

f the

bas

e x

the

heig

ht

-th

e su

rface

are

a of

a p

rism

= th

e su

m o

f the

are

a of

all

its fa

ces

-th

e vo

lum

e of

a c

ube

= l3

-th

e vo

lum

e of

a re

ctan

gula

r pris

m =

l x

b x

h

-th

e vo

lum

e of

a tr

iang

ular

pris

m =

(1 2b x

h) x

hei

ght o

f pris

m

• Fo

r con

vers

ions

, not

e:

-if

1 cm

= 1

0 m

m th

en 1

cm

3 = 1

000

mm

3 and

-if

1 m

= 1

00 c

m th

en 1

m3 =

1 0

00 0

00 c

m3 o

r 106

cm3

-an

obj

ect w

ith a

vol

ume

of 1

cm

3 w

ill d

ispl

ace

exac

tly 1

ml o

f wat

er

-an

obj

ect w

ith a

vol

ume

of 1

m3 w

ill d

ispl

ace

exac

tly 1

kl o

f wat

er.

• E

mph

asiz

e th

at th

e am

ount

of s

pace

insi

de a

pris

m is

cal

led

its c

apac

ity; a

nd

the

amou

nt o

f spa

ce o

ccup

ied

by a

pris

m is

cal

led

its v

olum

e.

• In

vest

igat

e th

e ne

ts o

f cub

es a

nd re

ctan

gula

r pris

ms

in o

rder

to d

educ

e fo

rmul

ae fo

r cal

cula

ting

thei

r sur

face

are

as.

exam

ple

of s

olvi

ng p

robl

ems

invo

lvin

g su

rfac

e ar

ea a

nd v

olum

e:

• C

alcu

late

the

volu

me

and

surfa

ce a

rea

of th

e pr

ism

if A

B =

8 c

m, B

C =

6 c

m a

nd

CF

= 16

cm

.A D

C F

EB

Tota

l tim

e fo

r su

rface

are

a an

d vo

lum

e:

5 ho

urs


109CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

dat

a ha

ndlin

g5.

1C

olle

ct,

orga

nize

an

d su

mm

ariz

eda

ta

Col

lect

dat

a•

Pos

e qu

estio

ns re

latin

g to

soc

ial,

econ

omic

, and

env

ironm

enta

l iss

ues

• S

elec

t app

ropr

iate

sou

rces

for

the

colle

ctio

n of

dat

a (in

clud

ing

peer

s, fa

mily

, new

spap

ers,

boo

ks,

mag

azin

es)

• D

istin

guis

h be

twee

n sa

mpl

es a

nd

popu

latio

ns, a

nd s

ugge

st a

ppro

pria

te

sam

ples

for i

nves

tigat

ion

• D

esig

n an

d us

e si

mpl

e qu

estio

nnai

res

to a

nsw

er q

uest

ions

w

ith m

ultip

le c

hoic

e re

spon

ses

org

aniz

e an

d su

mm

ariz

e da

ta•

Org

aniz

e (in

clud

ing

grou

ping

whe

re

appr

opria

te) a

nd re

cord

dat

a us

ing

-ta

lly m

arks

-ta

bles

-st

em-a

nd-le

af d

ispl

ays

• G

roup

dat

a in

to in

terv

als

• S

umm

ariz

e da

ta u

sing

mea

sure

s of

ce

ntra

l ten

denc

y, in

clud

ing:

-m

ean

-m

edia

n -

mod

e

•

Sum

mar

ize

data

usi

ng m

easu

res

of

disp

ersi

on, i

nclu

ding

: -

rang

e -

extre

mes

Wha

t is

diffe

rent

to G

rade

7?

The

follo

win

g ar

e ne

w in

Gra

de 8

• ex

trem

es

• br

oken

line

gra

phs

• di

sper

sion

of d

ata

• er

ror a

nd b

ias

in d

ata

dat

a se

ts a

nd c

onte

xts

Lear

ners

sho

uld

be e

xpos

ed to

a v

arie

ty o

f con

text

s th

at d

eal w

ith s

ocia

l and

en

viro

nmen

tal i

ssue

s, a

nd s

houl

d w

ork

with

giv

en d

ata

sets

, rep

rese

nted

in

a va

riety

of w

ays,

that

incl

ude

big

num

ber r

ange

s, p

erce

ntag

es a

nd d

ecim

al

fract

ions

. Lea

rner

s sh

ould

then

pra

ctis

e or

gani

zing

and

sum

mar

izin

g th

is d

ata,

an

alys

ing

and

inte

rpre

ting

the

data

, and

writ

ing

a re

port

abou

t the

dat

a.

Com

plet

e a

data

cyc

leLe

arne

rs s

houl

d co

mpl

ete

at le

ast o

ne d

ata

cycl

e fo

r the

yea

r, st

artin

g w

ith p

osin

g th

eir o

wn

ques

tions

, sel

ectin

g th

e so

urce

s an

d m

etho

d fo

r col

lect

ing,

reco

rdin

g,

orga

nizi

ng, r

epre

sent

ing,

ana

lysi

ng, s

umm

ariz

ing,

inte

rpre

ting

and

repo

rting

the

data

. Cha

lleng

e le

arne

rs to

thin

k ab

out w

hat k

inds

of q

uest

ions

and

dat

a ne

ed to

be

col

lect

ed to

be

repr

esen

ted

on a

his

togr

am, a

pie

cha

rt, a

bar

gra

ph, o

r a li

ne

grap

h.

Tota

l tim

e fo

r co

llect

ing

and

orga

nizi

ng

data

:

4 ho

urs.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.5

rep

rese

nt

data

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy to

dis

play

and

inte

rpre

t da

ta in

clud

ing:

-ba

r gra

phs

and

doub

le b

ar g

raph

s

-hi

stog

ram

s w

ith

give

n an

d ow

n in

terv

als

-pi

e ch

arts

-br

oken

-line

gra

phs

rep

rese

ntin

g da

ta

• D

raw

ing

pie

char

ts to

repr

esen

t dat

a do

not

hav

e to

be

accu

rate

ly d

raw

n w

ith a

co

mpa

ss a

nd p

rotra

ctor

, etc

. Lea

rner

s ca

n us

e an

y ro

und

obje

ct to

dra

w a

circ

le,

then

div

ide

the

circ

le in

to h

alve

s an

d qu

arte

rs a

nd e

ight

hs if

nee

ded,

as

a gu

ide

to e

stim

ate

the

prop

ortio

ns o

f the

circ

le th

at n

eed

to b

e sh

own

to re

pres

ent t

he

data

. Wha

t is

impo

rtant

is th

at th

e va

lues

or p

erce

ntag

es a

ssoc

iate

d w

ith th

e da

ta a

re s

how

n pr

opor

tiona

lly o

n th

e pi

e ch

art.

• D

raw

ing,

read

ing

and

inte

rpre

ting

pie

char

ts is

a u

sefu

l con

text

to re

-vis

it eq

uiva

lenc

e be

twee

n fra

ctio

ns a

nd p

erce

ntag

es, e

.g. 2

5% o

f the

dat

a is

re

pres

ente

d by

a 1 4 s

ecto

r of t

he c

ircle

.

• It

is a

con

text

in w

hich

lear

ners

can

find

per

cent

ages

of w

hole

num

bers

e.g

. if

25%

of 3

00 le

arne

rs li

ke ru

gby,

how

man

y (a

ctua

l num

ber)

lear

ners

like

rugb

y?

• H

isto

gram

s ar

e us

ed to

repr

esen

t gro

uped

dat

a sh

own

in in

terv

als

on th

e ho

rizon

tal a

xis

of th

e gr

aph.

Poi

nt o

ut th

e di

ffere

nces

bet

wee

n hi

stog

ram

s an

d ba

r gra

phs,

in p

artic

ular

bar

gra

phs

that

repr

esen

t dis

cret

e da

ta e

.g. f

avou

rite

spor

ts, c

ompa

red

to h

isto

gram

s th

at s

how

cat

egor

ies

in c

onse

cutiv

e, n

on-

over

lapp

ing

inte

rval

s, e

.g. t

est s

core

s ou

t of 1

00 s

how

n in

inte

rval

s of

10.

The

ba

rs o

n ba

r gra

phs

do n

ot h

ave

to to

uch

each

oth

er, w

hile

in a

his

togr

am th

ey

have

to to

uch

sinc

e th

ey s

how

con

secu

tive

inte

rval

s.

• B

roke

n-lin

e gr

aphs

refe

r to

data

gra

phs

that

repr

esen

t dat

a po

ints

join

ed b

y a

line

and

are

not t

he s

ame

as s

traig

ht li

ne g

raph

s th

at a

re d

raw

n us

ing

the

equa

tion

of th

e lin

e.

• B

roke

n-lin

e gr

aphs

are

use

d to

repr

esen

t dat

a th

at c

hang

es c

ontin

uous

ly o

ver

time,

e.g

. ave

rage

dai

ly te

mpe

ratu

re fo

r a m

onth

. Eac

h da

y’s

tem

pera

ture

is

repr

esen

ted

with

a p

oint

on

the

grap

h, a

nd o

nce

the

who

le m

onth

has

bee

n pl

otte

d, th

e po

ints

are

join

ed to

sho

w a

bro

ken-

line

grap

h.

• B

roke

n-lin

e gr

aphs

are

use

ful t

o re

ad ‘t

rend

s’ a

nd p

atte

rns

in th

e da

ta, f

or

pred

ictiv

e pu

rpos

es e

.g. w

ill th

e te

mpe

ratu

res

go u

p or

dow

n in

the

next

mon

th.

Tota

l tim

e fo

r re

pres

entin

g da

ta:

3 ho

urs


111CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.6

inte

rpre

t, an

alys

e an

d re

port

dat

a

inte

rpre

t dat

a

• C

ritic

ally

read

and

inte

rpre

t dat

a re

pres

ente

d in

:

-w

ords

-ba

r gra

phs

-do

uble

bar

gra

phs

-pi

e ch

arts

-hi

stog

ram

s

-br

oken

-line

gra

phs

ana

lyse

dat

a

• C

ritic

ally

ana

lyse

dat

a by

ans

wer

ing

ques

tions

rela

ted

to:

-da

ta c

ateg

orie

s, in

clud

ing

data

in

terv

als

-da

ta s

ourc

es a

nd c

onte

xts

-ce

ntra

l ten

denc

ies

– (m

ean,

mod

e,

med

ian)

-sc

ales

use

d on

gra

phs

-sa

mpl

es a

nd p

opul

atio

ns

-di

sper

sion

of d

ata

-er

ror a

nd b

ias

in th

e da

ta

rep

ort d

ata

• S

umm

ariz

e da

ta in

sho

rt pa

ragr

aphs

th

at in

clud

e

-dr

awin

g co

nclu

sion

s ab

out t

he d

ata

-m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

-id

entif

ying

sou

rces

of e

rror

and

bi

as in

the

data

-ch

oosi

ng a

ppro

pria

te s

umm

ary

stat

istic

s fo

r the

dat

a (m

ean,

m

edia

n, m

ode,

rang

e)

-th

e ro

le o

f ext

rem

es in

the

data

dev

elop

ing

criti

cal a

naly

sis

skill

s

• Le

arne

rs s

houl

d co

mpa

re th

e sa

me

data

repr

esen

ted

in d

iffer

ent w

ays

e.g.

in a

pi

e ch

art o

r a b

ar g

raph

or a

tabl

e, a

nd d

iscu

ss w

hat i

nfor

mat

ion

is s

how

n an

d w

hat i

s hi

dden

; the

y sh

ould

eva

luat

e w

hat f

orm

of r

epre

sent

atio

n w

orks

bes

t for

th

e gi

ven

data

.

• Le

arne

rs s

houl

d co

mpa

re g

raph

s on

the

sam

e to

pic

but w

here

dat

a ha

s be

en

colle

cted

from

diff

eren

t gro

ups

of p

eopl

e, a

t diff

eren

t tim

es, i

n di

ffere

nt p

lace

s or

in d

iffer

ent w

ays.

Her

e le

arne

rs s

houl

d di

scus

s di

ffere

nces

bet

wee

n th

e da

ta

with

an

awar

enes

s of

bia

s re

late

d to

the

impa

ct o

f dat

a so

urce

s an

d m

etho

ds o

f da

ta c

olle

ctio

n on

the

inte

rpre

tatio

n of

the

data

.

• Le

arne

rs s

houl

d co

mpa

re d

iffer

ent w

ays

of s

umm

ariz

ing

the

sam

e da

ta s

ets,

de

velo

ping

an

awar

enes

s of

how

dat

a re

porti

ng c

an b

e m

anip

ulat

ed; t

hey

shou

ld e

valu

ate

whi

ch s

umm

ary

stat

istic

s be

st re

pres

ent t

he d

ata.

• Le

arne

rs s

houl

d co

mpa

re g

raph

s of

the

sam

e da

ta, w

here

the

scal

es o

f th

e gr

aphs

are

diff

eren

t. H

ere

lear

ners

sho

uld

disc

uss

diffe

renc

es w

ith a

n aw

aren

ess

of h

ow re

pres

enta

tion

of d

ata

can

be m

anip

ulat

ed; t

hey

shou

ld

eval

uate

whi

ch fo

rm o

f rep

rese

ntat

ion

wor

ks b

est f

or th

e gi

ven

data

.

• Le

arne

rs s

houl

d co

mpa

re d

ata

on th

e sa

me

topi

c, w

here

one

set

of d

ata

has

extre

mes

, and

dis

cuss

diff

eren

ces

with

an

awar

enes

s of

the

effe

ct o

f the

ex

trem

es o

n th

e in

terp

reta

tion

of th

e da

ta, i

n pa

rticu

lar,

extre

mes

affe

ct th

e ra

nge.

• Le

arne

rs s

houl

d w

rite

repo

rts o

n th

e da

ta in

sho

rt pa

ragr

aphs

.

Tota

l tim

e fo

r an

alys

ing,

in

terp

retin

g an

d su

mm

aris

ing

data

:

3,5

hour

s



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• ca

lcul

atin

g an

d so

lvin

g pr

oble

ms

with

com

mon

frac

tions

and

dec

imal

frac

tions

• th

e Th

eore

m o

f Pyt

hago

ras

• ar

ea a

nd p

erim

eter

of 2

D s

hape

s

• su

rface

are

a an

d vo

lum

e of

3D

obj

ects

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

6,5

hour

s.


113CAPS

Gr

ad

e 8

– te

rm

4

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.2

Func

tions

and

re

latio

nshi

ps

inpu

t and

out

put v

alue

s•

Det

erm

ine

inpu

t val

ues,

out

put

valu

es o

r rul

es fo

r pat

tern

s an

d re

latio

nshi

ps u

sing

: -

flow

dia

gram

s -

tabl

es -

form

ulae

-eq

uatio

nseq

uiva

lent

form

s•

Det

erm

ine,

inte

rpre

t and

just

ify

equi

vale

nce

of d

iffer

ent d

escr

iptio

ns

of th

e sa

me

rela

tions

hip

or ru

le

pres

ente

d: -

verb

ally

-in

flow

dia

gram

s -

in ta

bles

-by

form

ulae

-by

equ

atio

ns

Func

tions

and

rela

tions

hips

wer

e al

so d

one

in T

erm

1. I

n th

is te

rm th

e fo

cus

is o

n us

ing

form

ulae

to fi

nd o

utpu

t val

ues

from

giv

en in

put v

alue

s, a

s w

ell a

s eq

uiva

lent

form

s of

des

crip

tions

of t

he s

ame

rela

tions

hip.

see

furt

her n

otes

and

exa

mpl

es in

ter

m 1

. ex

ampl

eU

se th

e fo

rmul

a fo

r the

are

a of

a re

ctan

gle:

A =

l x

b to

cal

cula

te th

e fo

llow

ing:

a)

The

area

, if t

he le

ngth

is 4

,5 c

m a

nd th

e w

idth

is 2

,5 c

m

b)

The

leng

th, i

f the

are

a is

240

cm

2 and

the

wid

th is

4 c

mc)

Th

e w

idth

, if t

he a

rea

is 1

4 cm

2 and

the

leng

th is

3,5

cm

Lear

ners

can

writ

e th

ese

as n

umbe

r sen

tenc

es, a

nd s

olve

by

insp

ectio

n.

Tim

e fo

r Fu

nctio

ns a

nd

Rel

atio

nshi

ps

in th

is te

rm:

6 ho

urs

2.4

alg

ebra

iceq

uatio

ns

equa

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-se

t up

equa

tions

to d

escr

ibe

prob

lem

situ

atio

ns -

anal

yse

and

inte

rpre

t equ

atio

ns

that

des

crib

e a

give

n si

tuat

ion

-so

lve

equa

tions

by

insp

ectio

n -

dete

rmin

e th

e nu

mer

ical

val

ue o

f an

exp

ress

ion

by s

ubst

itutio

n -

Iden

tify

varia

bles

and

con

stan

ts in

gi

ven

form

ulae

or e

quat

ions

• E

xten

d so

lvin

g eq

uatio

ns to

incl

ude:

-us

ing

addi

tive

and

mul

tiplic

ativ

e in

vers

es

-us

ing

law

s of

exp

onen

ts

• U

se s

ubst

itutio

n in

equ

atio

ns to

ge

nera

te ta

bles

of o

rder

ed p

airs

Alg

ebra

ic e

quat

ions

wer

e al

so d

one

in T

erm

s 1

and

2. In

this

term

the

focu

s is

on

usi

ng s

ubst

itutio

n in

equ

atio

ns to

gen

erat

e ta

bles

of o

rder

ed p

airs

.se

e fu

rthe

r not

es a

nd e

xam

ples

in t

erm

s 1

and

2.ex

ampl

es o

f gen

erat

ing

orde

red

pairs

a)

Com

plet

e th

e ta

ble

belo

w fo

r x a

nd y

val

ues

for t

he e

quat

ion:

y =

–3x

+ 2

x–3

–10

y–4

–10

b)

Com

plet

e th

e ta

ble

belo

w fo

r x a

nd y

val

ues

for t

he e

quat

ion:

y =

x2 –

2

x–3

–20

y–2

2

Tim

e fo

r A

lgeb

raic

eq

uatio

ns in

th

is te

rm:

3 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.5

G

raph

s

inte

rpre

ting

grap

hs

• R

evis

e th

e fo

llow

ing

done

in G

rade

7:

-A

naly

se a

nd in

terp

ret g

loba

l gra

phs

of p

robl

em s

ituat

ions

, with

a s

peci

al

focu

s on

the

fol

low

ing

trend

s an

d fe

atur

es:

♦lin

ear o

r non

-line

ar

♦co

nsta

nt, i

ncre

asin

g or

de

crea

sing

• E

xten

d th

e fo

cus

on fe

atur

es o

f gr

aphs

to in

clud

e:

-m

axim

um o

r min

imum

-di

scre

te o

r con

tinuo

us

dra

win

g gr

aphs

• D

raw

glo

bal g

raph

s fro

m g

iven

de

scrip

tions

of a

pro

blem

situ

atio

n,

iden

tifyi

ng fe

atur

es li

sted

abo

ve

• U

se ta

bles

of o

rder

ed p

airs

to p

lot

poin

ts a

nd d

raw

gra

phs

on th

e C

arte

sian

pla

ne

Wha

t is

diffe

rent

to G

rade

7?

• N

ew fe

atur

es o

f glo

bal g

raph

s: m

axim

um a

nd m

inim

um; d

iscr

ete

and

cont

inuo

us

• P

lotti

ng p

oint

s to

dra

w g

raph

s

Exa

mpl

es o

f con

text

s fo

r glo

bal g

raph

s in

clud

e:

• th

e re

latio

nshi

p be

twee

n tim

e an

d di

stan

ce tr

avel

led

• th

e re

latio

nshi

p be

twee

n te

mpe

ratu

re a

nd ti

me

over

whi

ch it

is m

easu

red

• th

e re

latio

nshi

p be

twee

n ra

infa

ll an

d tim

e ov

er w

hich

it is

mea

sure

d, e

tc.

exam

ples

of d

raw

ing

grap

hs b

y pl

ottin

g po

ints

a)

Com

plet

e th

e ta

ble

of o

rder

ed p

airs

bel

ow fo

r the

equ

atio

n: y

= x

+ 3

x–4

–3–2

–1 0

12

34

y

Now

, plo

t the

abo

ve c

o-or

dina

te p

oint

s on

the

Car

tesi

an p

lane

. Joi

n po

ints

to

form

a g

raph

.

b)

Com

plet

e th

e ta

ble

of o

rder

ed p

airs

bel

ow fo

r the

equ

atio

n: y

= x

2 + 3

x–4

–3–2

–1 0

12

34

y

Now

, plo

t the

abo

ve c

o-or

dina

te p

oint

s on

the

Car

tesi

an p

lane

. Joi

n po

ints

to

form

a g

raph

.

Tota

l tim

e fo

r gr

aphs

:

9 ho

urs


115CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(Geo

met

ry)

3.4

tran

sfor

mat

ion

Geo

met

ry

tran

sfor

mat

ions

• R

ecog

nize

, des

crib

e an

d pe

rform

tra

nsfo

rmat

ions

with

poi

nts

on a

co

ordi

nate

pla

ne, f

ocus

ing

on:

-re

flect

ing

a po

int i

n th

e y-

axis

or

x-a

xis

-tra

nsla

ting

a po

int w

ithin

and

ac

ross

qua

dran

ts

• R

ecog

nize

, des

crib

e an

d pe

rform

tra

nsfo

rmat

ions

with

tria

ngle

s on

a

co-o

rdin

ate

plan

e, fo

cusi

ng o

n th

e co

-ord

inat

es o

f the

ver

tices

whe

n:

-re

flect

ing

a tri

angl

e in

the

x-a

xis

or

y-ax

is

-tra

nsla

ting

a tri

angl

e w

ithin

and

ac

ross

qua

dran

ts

-ro

tatin

g a

trian

gle

arou

nd th

e or

igin

enla

rgem

ents

and

redu

ctio

ns•

Use

pro

porti

on to

des

crib

e th

e ef

fect

of

enl

arge

men

t or r

educ

tion

on a

rea

and

perim

eter

of g

eom

etric

figu

res

Wha

t is

diffe

rent

to G

rade

7?

• Tr

ansf

orm

atio

ns a

re d

one

on a

co-

ordi

nate

pla

ne

• C

o-or

dina

tes

of p

oint

s an

d ve

rtice

s

Co-

ordi

nate

pla

ne•

Doi

ng tr

ansf

orm

atio

ns o

n th

e co

-ord

inat

e pl

ane

is a

n op

portu

nity

to p

ract

ise

plot

ting

poin

ts w

ith o

rder

ed p

airs

, and

link

s up

with

dra

win

g al

gebr

aic

grap

hs.

• L

earn

ers

have

to le

arn

how

to p

lot p

oint

s on

the

co-o

rdin

ate

plan

e an

d re

ad

the

co-o

rdin

ates

of p

oint

s of

f the

x-a

xis

and

y-ax

is. T

his

is a

lso

done

with

al

gebr

aic

grap

hs.

• Le

arne

rs h

ave

to k

now

the

conv

entio

n fo

r writ

ing

orde

red

pairs

(x;y

)

• P

oint

out

the

diffe

renc

es b

etw

een

the

axes

in th

e fo

ur q

uadr

ants

.

Focu

s of

tran

sfor

mat

ions

• D

oing

tran

sfor

mat

ions

on

a co

-ord

inat

e pl

ane

focu

ses

atte

ntio

n on

the

co-

ordi

nate

s of

poi

nts

and

verti

ces

of s

hape

s.

• Le

arne

rs s

houl

d re

cogn

ize

that

tran

slat

ions

, refl

ectio

ns a

nd ro

tatio

ns o

nly

chan

ge th

e po

sitio

n of

the

figur

e, a

nd n

ot it

s sh

ape

or s

ize.

• Le

arne

rs s

houl

d re

cogn

ize

that

the

abov

e tra

nsfo

rmat

ions

pro

duce

con

grue

nt

figur

es.

• Le

arne

rs d

o no

t hav

e to

lear

n ge

nera

l rul

es fo

r the

tran

sfor

mat

ions

at t

his

stag

e, b

ut s

houl

d ex

plor

e th

e w

ay th

e co

-ord

inat

es o

f poi

nts

chan

ge w

hen

perfo

rmin

g di

ffere

nt tr

ansf

orm

atio

ns w

ith li

nes

or s

hape

s.

• Le

arne

rs s

houl

d re

cogn

ize

that

enl

arge

men

ts a

nd re

duct

ions

cha

nge

the

size

of

figu

res

by in

crea

sing

or d

ecre

asin

g th

e le

ngth

of s

ides

, but

kee

ping

the

angl

es th

e sa

me,

pro

duce

s si

mila

r rat

her t

han

cong

ruen

t figu

res.

• Le

arne

rs s

houl

d al

so b

e ab

le to

wor

k ou

t the

fact

or o

f enl

arge

men

t or

redu

ctio

n of

a fi

gure

.ex

ampl

es o

f tra

nsfo

rmat

ion

prob

lem

s•

Plo

t poi

nt A

(4;3

) and

A',

its im

age,

afte

r refl

ectio

n in

:

a)

the

x-a

xis

b)

the

y-a

xis.

• W

rite

dow

n th

e co

-ord

inat

es o

f T' i

f T(–

2;3)

is tr

ansl

ated

4 u

nits

dow

nwar

ds.

• Th

e pe

rimet

er o

f squ

are

AB

CD

= 4

8cm

a)

Writ

e do

wn

the

perim

eter

of t

he s

quar

e if

the

leng

th o

f eac

h si

de is

do

uble

d.

b)

Will

the

area

of t

he e

nlar

ged

squa

re b

e tw

ice

or fo

ur ti

mes

that

of t

he

orig

inal

squ

are?

Tota

l tim

e fo

r Tr

ansf

orm

a-tio

ns:

6 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

3.2

Geo

met

ry o

f 3d

ob

ject

s

Cla

ssify

ing

3d o

bjec

ts

• D

escr

ibe,

nam

e an

d co

mpa

re th

e 5

Pla

toni

c so

lids

in te

rms

of th

e sh

ape

and

num

ber o

f fac

es, t

he n

umbe

r of

verti

ces

and

the

num

ber o

f edg

es

Bui

ldin

g 3d

mod

els

• R

evis

e us

ing

nets

to m

ake

mod

els

of

geom

etric

sol

ids,

incl

udin

g:

-cu

bes

-pr

ism

s

-py

ram

ids

Wha

t is

diffe

rent

to G

rade

7?

• N

amin

g an

d co

mpa

ring

Pla

toni

c so

lids

• N

ets

of p

yram

ids

Plat

onic

sol

ids

• P

lato

nic

solid

s ar

e a

spec

ial g

roup

of p

olyh

edra

that

hav

e fa

ces

that

are

co

ngru

ent r

egul

ar p

olyg

ons.

• Th

ere

are

only

5 P

lato

nic

solid

s:

-Te

trahe

dron

-H

exah

edro

n (c

ube)

-O

ctah

edro

n

-D

odec

ahed

ron

-Ic

osah

edro

ns

• Th

e na

me

of e

ach

Pla

toni

c so

lid is

der

ived

from

its

num

ber o

f fac

es.

• P

lato

nic

solid

s pr

ovid

e an

inte

rest

ing

cont

ext i

n w

hich

to in

vest

igat

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

r of f

aces

, ver

tices

and

edg

es. B

y lis

ting

thes

e pr

oper

ties

for a

ll th

e P

lato

nic

solid

s, le

arne

rs c

an in

vest

igat

e th

e pa

ttern

that

em

erge

s, to

com

e up

with

the

gene

ral r

ule:

V –

E +

F =

2, w

here

V =

num

ber

of v

ertic

es; E

= n

umbe

r of e

dges

; F =

num

ber o

f fac

es

usi

ng a

nd c

onst

ruct

ing

nets

• U

sing

and

con

stru

ctin

g ne

ts a

re u

sefu

l con

text

s fo

r exp

lorin

g or

con

solid

atin

g pr

oper

ties

of p

olyh

edra

.

• Le

arne

rs s

houl

d re

cogn

ize

the

nets

of d

iffer

ent s

olid

s.

• Le

arne

rs s

houl

d m

ake

sket

ches

of t

he n

ets

usin

g th

eir k

now

ledg

e of

the

shap

e an

d nu

mbe

r of f

aces

of t

he s

olid

s, b

efor

e dr

awin

g an

d cu

tting

out

the

nets

to b

uild

mod

els.

• S

ince

lear

ners

hav

e m

ore

know

ledg

e ab

out t

he s

ize

of a

ngle

s in

equ

ilate

ral

trian

gles

, and

can

mea

sure

ang

les,

thei

r con

stru

ctio

ns o

f net

s sh

ould

be

mor

e ac

cura

te.

• Le

arne

rs h

ave

to w

ork

out t

he re

lativ

e po

sitio

n of

face

s of

the

nets

in o

rder

to

build

the

3D m

odel

.

Tota

l tim

e fo

r ge

omet

ry o

f 3D

obj

ects

:

7 ho

urs


117CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

dat

a ha

ndlin

g5.

4 P

roba

bilit

yPr

obab

ility

• C

onsi

der a

sim

ple

situ

atio

n (w

ith

equa

lly li

kely

out

com

es) t

hat c

an b

e de

scrib

ed u

sing

pro

babi

lity

and:

-lis

t all

the

poss

ible

out

com

es

-de

term

ine

the

prob

abili

ty o

f eac

h po

ssib

le o

utco

me

usin

g th

e de

finiti

on o

f pro

babi

lity

-pr

edic

t, w

ith re

ason

s, th

e re

lativ

e fre

quen

cy o

f the

pos

sibl

e ou

tcom

es

for a

ser

ies

of tr

ials

bas

ed o

n pr

obab

ility

-co

mpa

re re

lativ

e fre

quen

cy w

ith

prob

abili

ty a

nd e

xpla

in p

ossi

ble

diffe

renc

es

Prob

abili

ty e

xper

imen

tsIn

the

Inte

rmed

iate

Pha

se a

nd G

rade

7 le

arne

rs d

id p

roba

bilit

y ex

perim

ents

with

co

ins,

dic

e an

d sp

inne

rs. I

n G

rade

8 d

oing

act

ual t

rials

of e

xper

imen

ts b

ecom

e le

ss im

porta

nt, a

nd le

arne

rs s

houl

d co

nsid

er p

roba

bilit

y fo

r hyp

othe

tical

eve

nts

e.g.

the

prob

abili

ty o

f whi

te a

s a

succ

essf

ul o

utco

me

on a

roul

ette

tabl

e, o

r the

pr

obab

ility

of g

ettin

g a

Coc

a C

ola

at th

e sh

op if

you

kno

w w

hat t

he to

tal n

umbe

r of

drin

ks is

that

they

sto

ck a

nd h

ow m

any

cans

of C

oca

Col

a th

ey h

ave.

Com

parin

g re

lativ

e fr

eque

ncy

and

prob

abili

ty

• Th

e re

lativ

e fre

quen

cy is

the

obse

rved

num

ber o

f suc

cess

ful o

utco

mes

for a

fin

ite s

ampl

e of

tria

ls.

• Fo

r exa

mpl

e, if

you

toss

a c

oin

50 ti

mes

, the

resu

lts a

re 2

7 he

ads

and

23 ta

ils.

Defi

ne a

hea

d as

a s

ucce

ssfu

l out

com

e. T

he re

lativ

e fre

quen

cy o

f hea

ds is

: 27

50

= 5

4%

• Th

e pr

obab

ility

of a

hea

d is

50%

(one

of t

wo

likel

y ou

tcom

es).

The

diffe

renc

e be

twee

n th

e re

lativ

e fre

quen

cy o

f 54%

and

the

prob

abili

ty o

f 50%

is d

ue to

sm

all s

ampl

e si

ze.

• Th

e m

ore

trial

s yo

u do

, the

clo

ser t

he re

lativ

e fre

quen

cy g

ets

to th

e pr

obab

ility

. Th

is c

an b

e co

mpa

red

in c

lass

by

com

bini

ng re

sults

from

tria

ls d

one

in g

roup

s or

pai

rs.

Tota

l tim

e fo

r pr

obab

ility

:

4,5

hour

s

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• fu

nctio

ns a

nd re

latio

nshi

ps

• al

gebr

aic

equa

tions

• gr

aphs

• tra

nsfo

rmat

ion

geom

etry

• ge

omet

ry o

f 3D

obj

ects

• pr

obab

ility

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

9, 5

hou

rs



time alloCation Per term: Grade 9



Whole numbers 4,5 hours

Construction of geometric figures

9 hours


5 hours

Transformation geometry

9 hours

Integers 4,5 hours


9 hours


9 hours

Geometry of 3D objects

9 hours

Common fractions 4,5 hours

Geometry of straight-lines

9 hours

Algebraic equations

9 hours

Collect, organize and summarize data

4 hours

Decimal fractions 4,5 hours

Theorem of Pythagoras

5 hours Graphs 12

hours Represent data 3 hours

Exponents 5 hours

Area and Perimeter of 2D shapes

5 hours

Surface Area and volume of 3D objects

5 hours


3,5 hours


4,5 hours Probability 4,5

hours


4 hours


4,5 hours

Algebraic equations

4 hours

revision/assessment

5 hours

revision/assessment

8 hours

revision/assessment

5 hours

revision/assessment

12 hours



119CAPS

3.3.3Clarifi

catio

nofcon

tentfo

rGrade9

Gr

ad

e 9

– te

rm

1

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, op

erat

ions

and

re

latio

nshi

ps

1.1

Who

le

num

bers

Prop

ertie

s of

num

bers

• D

escr

ibe

the

real

num

ber s

yste

m

by re

cogn

isin

g, d

efini

ng a

nd

dist

ingu

ishi

ng p

rope

rties

of:

-na

tura

l num

bers

-w

hole

num

bers

-in

tege

rs

-ra

tiona

l num

bers

-irr

atio

nal n

umbe

rs

Cal

cula

tions

usi

ng w

hole

num

bers

Rev

ise:

• C

alcu

latio

ns u

sing

all

four

ope

ratio

ns

on w

hole

num

bers

, est

imat

ing

and

usin

g ca

lcul

ator

s w

here

app

ropr

iate

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of s

trate

gies

to p

erfo

rm

and

chec

k w

ritte

n an

d m

enta

l ca

lcul

atio

ns o

f who

le n

umbe

rs

incl

udin

g:

• es

timat

ion

• ad

ding

, sub

tract

ing

and

mul

tiply

ing

in

colu

mns

• lo

ng d

ivis

ion

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• us

ing

a ca

lcul

ator

mul

tiple

s an

d fa

ctor

s

Use

prim

e fa

ctor

isat

ion

of n

umbe

rs to

fin

d LC

M a

nd H

CF

Wha

t is

diffe

rent

to G

rade

8?

In G

rade

9 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r w

hole

num

bers

, dev

elop

ed in

Gra

de 8

.

• Th

e fo

cus

in G

rade

9 s

houl

d be

on

deve

lopi

ng a

n un

ders

tand

ing

of d

iffer

ent

num

ber s

yste

ms

and

the

prop

ertie

s of

ope

ratio

ns th

at a

pply

for d

iffer

ent n

umbe

r sy

stem

s.

• Th

e co

ntex

ts fo

r sol

ving

pro

blem

s sh

ould

be

mor

e co

mpl

ex a

nd v

arie

d, in

volv

ing

who

le n

umbe

rs, i

nteg

ers

and

ratio

nal n

umbe

rs. F

inan

cial

con

text

s ar

e es

peci

ally

ric

h in

this

rega

rd.

• Le

arne

rs s

houl

d be

giv

en a

cle

ar in

dica

tion

of w

hen

the

use

of c

alcu

lato

rs is

pe

rmis

sibl

e or

not

. Cal

cula

tors

sho

uld

be u

sed

rout

inel

y fo

r cal

cula

tions

with

big

nu

mbe

rs a

nd w

here

kno

wle

dge

of n

umbe

r fac

ts o

r con

cept

s ar

e no

t exp

licitl

y as

sess

ed. H

owev

er, g

uard

aga

inst

lear

ners

bec

omin

g de

pend

ent o

n ca

lcul

ator

s fo

r all

calc

ulat

ions

. Cal

cula

tors

rem

ain

a us

eful

tool

for c

heck

ing

solu

tions

.

• C

ompe

tenc

y in

find

ing

mul

tiple

s an

d fa

ctor

s, a

nd p

rime

fact

oris

atio

n of

who

le

num

bers

, rem

ains

impo

rtant

for d

evel

opin

g co

mpe

tenc

y in

fact

oris

ing

alge

brai

c ex

pres

sion

s an

d so

lvin

g al

gebr

aic

equa

tions

.

Prop

ertie

s of

num

bers

• B

y di

stin

guis

hing

the

prop

ertie

s of

diff

eren

t num

ber s

yste

ms,

lear

ners

sho

uld

reco

gniz

e th

at n

atur

al n

umbe

rs is

a s

ubse

t of w

hole

num

bers

, whi

ch in

turn

is

a su

bset

of i

nteg

ers,

whi

ch in

turn

is a

sub

set o

f rat

iona

l num

bers

. All

of th

ese

num

bers

form

par

t of t

he re

al n

umbe

r sys

tem

.

• N

ote

that

0 m

ay s

omet

imes

be

incl

uded

in th

e se

t of n

atur

al n

umbe

rs.

• Le

arne

rs s

houl

d re

cogn

ize

the

follo

win

g di

stin

guis

hing

feat

ures

of t

he n

umbe

r sy

stem

s:

-in

tege

rs e

xten

d th

e na

tura

l and

who

le n

umbe

r sys

tem

s by

incl

udin

g th

e op

erat

ion

a –

b, w

here

a <

b.

-ra

tiona

l num

bers

ext

end

the

set o

f int

eger

s by

incl

udin

g th

e op

erat

ion

a b w

here

a <

b

-ra

tiona

l num

bers

are

defi

ned

as n

umbe

rs th

at c

an b

e w

ritte

n in

the

form

a b w

here

a a

nd b

are

inte

gers

and

b ≠

0

-si

nce

inte

gers

are

a s

ubse

t of r

atio

nal n

umbe

rs, e

very

inte

ger,

can

be

expr

esse

d as

a ra

tiona

l num

ber a b

Tota

l tim

e fo

r who

le

num

bers

4, 5

hou

rs.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, op

erat

ions

and

re

latio

nshi

ps

1.1

Who

le

num

bers

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

-ra

tio a

nd ra

te

-di

rect

and

indi

rect

pro

porti

on

-irr

atio

nal n

umbe

rs a

re n

umbe

rs th

at c

anno

t be

expr

esse

d as

ratio

nal

num

bers

in th

e fo

rm a b

-Pi

(π) i

s an

irra

tiona

l num

ber,

even

thou

gh w

e us

e 22

7

or 3

,14

as ra

tiona

l nu

mbe

r app

roxi

mat

ions

for π

in c

alcu

latio

ns

rat

io a

nd ra

te p

robl

ems

• In

clud

e pr

oble

ms

invo

lvin

g sp

eed,

dis

tanc

e an

d tim

e. L

earn

ers

shou

ld b

e fa

mili

ar w

ith th

e fo

llow

ing

form

ulae

for t

hese

cal

cula

tions

:

a)

spee

d =

dist

ance

tim

e

b)

dist

ance

= s

peed

x ti

me

c)

time

= di

stan

ce

spee

d

• S

peed

is u

sual

ly g

iven

as

cons

tant

spe

ed o

r ave

rage

spe

ed.

• M

ake

sure

lear

ners

reco

gniz

e an

d ar

e ab

le to

con

vert

corr

ectly

bet

wee

n un

its

for t

ime

and

dist

ance

.

exam

ples

a)

A ca

r tra

velli

ng a

t a c

onst

ant s

peed

trav

els

60 k

m in

18

min

utes

. How

far,

trave

lling

at t

he s

ame

cons

tant

spe

ed, w

ill th

e ca

r tra

vel i

n 1

hour

12

min

utes

?

b)

A ca

r tra

velli

ng a

t an

aver

age

spee

d of

100

km

/h c

over

s a

certa

in d

ista

nce

in

3 ho

urs

20 m

inut

es. A

t wha

t con

stan

t spe

ed m

ust t

he c

ar tr

avel

to c

over

the

sam

e di

stan

ce in

2 h

ours

40

min

utes

?

dire

ct a

nd in

dire

ct p

ropo

rtio

n

Lear

ners

sho

uld

be fa

mili

ar w

ith th

e fo

llow

ing

rela

tions

hips

:

• x

is d

irect

ly p

ropo

rtion

al to

y if

x y = c

onst

ant

• x

and

y ar

e di

rect

ly p

ropo

tiona

l if,

as th

e va

lue

of x

incr

ease

s th

e va

lue

of y

in

crea

ses

in th

e sa

me

prop

ortio

n, a

nd a

s th

e va

lue

of x

dec

reas

es th

e va

lue

of y

de

crea

ses

in th

e sa

me

prop

ortio

n

• Th

e di

rect

pro

porti

onal

rela

tions

hip

is re

pres

ente

d by

a s

traig

ht li

ne g

raph

• x

is in

dire

ctly

or i

nver

sely

pro

porti

onal

to y

if x

x y

= a

con

stan

t. In

oth

er w

ords

y

= c x

• x

and

y ar

e in

dire

ctly

pro

potio

nal i

f, as

the

valu

e of

x in

crea

ses

the

valu

e of

y

decr

ease

s an

d as

the

valu

e of

x d

ecre

ases

the

valu

e of

y in

crea

ses

• an

indi

rect

pro

porti

onal

rela

tions

hip

is re

pres

ente

d by

a n

on-li

near

cur

ve


121CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

num

bers

, op

erat

ions

and

re

latio

nshi

ps

1.1

Who

le

num

bers

• S

olve

pro

blem

s th

at in

volv

e w

hole

nu

mbe

rs, p

erce

ntag

es a

nd d

ecim

al

fract

ions

in fi

nanc

ial c

onte

xts

such

as

:

-pr

ofit,

loss

, dis

coun

t and

VAT

-bu

dget

s

-ac

coun

ts a

nd lo

ans

-si

mpl

e in

tere

st a

nd h

ire p

urch

ase

-ex

chan

ge ra

tes

and

com

mis

sion

-re

ntal

s

-co

mpo

und

inte

rest

Fina

ncia

l con

text

s

• O

nce

lear

ners

hav

e do

ne s

uffic

ient

cal

cula

tions

for s

impl

e an

d co

mpo

und

inte

rest

thro

ugh

repe

ated

cal

cula

tions

, the

y co

uld

use

give

n fo

rmul

ae fo

r the

se

calc

ulat

ions

.

exam

ples

a)

Cal

cula

te th

e si

mpl

e in

tere

st o

n R

600

at 7

% p

.a fo

r 3 y

ears

usi

ng th

e fo

rmul

a SI

= P.

n.r

100 o

r SI =

P.n

.i fo

r i =

r

10

0 .

b)

R80

0 in

vest

ed a

t r%

per

ann

um s

impl

e in

tere

st fo

r a p

erio

d of

3 y

ears

yi

elds

R16

8. C

alcu

late

the

valu

e of

r

c)

How

long

will

it ta

ke fo

r R3

000

inve

sted

at 6

% p

er a

nnum

sim

ple

inte

rest

to

grow

to R

4 26

0?

d)

Tem

oso

borr

owed

R50

0 fro

m th

e ba

nk fo

r 3 y

ears

at 8

% p

.a. c

ompo

und

inte

rest

. With

out u

sing

a fo

rmul

a, c

alcu

late

how

muc

h Te

mos

o ow

es th

e ba

nk a

t the

end

of t

hree

yea

rs.

e)

Use

the

form

ula

A =

P(1

+

r

100 )n to

cal

cula

te th

e co

mpo

und

inte

rest

on

a lo

an o

f R3

450

at 6

,5%

per

ann

um fo

r 5 y

ears

.

1.3

inte

gers

Cal

cula

tions

with

inte

gers

• R

evis

e:

-pe

rform

cal

cula

tions

invo

lvin

g al

l fo

ur o

pera

tions

with

inte

gers

-pe

rform

cal

cula

tions

invo

lvin

g al

l fo

ur o

pera

tions

with

num

bers

that

in

volv

e th

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of in

tege

rs

Prop

ertie

s of

inte

gers

• R

evis

e:

-co

mm

utat

ive,

ass

ocia

tive

and

dist

ribut

ive

prop

ertie

s of

add

ition

an

d m

ultip

licat

ion

for i

nteg

ers

-ad

ditiv

e an

d m

ultip

licat

ive

inve

rses

fo

r int

eger

s

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

mul

tiple

ope

ratio

ns w

ith in

tege

rs

Wha

t is

diffe

rent

to G

rade

8?

In G

rade

9 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r in

tege

rs, d

evel

oped

in G

rade

8.

In G

rade

9, l

earn

ers

wor

k w

ith in

tege

rs m

ostly

as

coef

ficie

nts

in a

lgeb

raic

ex

pres

sion

s an

d eq

uatio

ns. T

hey

are

expe

cted

to b

e co

mpe

tent

in p

erfo

rmin

g al

l fo

ur o

pera

tions

with

inte

gers

and

usi

ng th

e pr

oper

ties

of in

tege

rs a

ppro

pria

tely

w

here

nec

essa

ry.

Tota

l tim

e fo

r In

tege

rs:

4,5

hour

s



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.4

Com

mon

fr

actio

ns

Cal

cula

tions

usi

ng fr

actio

ns

• A

ll fo

ur o

pera

tions

with

com

mon

fra

ctio

ns a

nd m

ixed

num

bers

• A

ll fo

ur o

pera

tions

, with

num

bers

that

in

volv

e th

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of c

omm

on

fract

ions

Cal

cula

tion

tech

niqu

es

• R

evis

e:

-C

onve

rt m

ixed

num

bers

to

com

mon

frac

tions

in o

rder

to

perfo

rm c

alcu

latio

ns w

ith th

em

-U

se k

now

ledg

e of

mul

tiple

s an

d fa

ctor

s to

writ

e fra

ctio

ns in

the

sim

ples

t for

m b

efor

e or

afte

r ca

lcul

atio

ns

-U

se k

now

ledg

e of

equ

ival

ent

fract

ions

to a

dd a

nd s

ubtra

ct

com

mon

frac

tions

-U

se k

now

ledg

e of

reci

proc

al

rela

tions

hips

to d

ivid

e co

mm

on

fract

ions

solv

ing

prob

lem

s

Sol

ve p

robl

ems

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, m

ixed

num

bers

and

pe

rcen

tage

s

equi

vale

nt fo

rms

• R

evis

e eq

uiva

lent

form

s be

twee

n:

-co

mm

on fr

actio

ns w

here

one

de

nom

inat

or is

a m

ultip

le o

f an

othe

r

-co

mm

on fr

actio

n an

d de

cim

al

fract

ion

form

s of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

Wha

t is

diffe

rent

to G

rade

8?

In G

rade

9 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r co

mm

on fr

actio

ns, d

evel

oped

in G

rade

8.

In G

rade

9, l

earn

ers

wor

k w

ith c

omm

on fr

actio

ns m

ostly

as

coef

ficie

nts

in a

lgeb

ra-

ic e

xpre

ssio

ns a

nd e

quat

ions

. The

y ar

e ex

pect

ed to

be

com

pete

nt in

per

form

ing

mul

tiple

ope

ratio

ns u

sing

com

mon

frac

tions

and

mix

ed n

umbe

rs, a

pply

ing

prop

er-

ties

of ra

tiona

l num

bers

app

ropr

iate

ly. T

hey

are

also

exp

ecte

d to

reco

gniz

e an

d us

e eq

uiva

lent

form

s fo

r com

mon

frac

tions

app

ropr

iate

ly in

cal

cula

tions

and

whe

n si

mpl

ifyin

g al

gebr

aic

fract

ions

.

Tota

l tim

e fo

r com

mon

fra

ctio

ns:

4,5

hour

s


123CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.5

dec

imal

fr

actio

ns

Cal

cula

tions

with

dec

imal

frac

tions

• m

ultip

le o

pera

tions

with

dec

imal

fra

ctio

ns, u

sing

a c

alcu

lato

r whe

re

appr

opria

te

• m

ultip

le o

pera

tions

, with

or w

ithou

t br

acke

ts, w

ith n

umbe

rs th

at in

volv

e th

e sq

uare

s, c

ubes

, squ

are

root

s an

d cu

be ro

ots

of d

ecim

al fr

actio

ns

Cal

cula

tion

tech

niqu

es

• U

se k

now

ledg

e of

pla

ce v

alue

s to

es

timat

e th

e nu

mbe

r of d

ecim

al

plac

es in

the

resu

lt be

fore

per

form

ing

calc

ulat

ions

• U

se ro

undi

ng o

ff an

d a

calc

ulat

or to

ch

eck

resu

lts w

here

app

ropr

iate

solv

ing

prob

lem

s

Sol

ve p

robl

ems

in c

onte

xt in

volv

ing

deci

mal

frac

tions

equi

vale

nt fo

rms

• R

evis

e eq

uiva

lent

form

s be

twee

n:

-co

mm

on fr

actio

n an

d de

cim

al

fract

ion

form

s of

the

sam

e nu

mbe

r

-co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

Wha

t is

diffe

rent

to G

rade

8?

In G

rade

9 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r de

cim

al fr

actio

ns, d

evel

oped

in G

rade

8.

In G

rade

9, l

earn

ers

wor

k w

ith d

ecim

al fr

actio

ns m

ostly

as

coef

ficie

nts

in a

lgeb

raic

ex

pres

sion

s an

d eq

uatio

ns. T

hey

are

expe

cted

to b

e co

mpe

tent

in p

erfo

rmin

g m

ultip

le o

pera

tions

usi

ng d

ecim

al fr

actio

ns a

nd m

ixed

num

bers

, app

lyin

g pr

oper

-tie

s of

ratio

nal n

umbe

rs a

ppro

pria

tely.

The

y ar

e al

so e

xpec

ted

to re

cogn

ize

and

use

equi

vale

nt fo

rms

for d

ecim

al fr

actio

ns a

ppro

pria

tely

in c

alcu

latio

ns.

Tota

l tim

e fo

r dec

imal

fra

ctio

ns:

4,5

hour

s



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.2

expo

nent

sC

ompa

ring

and

repr

esen

ting

num

bers

in e

xpon

entia

l for

m

• R

evis

e:

-co

mpa

re a

nd r

epre

sent

inte

gers

in

expo

nent

ial f

orm

-co

mpa

re a

nd re

pres

ent n

umbe

rs in

sc

ient

ific

nota

tion

• E

xten

d sc

ient

ific

nota

tion

to in

clud

e ne

gativ

e ex

pone

nts

Cal

cula

tions

usi

ng n

umbe

rs in

ex

pone

ntia

l for

m

• R

evis

e th

e fo

llow

ing

gene

ral l

aws

of

expo

nent

s:

-am

x a

n = a

m +

n

-am

÷ a

n = a

m -

n , if

m>n

-(a

m)n =

am

x n

-(a

x t)

n = a

n x

t n

-a0 =

1

• E

xten

d th

e ge

nera

l law

s of

exp

onen

ts

to in

clud

e:

-in

tege

r exp

onen

ts

-a–m

=

1 am

• P

erfo

rm c

alcu

latio

ns in

volv

ing

all

four

ope

ratio

ns u

sing

num

bers

in

expo

nent

ial f

orm

solv

ing

prob

lem

s

• S

olve

pro

blem

s in

con

text

s in

volv

ing

num

bers

in e

xpon

entia

l for

m,

incl

udin

g sc

ient

ific

nota

tion

Wha

t is

diffe

rent

to G

rade

8?

• A

dditi

onal

law

s of

exp

onen

ts in

volv

ing

inte

ger e

xpon

ents

• S

cien

tific

nota

tion

for n

umbe

rs, i

nclu

ding

neg

ativ

e ex

pone

nts

In G

rade

9 le

arne

rs c

onso

lidat

e nu

mbe

r kno

wle

dge

and

calc

ulat

ion

tech

niqu

es fo

r ex

pone

nts,

dev

elop

ed in

Gra

de 8

.

law

s of

exp

onen

ts

• Th

e la

ws

of e

xpon

ents

sho

uld

be in

trodu

ced

thro

ugh

a ra

nge

of n

umer

ic

exam

ples

firs

t, th

en v

aria

bles

can

be

used

.

• Th

e fo

llow

ing

law

s of

exp

onen

ts s

houl

d be

kno

wn,

whe

re m

and

n a

re in

tege

rs

and

a an

d t a

re n

ot e

qual

to 0

:

am x

an =

am

+nam

÷ a

n = a

m -

n

exam

ples

exam

ples

a)

23 x 2

4 = 2

3+4 =

27 =

128

a) 3

5 ÷ 3

2 = 3

3 = 2

7

b)

x3 x x

4 = x

3+4 =

x7

b) x

5 ÷ x

3 = x

2

(am)n =

am

x n

(a x

t)n =

an x

t n

exam

ples

exam

ples

a)

(23 )2 =

26 =

64

a) (

3x2 )3 =

33 x

6 = 2

7x6

a0 = 1

a–m =

1 am

exam

ples

exam

ples

a)

(37)

0 = 1

a)

5–3

= 1 53

= 1 75

b)

(4x2 )0 =

1

b) 7

3 ÷ 7

5 = 7

–2 =

1 72 =

1 49

• m

ake

sure

lear

ners

und

erst

and

thes

e la

ws

read

ing

from

bot

h si

des

of th

e eq

ual

sign

i.e.

if th

e LH

S =

RH

S, t

hen

the

RH

S =

LH

S

Tota

l tim

e fo

r ex

pone

nts:

5 ho

urs


125CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

• Th

e la

w a

0 = 1

can

be

deriv

ed b

y us

ing

the

law

of e

xpon

ents

for d

ivis

ion

in a

few

ex

ampl

es e

.g. a

4 ÷

a4 =

a x

a x

a x

a a

x a

x a

x a =

1, t

here

fore

a4–

4 = a0 =

1

• Le

arne

rs s

houl

d be

abl

e to

use

the

law

s of

exp

onen

ts in

cal

cula

tions

and

for

solv

ing

sim

ple

expo

nent

ial e

quat

ions

as

wel

l as

expa

ndin

g an

d si

mpl

ifyin

g al

gebr

aic

expr

essi

ons.

• Lo

ok o

ut fo

r the

follo

win

g co

mm

on m

isco

ncep

tions

whe

re:

-Le

arne

rs m

ultip

ly u

nlik

e ba

ses

and

add

the

expo

nent

s

Exa

mpl

e: x

m x

yn =

(xy)

m +

n

25 x

27 = 4

9 inst

ead

of th

e co

rrec

t ans

wer

29

-Le

arne

rs fo

rget

the

mid

dle

term

of t

he b

inom

ial e

.g. (

x +

y)m

= x

m +

ym

-Le

arne

rs c

onfu

se a

ddin

g th

e ex

pone

nts

and

addi

ng th

e te

rms,

e.g

. xm

+ x

n =

xm +

n

-Le

arne

rs c

onfu

se th

e ex

pone

nt o

f the

var

iabl

e an

d th

e co

effic

ient

e.g

. 2x

–3 =

1

2x3 i

nste

ad o

f the

cor

rect

ans

wer

2 1

x2

Cal

cula

tions

and

sim

ple

equa

tions

usi

ng n

umbe

rs in

exp

onen

tial f

orm

• Th

e ca

lcul

atio

ns a

nd e

quat

ions

sho

uld

prov

ide

oppo

rtuni

ties

to a

pply

the

law

s of

ex

pone

nts

and

shou

ld n

ot b

e un

duly

com

plex

.

exam

ples

a)

Cal

cula

te: 2

–1 x

63 x

3–2

b)

Sim

plify

: (–2

x2 )(–2

x)–2

c)

Sol

ve x

: 3x =

9

d)

Sol

ve x

: 2x =

1 4

e)

5x+1 =

1

Scientificno

tatio

n

• W

hen

writ

ing

num

bers

in s

cien

tific

nota

tion,

lear

ners

hav

e to

und

erst

and

the

rela

tions

hip

betw

een

the

num

ber o

f dec

imal

pla

ces

and

the

inde

x of

10.

exam

ple:

25

= 2,

5 x

101 ;

and

250

= 2,

5 x

102

• S

cien

tific

nota

tion

that

ext

ends

to n

egat

ive

expo

nent

s in

clud

es w

ritin

g ve

ry s

mal

l nu

mbe

rs in

sci

entifi

c no

tatio

n.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

1.2

expo

nent

sex

ampl

e: 2

5 m

illio

nth

= 2,

5 x

10–5

• Le

arne

rs p

ract

ise

writ

ing

smal

l and

larg

e nu

mbe

rs in

sci

entifi

c no

tatio

n, w

hich

th

ey m

ight

alre

ady

have

enc

ount

ered

in N

atur

al S

cien

ce. I

t is

usef

ul to

refe

r to

thes

e co

ntex

ts w

hen

disc

ussi

ng s

cien

tific

nota

tion.

• C

alcu

latio

ns c

an b

e do

ne w

ith o

r with

out a

cal

cula

tor.

exam

ples

a)

Cal

cula

te: 2

,6 x

105 x

9 x

107

with

out u

sing

a c

alcu

lato

r and

giv

e an

swer

in

scie

ntifi

c no

tatio

n.

b)

Writ

e in

sci

entifi

c no

tatio

n: 0

,000

53

c)

Cal

cula

te: 5

,8 x

10–4

+ 2

,3 x

10–5

with

out u

sing

a c

alcu

lato

r.

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

an

d ge

omet

ric p

atte

rns

look

ing

for

rela

tions

hips

bet

wee

n nu

mbe

rs

incl

udin

g pa

ttern

s:

-re

pres

ente

d in

phy

sica

l or

diag

ram

fo

rm

-no

t lim

ited

to s

eque

nces

invo

lvin

g a

cons

tant

diff

eren

ce o

r rat

io

-of

lear

ner’s

ow

n cr

eatio

n

-re

pres

ente

d in

tabl

es

-re

pres

ente

d al

gebr

aica

lly

• D

escr

ibe

and

just

ify th

e ge

nera

l rul

es

for o

bser

ved

rela

tions

hips

bet

wee

n nu

mbe

rs in

ow

n w

ords

or i

n al

gebr

aic

lang

uage

Wha

t is

diffe

rent

to G

rade

8?

Lear

ners

con

solid

ate

wor

k in

volv

ing

num

eric

and

geo

met

ric p

atte

rns

done

in

Gra

de 8

.

• In

vest

igat

ing

num

ber p

atte

rns

is a

n op

portu

nity

to g

ener

aliz

e –

to g

ive

gene

ral

alge

brai

c de

scrip

tions

of t

he re

latio

nshi

p be

twee

n te

rms

and

thie

r pos

ition

in a

se

quen

ce a

nd to

just

ify s

olut

ions

.

Kin

ds o

f num

eric

pat

tern

s

• G

iven

a s

eque

nce

of n

umbe

rs, l

earn

ers

have

to id

entif

y a

patte

rn o

r re

latio

nshi

p be

twee

n co

nsec

utiv

e te

rms

in o

rder

to e

xten

d th

e pa

ttern

.

exam

ples

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

the

sequ

ence

s be

low

. Use

this

rule

to g

ive

the

next

thre

e nu

mbe

rs in

the

sequ

ence

:

a)

–1; –

1,5;

–2;

–2,

5 ...

...

Her

e le

arne

rs s

houl

d id

entif

y th

e co

nsta

nt d

iffer

ence

bet

wee

n co

nsec

utiv

e te

rms

in o

rder

to e

xten

d th

e pa

ttern

. Thi

s pa

ttern

can

be

desc

ribed

in

lear

ners

’ ow

n w

ords

as

‘add

ing

–0,5

’ or ‘

coun

ting

in –

0,5s

’ or

‘add

–0,

5 to

th

e pr

evio

us n

umbe

r in

the

patte

rn’.

b)

2; –

1; 0

,5; –

0,25

; 0,

125

... ..

.

Her

e le

arne

rs s

houl

d id

entif

y th

e co

nsta

nt ra

tio b

etw

een

cons

ecut

ive

term

s. T

his

patte

rn c

an b

e de

scrib

ed in

lear

ners

’ ow

n w

ords

as

‘mul

tiply

the

prev

ious

num

ber b

y –0

,5 ’.

Tota

l tim

e fo

r nu

mer

ic a

nd

geom

etric

pa

ttern

s

4, 5

hou

rs


127CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

c)

1; 0

; –2;

–5;

–9;

–14

... .

This

pat

tern

has

nei

ther

a c

onst

ant d

iffer

ence

nor

con

stan

t rat

io. T

his

patte

rn c

an b

e de

scrib

ed in

lear

ners

’ ow

n w

ords

as

‘sub

tract

1 m

ore

than

w

as s

ubtra

cted

to g

et th

e pr

evio

us te

rm’.

Usi

ng th

is ru

le, t

he n

ext 3

term

s w

ill b

e –2

0, –

27, –

35

• G

iven

a s

eque

nce

of n

umbe

rs, l

earn

ers

have

to id

entif

y a

patte

rn o

r re

latio

nshi

p be

twee

n th

e te

rm a

nd it

s po

sitio

n in

the

sequ

ence

. Thi

s en

able

s le

arne

rs to

pre

dict

a te

rm in

a s

eque

nce

base

d on

the

posi

tion

of th

at

term

in th

e se

quen

ce. I

t is

usef

ul fo

r lea

rner

s to

repr

esen

t the

se s

eque

nces

in

tabl

es s

o th

at th

ey c

an c

onsi

der t

he p

ositi

on o

f the

term

.

exam

ples

a)

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

this

se

quen

ce:

2; 5

; 10;

17

... .

Use

you

r rul

e to

find

the

10th

term

in th

is

sequ

ence

.

Firs

tly, l

earn

ers

have

to u

nder

stan

d th

at th

e ‘1

0th

term

’ ref

ers

to p

ositi

on 1

0 in

th

e nu

mbe

r seq

uenc

e. T

hey

have

to fi

nd a

rule

in o

rder

to d

eter

min

e th

e 10

th

term

, rat

her t

han

cont

inui

ng th

e se

quen

ce u

p to

the

tent

h te

rm.

This

seq

uenc

e ca

n be

repr

esen

ted

in th

e fo

llow

ing

tabl

e:

Pos

ition

in s

eque

nce

12

34

10

Term

25

1017

?

Lear

ners

hav

e to

reco

gniz

e th

at e

ach

term

in th

e bo

ttom

row

is o

btai

ned

by s

quar

ing

the

posi

tion

num

ber i

n th

e to

p ro

w a

nd a

ddin

g. T

hus

the

10th

te

rm w

ill b

e '1

0 sq

uare

d +1

’ or 1

02 + 1

whi

ch is

101

. Usi

ng th

e sa

me

rule

, le

arne

rs c

an a

lso

be a

sked

wha

t ter

m n

umbe

r or p

ositi

on w

ill b

e 62

6? If

the

term

is o

btai

ned

by s

quar

ing

the

posi

tion

num

ber o

f the

term

and

add

ing

1,

then

the

posi

tion

num

ber c

an b

e ob

tain

ed b

y su

btra

ctin

g, th

en fi

ndin

g th

e sq

uare

root

of t

he te

rm. H

ence

, 626

will

be

the

25th

term

in th

e se

quen

ce

sinc

e 62

6 –

1 =

625

and

625

√ =

25.

b)

Pro

vide

a ru

le to

des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

this

se

quen

ce: –

2; –

5;–8

; –11

... .

Use

this

rule

to fi

nd th

e 20

th te

rm in

this

se

quen

ce.

If le

arne

rs c

onsi

der o

nly

the

rela

tions

hip

betw

een

cons

ecut

ive

term

s, th

en

they

can

con

tinue

the

patte

rn (‘

add

-3 to

pre

viou

s nu

mbe

r’) u

p to

the

20th

te

rm to

find

the

answ

er. H

owev

er, i

f the

y lo

ok fo

r a re

latio

nshi

p or

rule

be

twee

n th

e te

rm a

nd th

e po

sitio

n of

the

term

, the

y ca

n pr

edic

t the

ans

wer

w

ithou

t con

tinui

ng th

e pa

ttern

. Usi

ng n

umbe

r sen

tenc

es c

an b

e us

eful

to

find

the

rule

:



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

1st t

erm

: –2

= –3

(1) +

1

2nd

term

: –5

= –3

(2) +

1

3rd

term

: –8

= –3

(3) +

1

4th

term

: –11

= –

3(4)

+ 1

The

num

ber i

n th

e br

acke

ts c

orre

spon

ds to

the

posi

tion

of th

e te

rm. H

ence

, th

e 20

th te

rm w

ill b

e: –

3(20

) + 1

= –

59

The

rule

in le

arne

rs’ o

wn

wor

ds c

an b

e w

ritte

n as

–3

x th

e po

sitio

n of

the

term

+ 1

’ or –

3n +

1, w

here

n is

the

posi

tion

of th

e te

rm.

• Th

ese

type

s of

num

eric

pat

tern

s de

velo

p an

und

erst

andi

ng o

f fun

ctio

nal

rela

tions

hips

, in

whi

ch th

ere

is a

dep

ende

nt v

aria

ble

(pos

ition

of t

he te

rm) a

nd

an in

depe

nden

t var

iabl

e (th

e te

rm it

self)

, and

whe

re y

ou h

ave

a un

ique

out

put

for a

ny g

iven

inpu

t val

ue.

Kin

ds o

f geo

met

ric p

atte

rns

• G

eom

etric

pat

tern

s ar

e nu

mbe

r pat

tern

s re

pres

ente

d di

agra

mm

atic

ally.

The

di

agra

mm

atic

repr

esen

tatio

n re

veal

s th

e st

ruct

ure

of th

e nu

mbe

r pat

tern

.

• H

ence

, rep

rese

ntin

g th

e nu

mbe

r pat

tern

s in

tabl

es, m

akes

it e

asie

r for

lear

ners

to

des

crib

e th

e ge

nera

l rul

e fo

r the

pat

tern

.

exam

ple

Con

side

r th

is p

atte

rn f

or b

uild

ing

hexa

gons

usi

ng m

atch

stic

ks.

How

man

y m

atch

stic

ks w

ill b

e us

ed t

o bu

ild t

he 1

0th

hexa

gon?

Pro

vide

an

expr

essi

on t

o de

scrib

e th

e ge

nera

l ter

m fo

r thi

s nu

mbe

r seq

uenc

e.

The

rule

for t

he p

atte

rn is

con

tain

ed in

the

stru

ctur

e (c

onst

ruct

ion)

of t

he

succ

essi

ve h

exag

onal

sha

pes:

(1)

add

on 1

mat

chst

ick

per s

ide

(2)

ther

e ar

e 6

side

s, s

o

(3)

add

on 6

mat

chst

icks

per

hex

agon

as

you

proc

eed

from

a g

iven

hex

agon

to

the

next

one

.

So,

for t

he 2

nd h

exag

on, y

ou h

ave

2 x

6 m

atch

es; f

or th

e 3r

d he

xago

n yo

u ha

ve

3 x

6 m

atch

es. U

sing

this

pat

tern

for b

uild

ing

hexa

gons

, the

10t

h he

xago

n w

ill

have

10

x 6m

atch

es.


129CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns a

nd

alge

bra

2.1

num

eric

and

ge

omet

ric

patte

rns

Lear

ners

can

als

o us

e a

tabl

e to

reco

rd th

e nu

mbe

r of m

atch

es u

sed

for e

ach

hexa

gon.

Thi

s w

ay th

e nu

mbe

r pat

tern

is re

late

d to

the

num

ber o

f mat

ches

use

d fo

r eac

h ne

w h

exag

on.

Posi

tion

of h

exag

on in

pat

tern

12

34

56

10

num

ber o

f mat

ches

612

18

The

nth te

rm fo

r thi

s se

quen

ce c

an b

e w

ritte

n as

6n

or 6

+ (n

– 1

)6

2.2

Func

tions

an

d re

latio

nshi

ps

inpu

t and

out

put v

alue

s

• D

eter

min

e in

put v

alue

s, o

utpu

t va

lues

or r

ules

for p

atte

rns

and

rela

tions

hips

usi

ng:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

-eq

uatio

ns

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

eq

uiva

lenc

e of

diff

eren

t des

crip

tions

of

the

sam

e re

latio

nshi

p or

rule

pr

esen

ted:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

equ

atio

ns

-by

gra

phs

on a

Car

tesi

an p

lane

Wha

t is

diffe

rent

to G

rade

8?

Lear

ners

con

solid

ate

wor

k w

ith in

put a

nd o

utpu

t val

ues

done

in G

rade

8. T

hey

shou

ld c

ontin

ue to

find

inpu

t or o

utpu

t val

ues

in fl

ow d

iagr

ams,

tabl

es, f

orm

ulae

an

d eq

uatio

ns.

Func

tions

and

rela

tions

hips

are

don

e ag

ain

in t

erm

3.

In th

is p

hase

, it i

s us

eful

to s

tart

spec

ifyin

g w

heth

er th

e in

put v

alue

s ar

e na

tura

l nu

mbe

rs, o

r int

eger

s or

ratio

nal n

umbe

rs. T

his

build

s le

arne

rs’ a

war

enes

s of

the

dom

ain

of in

put v

alue

s. H

ence

, to

find

outp

ut v

alue

s, le

arne

rs s

houl

d be

giv

en th

e ru

le/fo

rmul

a as

wel

l as

the

dom

ain

of th

e in

put v

alue

s.

Lear

ners

sho

uld

begi

n to

reco

gniz

e eq

uiva

lent

repr

esen

tatio

ns o

f the

sam

e re

la-

tions

hips

sho

wn

as a

n eq

uatio

n, a

set

of o

rder

ed p

airs

in a

tabl

e or

on

a gr

aph.

exam

ples

a)

If th

e ru

le fo

r find

ing

in th

e ta

ble

belo

w is

: y =

1 2 x +

1, d

eter

min

e th

e va

lues

of y

fo

r the

giv

en x

val

ues:

x0

12

410

5010

0

y

b)

Des

crib

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

rs in

the

top

row

and

thos

e in

the

botto

m ro

w in

the

tabl

e. T

hen

writ

e do

wn

the

valu

e of

m a

nd n

x–2

–10

12

12n

y–7

–5–3

–11

m27

In ta

bles

suc

h as

thes

e, m

ore

than

one

rule

mig

ht b

e po

ssib

le to

des

crib

e th

e re

latio

nshi

p be

twee

n x

and

y va

lues

. The

rule

s ar

e ac

cept

able

if th

ey m

atch

the

give

n in

put v

alue

s to

the

corr

espo

ndin

g ou

tput

val

ues.

For

exa

mpl

e, th

e ru

le

y =

2x –

3 d

escr

ibes

the

rela

tions

hip

betw

een

the

give

n va

lues

for x

and

y. T

o fin

d m

and

n, l

earn

ers

have

to s

ubst

itute

the

corr

espo

ndin

g va

lues

for x

or y

into

the

rule

and

sol

ve th

e eq

uatio

n by

insp

ectio

n.



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.3

alg

ebra

ic

expr

essi

ons

alg

ebra

ic la

ngua

ge

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-R

ecog

nize

and

iden

tify

conv

entio

ns

for w

ritin

g al

gebr

aic

expr

essi

ons

-Id

entif

y an

d cl

assi

fy li

ke a

nd u

nlik

e te

rms

in a

lgeb

raic

exp

ress

ions

-R

ecog

nize

and

iden

tify

coef

ficie

nts

and

expo

nent

s in

alg

ebra

ic

expr

essi

ons

• R

ecog

nize

and

diff

eren

tiate

bet

wee

n m

onom

ials

, bin

omia

ls a

nd tr

inom

ials

expa

nd a

nd s

impl

ify a

lgeb

raic

ex

pres

sion

s

• R

evis

e th

e fo

llow

ing

done

in G

rade

8,

usi

ng th

e co

mm

utat

ive,

ass

ocia

tive

and

dist

ribut

ive

law

s fo

r rat

iona

l nu

mbe

rs a

nd la

ws

of e

xpon

ents

to:

-ad

d an

d su

btra

ct li

ke te

rms

in

alge

brai

c ex

pres

sion

s

-m

ultip

ly in

tege

rs a

nd m

onom

ials

by

: ♦m

onom

ials

♦bi

nom

ials

♦tri

nom

ials

-di

vide

the

follo

win

g by

inte

gers

or

mon

omia

ls:

♦m

onom

ials

♦bi

nom

ials

♦tri

nom

ials

-S

impl

ify a

lgeb

raic

exp

ress

ions

in

volv

ing

the

abov

e op

erat

ions

Wha

t is

diffe

rent

to G

rade

8?

• A

lgeb

raic

man

ipul

atio

ns w

hich

incl

ude:

-m

ultip

ly in

tege

rs a

nd m

onom

ials

by

poly

nom

ials

-di

vide

pol

ynom

ials

by

inte

gers

or m

onom

ials

-th

e pr

oduc

t of t

wo

bino

mia

ls

-th

e sq

uare

of a

bin

omia

l

alg

ebra

ic e

xpre

ssio

ns a

re d

one

agai

n in

ter

m 3

. in

this

term

the

focu

s is

on

expa

ndin

g an

d si

mpl

ifyin

g al

gebr

aic

expr

essi

ons.

in t

erm

3 th

e fo

cus

is o

n fa

ctor

izin

g ex

pres

sion

s.

man

ipul

atin

g al

gebr

aic

expr

essi

ons

• m

ake

sure

lear

ners

und

erst

and

that

the

rule

s fo

r ope

ratin

g w

ith in

tege

rs a

nd

ratio

nal n

umbe

rs, i

nclu

ding

law

s of

exp

onen

ts, a

pply

equ

ally

whe

n nu

mbe

rs a

re

repl

aced

with

var

iabl

es. T

he v

aria

bles

are

num

bers

of a

giv

en ty

pe (e

.g. i

nteg

ers

or ra

tiona

l num

bers

) in

gene

raliz

ed fo

rm.

• W

hen

mul

tiply

ing

or d

ivid

ing

expr

essi

ons,

mak

e su

re le

arne

rs u

nder

stan

d ho

w

the

dist

ribut

ive

rule

wor

ks.

• Th

e as

soci

ativ

e ru

le a

llow

s fo

r gro

upin

g of

like

term

s w

hen

addi

ng.

Look

out

for t

he fo

llow

ing

com

mon

mis

conc

eptio

ns:

•x

+ x

= 2

x an

d n

ot

x2 . N

ote

the

conv

entio

n is

to w

rite

2x ra

ther

than

x2

•x2 +

x2 =

2x2

and

no

t 2x

4

•a

+b =

a +

b a

nd n

ot

ab

•(–

2x2 )

3 =

–8x

6 an

d n

ot

–6x5

•-x

(3x +

1) =

–3x

2 –

1 and

no

t –3

x2 +

1

•6x

2 +

11

x2 x2

= 6

+

and

no

t 6 +

1

• If

x =

2 then

–3x

2 =

–3(2

)2 =

3 x

4 =

12 a

nd n

ot

(–6)

2

• If

x =

–2 th

en –

x2 –

x =

–(–2

)2 –

(–2)

= –4

+ 2

= –

2 an

d n

ot

4 +

2 =

6

•25

x2 –9

x2 =

16x2

= 4x

and

no

t 5x

– 3

x =

2x√

√

•(x

+ 2

)2 =

x2 +

4x +

4 a

nd n

ot

x2 +

4

Tim

e fo

r al

gebr

aic

expr

essi

ons

in

this

term

:

4,5

hour

s


131CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.3

alg

ebra

ic

expr

essi

ons

-D

eter

min

e th

e sq

uare

s, c

ubes

, sq

uare

root

s an

d cu

be ro

ots

of

sing

le a

lgeb

raic

term

s or

like

al

gebr

aic

term

s

-D

eter

min

e th

e nu

mer

ical

val

ue

of a

lgeb

raic

exp

ress

ions

by

subs

titut

ion

• E

xten

d th

e ab

ove

alge

brai

c m

anip

ulat

ions

to in

clud

e:

-m

ultip

ly in

tege

rs a

nd m

onom

ials

by

poly

nom

ials

-di

vide

pol

ynom

ials

by

inte

gers

or

mon

omia

ls

-th

e pr

oduc

t of t

wo

bino

mia

ls

-th

e sq

uare

of a

bin

omia

l

Fact

oriz

e al

gebr

aic

expr

essi

ons

• Fa

ctor

ize

alge

brai

c ex

pres

sion

s th

at

invo

lve:

-co

mm

on fa

ctor

s

-di

ffere

nce

of tw

o sq

uare

s

-tri

nom

ials

of t

he fo

rm:

♦x2 +

bx

+ c

♦ax

2 + b

x +

c, w

here

a is

a c

omm

on

fact

or

• S

impl

ify a

lgeb

raic

exp

ress

ions

that

in

volv

e th

e ab

ove

fact

oris

atio

n pr

oces

ses

• S

impl

ify a

lgeb

raic

frac

tions

usi

ng

fact

oris

atio

n

exam

ples

of e

xpan

ding

and

sim

plify

ing

expr

essi

ons

a)

Sim

plify

: –3(

x3 + 2

x2 – x

) – x

2 (3x

+ 1)

[mul

tiply

inte

ger a

nd m

onom

ial b

y po

lyno

mia

l]

b)

Det

erm

ine/

expa

nd: (

x +

2)(x

– 3

) [m

ultip

ly b

inom

ial b

y bi

nom

ial]

c)

Det

erm

ine/

expa

nd: (

x +

2)(x

– 2

) [m

ultip

ly b

inom

ial b

y bi

nom

ial]

d)

Det

erm

ine/

expa

nd: (

x +

3)2 [

mul

tiply

out

a p

erfe

ct s

quar

e]

e)

Sim

plify

: 2(x

– 3

)2 –3(

x +

1)(2

x –

5) [m

ultip

le c

alcu

latio

ns in

volv

ing

prod

uct o

f bi

nom

ials

]

f) If

x =

–2 d

eter

min

e th

e nu

mer

ical

val

ue o

f 3x2 –

4x

+ 5

[usi

ng s

ubst

itutio

n]

g)

Sim

plify

: 6x4 –

8x3

– 2x

2 +

42x

2 fo

r x ≠

0 fo

r [d

ivid

e po

lyno

mia

l by

mon

omia

l; re

min

d le

arne

rs th

at d

enom

inat

or c

anno

t be

0]

h)

Sim

plify

: 8x

3 –

(–x3 )(

2x)

-x2

for x

≠ 0

for

[ca

lcul

atio

ns in

volv

ing

mul

tiple

op

erat

ions

; rem

ind

lear

ners

that

den

omin

ator

can

not b

e ]

i) D

eter

min

e:

36x4

√ [s

quar

e ro

ot o

f mon

omia

l]

It m

ight

hel

p to

rem

ind

lear

ners

that

thes

e va

riabl

es (o

r x in

this

cas

e) re

pres

ent

num

bers

of a

par

ticul

ar ty

pe –

thes

e m

ay b

e ra

tiona

l, or

inte

gers

, or p

erha

ps

who

le n

umbe

rs; s

uch

a re

min

der a

lso

then

impl

ies

that

all

the

asso

ciat

ed ru

les

or p

rope

rties

of t

hese

num

bers

app

ly h

ere.

So,

in th

e ab

ove

exam

ple,

if x

is a

n in

tege

r, th

en x

= a

or x

= –

a be

caus

e a4 =

(–a)

4



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.4

alg

ebra

ic

equa

tions

equa

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-S

et u

p eq

uatio

ns to

des

crib

e pr

oble

m s

ituat

ions

-A

naly

se a

nd in

terp

ret e

quat

ions

th

at d

escr

ibe

a gi

ven

situ

atio

n

-S

olve

equ

atio

ns b

y:

♦in

spec

tion

♦us

ing

addi

tive

and

mul

tiplic

ativ

e in

vers

es

♦us

ing

law

s of

exp

onen

ts

-D

eter

min

e th

e nu

mer

ical

val

ue o

f an

exp

ress

ion

by s

ubst

itutio

n.

-U

se s

ubst

itutio

n in

equ

atio

ns to

ge

nera

te ta

bles

of o

rder

ed p

airs

• E

xten

d so

lvin

g eq

uatio

ns to

incl

ude:

-us

ing

fact

oris

atio

n

-eq

uatio

ns o

f the

form

: a p

rodu

ct o

f fa

ctor

s =

0

Wha

t is

diffe

rent

to G

rade

8?

• S

olvi

ng e

quat

ions

usi

ng fa

ctor

izat

ion

• S

olvi

ng e

quat

ions

of t

he fo

rm: a

pro

duct

of f

acto

rs =

0

alg

ebra

ic e

quat

ions

are

don

e ag

ain

in t

erm

3. i

n th

is te

rm th

e fo

cus

is o

n co

nsol

idat

ing

solv

ing

equa

tions

usi

ng a

dditi

ve a

nd m

ultip

licat

ive

inve

rses

an

d th

e la

ws

of e

xpon

ents

. in

term

3, t

he fo

cus

is o

n so

lvin

g eq

uatio

ns a

fter

fact

oriz

ing

as w

ell a

s ge

nera

ting

tabl

es o

f ord

ered

pai

rs fo

r lin

ear e

quat

ions

. Le

arne

rs h

ave

oppo

rtuni

ties

to w

rite

and

solv

e eq

uatio

ns w

hen

they

writ

e ge

nera

l ru

les

to d

escr

ibe

rela

tions

hips

bet

wee

n nu

mbe

rs in

num

ber p

atte

rns,

and

whe

n th

ey fi

nd in

put o

r out

put v

alue

s fo

r giv

en ru

les

in fl

ow d

iagr

ams,

tabl

es a

nd

form

ulae

.

In G

rade

9, l

earn

ers

can

be g

iven

equ

atio

ns w

here

they

hav

e to

exp

and,

sim

plify

or

fact

oriz

e ex

pres

sion

s fir

st, b

efor

e so

lvin

g th

e eq

uatio

n.

For e

quat

ions

of t

he fo

rm: a

pro

duct

of t

wo

fact

ors

= 0,

lear

ners

hav

e to

un

ders

tand

that

if th

e pr

oduc

t of t

wo

fact

ors

equa

ls 0

, the

n at

leas

t one

of t

he

fact

ors

mus

t be

equa

l to

0. H

ence

to s

olve

the

equa

tion,

eac

h fa

ctor

mus

t be

writ

ten

as a

n eq

uatio

n eq

ual t

o 0,

and

ther

efor

e m

ore

than

one

sol

utio

n fo

r x is

po

ssib

le.

Whe

n w

orki

ng w

ith a

lgeb

raic

frac

tions

, lea

rner

s m

ust b

e re

min

ded

that

the

deno

min

ator

can

not e

qual

0, s

o an

y va

lue

of x

that

mak

es th

e de

nom

inat

or 0

ca

nnot

be

a so

lutio

n to

the

equa

tion.

exam

ples

of e

quat

ions

a)

Sol

ve: x

if

3(x –

2) =

x +

2

3x

– 6

= x

+ 2

(e

xpan

d LH

S fi

rst)

3x

– x

– 6

= x

– x

+ 2

(add

–x

to b

oth

side

s of

the

equa

tion)

th

eref

ore

2x –

6 =

2

2x

– 6

+ 6

= 2

+ 6

(add

6 to

bot

h si

des

of th

e eq

uatio

n)

th

eref

ore

2x =

8

2x

2 =

8 2 (d

ivid

e bo

th s

ides

of t

he e

quat

ion

by 2

)

x

= 4

b)

Sol

ve x

if (x

– 1

)(x

+ 3)

= 0

x –

1= 0

or x

+ 3

= 0

(at l

east

one

fact

or m

ust b

e eq

ual t

o 0)

Thus

x =

1 (a

dd –

1 to

bot

h si

des

of th

e eq

uatio

n) o

r x =

–3

(add

–3

to b

oth

side

s of

the

equa

tion)

Tim

e fo

r al

gebr

aic

equa

tions

in

this

term

:

4 ho

urs


133CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.4

alg

ebra

ic

equa

tions

c)

Sol

ve x

if x 3 +

2x –

14

= 1

4x +

3(2

x –

1) =

12

(mul

tiply

eac

h te

rm o

n bo

th s

ides

of t

he e

quat

ion

by th

e LC

M, 1

2)

4x +

6x

–3 =

12

(exp

and

expr

essi

on o

n LH

S)

10x

= 15

(add

3 to

bot

h si

des

of th

e eq

uatio

n, a

nd a

dd li

ke te

rms

to th

e LH

S)

x =

2 3 (di

vide

bot

h si

des

of th

e eq

uatio

n by

10)

d)

If y

= 2x

2 + 4

x +

3, c

alcu

late

y w

hen

x =

–2

e)

Than

di is

6 y

ears

old

er th

an S

ophi

e. In

3 y

ears

tim

e Th

andi

will

be

twic

e as

old

as

Sop

hie.

How

old

is T

hand

i now

?

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• th

e pr

oper

ties

of d

iffer

ent n

umbe

r sys

tem

s

• ca

lcul

atin

g an

d so

lvin

g pr

oble

ms

with

who

le n

umbe

rs, i

nteg

ers,

com

mon

frac

tions

and

dec

imal

frac

tions

, num

bers

in e

xpon

entia

l for

m

• nu

mer

ic a

nd g

eom

etric

pat

tern

s

• fu

nctio

ns a

nd re

latio

nshi

ps

• al

gebr

aic

expr

essi

ons

• al

gebr

aic

equa

tions

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

7 ho

urs



Gr

ad

e 9

– te

rm

2

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.5

Con

stru

ctio

n of

geo

met

ric

figures

Con

stru

ctio

ns

• A

ccur

atel

y co

nstru

ct g

eom

etric

fig

ures

app

ropr

iate

ly u

sing

a

com

pass

, rul

er a

nd p

rotra

ctor

, in

clud

ing

bise

ctin

g an

gles

of a

tri

angl

e

• C

onst

ruct

ang

les

of 4

5°, 3

0°, 6

0°an

d an

d th

eir m

ultip

les

with

out u

sing

a

prot

ract

or

inve

stig

atin

g pr

oper

ties

of g

eom

etric

fig

ures

• B

y co

nstru

ctio

n, in

vest

igat

e th

e an

gles

in a

tria

ngle

, foc

usin

g on

the

rela

tions

hip

betw

een

the

exte

rior

angl

e of

a tr

iang

le a

nd it

s in

terio

r an

gles

• B

y co

nstru

ctio

n, e

xplo

re th

e m

inim

um

cond

ition

s fo

r tw

o tri

angl

es to

be

cong

ruen

t

• B

y co

nstru

ctio

n, in

vest

igat

e si

des,

ang

les

and

diag

onal

s in

qu

adril

ater

als,

focu

sing

on

the

diag

onal

s of

rect

angl

es, s

quar

es,

para

llelo

gram

s, rh

ombi

and

kite

s

• B

y co

nstru

ctio

n ex

plor

e th

e su

m o

f th

e in

terio

r ang

les

of p

olyg

ons

Wha

t is

diffe

rent

to G

rade

8?

• B

isec

ting

angl

es in

a tr

iang

le

• C

onst

ruct

ing

30°

with

out a

pro

tract

or

• In

vest

igat

ion

of n

ew p

rope

rties

of t

riang

les,

qua

drila

tera

ls a

nd p

olyg

ons

Con

stru

ctio

ns

• C

onst

ruct

ions

pro

vide

a u

sefu

l con

text

to e

xplo

re o

r con

solid

ate

know

ledg

e of

an

gles

and

sha

pes.

• m

ake

sure

lear

ners

are

com

pete

nt a

nd c

omfo

rtabl

e in

the

use

of a

com

pass

and

kn

ow h

ow to

mea

sure

and

read

ang

le s

izes

on

a pr

otra

ctor

• R

evis

e th

e co

nstru

ctio

ns o

f ang

les

if ne

cess

ary,

bef

ore

proc

eedi

ng w

ith th

e ne

w

cons

truct

ions

.

• S

tart

with

the

cons

truct

ions

of l

ines

, so

that

lear

ners

can

firs

t exp

lore

ang

le

rela

tions

hips

on

stra

ight

line

s.

• W

hen

cons

truct

ing

trian

gles

lear

ners

sho

uld

draw

on

know

n pr

oper

ties

and

cons

truct

ion

of c

ircle

s.

• C

onst

ruct

ion

of s

peci

al a

ngle

s w

ithou

t pro

tract

ors

are

done

by:

-bi

sect

ing

a rig

ht-a

ngle

to g

et 4

5°

-dr

awin

g an

equ

ilate

ral t

riang

le to

get

60°

-bi

sect

ing

the

angl

es o

f an

equi

late

ral t

riang

le to

get

30°

Tota

l tim

e fo

r co

nstru

ctio

ns

of g

eom

etric

fig

ures

:

9 ho

urs


135CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

Cla

ssify

ing

2d s

hape

s

• R

evis

e pr

oper

ties

and

defin

ition

s of

tri

angl

es in

term

s of

thei

r sid

es a

nd

angl

es, d

istin

guis

hing

bet

wee

n:

-eq

uila

tera

l tria

ngle

s

-is

osce

les

trian

gles

-rig

ht-a

ngle

d tri

angl

es

• R

evis

e an

d w

rite

clea

r defi

nitio

ns o

f qu

adril

ater

als

in te

rms

of th

eir s

ides

, an

gles

and

dia

gona

ls, d

istin

guis

hing

be

twee

n:

-pa

ralle

logr

am

-re

ctan

gle

-sq

uare

-rh

ombu

s

-tra

pezi

um

-ki

te

Wha

t is

diffe

rent

to G

rade

8?

• P

rope

rties

of d

iago

nals

of q

uadr

ilate

rals

• m

inim

um c

ondi

tions

for c

ongr

uent

and

sim

ilar t

riang

les

• Tr

iang

les

• C

onst

ruct

ions

ser

ve a

s a

usef

ul c

onte

xt fo

r exp

lorin

g pr

oper

ties

of tr

iang

les.

S

ee 3

.5 C

onst

ruct

ion

of G

eom

etric

figu

res.

• P

rope

rties

of t

riang

les

lear

ners

sho

uld

know

:

-th

e su

m o

f the

inte

rior a

ngle

s of

tria

ngle

s =

180°

-an

equ

ilate

ral t

riang

le h

as a

ll si

des

equa

l and

all

inte

rior a

ngle

s =

60°

-an

isos

cele

s tri

angl

e ha

s at

leas

t tw

o eq

ual s

ides

and

its

base

ang

les

are

equa

l

-a

right

-ang

led

trian

gle

has

one

angl

e th

at is

a ri

ght-a

ngle

-th

e si

de o

ppos

ite th

e rig

ht-a

ngle

in a

righ

t-ang

led

trian

gle,

is c

alle

d th

e hy

pote

nuse

-in

a ri

ght-a

ngle

d tri

angl

e, th

e sq

uare

of t

he h

ypot

enus

e is

equ

al to

the

sum

of

the

squa

res

of th

e ot

her t

wo

side

s (T

heor

em o

f Pyt

hago

ras)

.

-th

e ex

terio

r ang

le o

f a tr

iang

le =

the

sum

of t

he o

ppos

ite tw

o in

terio

r ang

les

Qua

drila

tera

ls

• C

onst

ruct

ions

ser

ve a

s a

usef

ul c

onte

xt fo

r exp

lorin

g pr

oper

ties

of tr

iang

les.

S

ee n

otes

on

cons

truct

ions

abo

ve.

• Th

e cl

assi

ficat

ion

of q

uadr

ilate

rals

sho

uld

incl

ude

the

reco

gniti

on th

at:

-re

ctan

gles

and

rhom

bi a

re s

peci

al k

inds

of p

aral

lelo

gram

s

-a

squa

re is

a s

peci

al k

ind

of re

ctan

gle

and

rhom

bus.

Prop

ertie

s of

qua

drila

tera

ls le

arne

rs s

houl

d kn

ow:

• th

e su

m o

f the

inte

rior a

ngle

s of

qua

drila

tera

ls =

360

°

• th

e op

posi

te s

ides

of p

aral

lelo

gram

s ar

e pa

ralle

l and

equ

al

• th

e op

posi

te a

ngle

s of

par

alle

logr

ams

are

equa

l

• th

e op

posi

te a

ngle

s of

a rh

ombu

s ar

e eq

ual

• th

e op

posi

te s

ides

of a

rhom

bus

are

para

llel a

nd e

qual

• th

e si

ze o

f eac

h an

gle

of re

ctan

gles

and

squ

ares

is 9

0°

• a

trape

zium

has

one

pai

r of o

ppos

ite s

ides

par

alle

l

Tota

l tim

e fo

r ge

omet

ry o

f 2D

sha

pes

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

sim

ilar a

nd c

ongr

uent

tria

ngle

s

• Th

roug

h in

vest

igat

ion,

est

ablis

h th

e m

inim

um c

ondi

tions

for c

ongr

uent

tri

angl

es

• Th

roug

h in

vest

igat

ion,

est

ablis

h th

e m

inim

um c

ondi

tions

for s

imila

r tri

angl

es

solv

ing

prob

lem

s

Sol

ve g

eom

etric

pro

blem

s in

volv

ing

unkn

own

side

s an

d an

gles

in

trian

gles

and

qua

drila

tera

ls, u

sing

kn

own

prop

ertie

s of

tria

ngle

s an

d qu

adril

ater

als,

as

wel

l as

prop

ertie

s of

co

ngru

ent a

nd s

imila

r tria

ngle

s.

• a

kite

has

two

pairs

of a

djac

ent s

ides

equ

al

• th

e di

agon

als

of a

squ

are,

rect

angl

e, p

aral

lelo

gram

and

rhom

bus

bise

ct e

ach

othe

r

• th

e di

agon

als

of a

squ

are,

rhom

bus

and

kite

are

per

pend

icul

ar

Con

grue

nt tr

iang

les

• C

onst

ruct

ions

are

a u

sefu

l con

text

for e

stab

lishi

ng th

e m

inim

um c

ondi

tions

for

two

trian

gles

to b

e co

ngru

ent.

See

not

es o

n C

onst

ruct

ions

abo

ve.

• C

ondi

tions

for t

wo

trian

gles

to b

e co

ngru

ent:

-th

ree

corr

espo

ndin

g si

des

are

equa

l (S

,S,S

)

-tw

o co

rres

pond

ing

side

s an

d th

e in

clud

ed a

ngle

are

equ

al (S

,A,S

)

-tw

o co

rres

pond

ing

angl

es a

nd a

cor

resp

ondi

ng s

ide

are

equa

l (A

,A,S

)

-rig

ht-a

ngle

, hyp

oten

use

and

one

othe

r cor

resp

ondi

ng s

ide

are

equa

l (R

,H,S

)

sim

ilar t

riang

les

• C

onst

ruct

ions

are

a u

sefu

l con

text

for e

stab

lishi

ng th

e m

inim

um c

ondi

tions

for

two

trian

gles

to b

e si

mila

r. S

ee n

otes

on

Con

stru

ctio

ns a

bove

.

• C

ondi

tion

for t

wo

trian

gles

to b

e si

mila

r: co

rres

pond

ing

angl

es a

re e

qual

and

co

rres

pond

ing

side

s ar

e pr

opor

tiona

l

solv

ing

prob

lem

s

• Le

arne

rs c

an s

olve

geo

met

ric p

robl

ems

to fi

nd u

nkno

wn

side

s an

d an

gles

in

tria

ngle

s an

d qu

adril

ater

als,

usi

ng k

now

n de

finiti

ons

as w

ell a

s an

gle

rela

tions

hips

on

stra

ight

line

s.

• Fo

r rig

ht-a

ngle

d tri

angl

es, l

earn

ers

can

also

use

the

Theo

rem

of P

ytha

gora

s to

fin

d un

know

n le

ngth

s.

• Le

arne

rs s

houl

d gi

ve re

ason

s an

d ju

stify

thei

r sol

utio

ns fo

r eve

ry w

ritte

n st

atem

ent.

• N

ote

that

sol

ving

geo

met

ric p

robl

ems

is a

n op

portu

nity

to p

ract

ise

solv

ing

equa

tions

.

exam

ple:

In ∆

AB

C, Â

= x

, ang

le B

= 5

0° a

nd a

ngle

Ĉ =

80°

. Wha

t is

the

size

of Â

?

Lear

ners

can

sol

ve x

in th

e fo

llow

ing

equa

tion:

x +

50°

+ 80

° =

180°

(bec

ause

the

sum

of t

he a

ngle

s in

a tr

iang

le =

180

°)

x =

180°

– 1

30°

(add

–50

° an

d –8

0° to

bot

h si

des

of th

e eq

uatio

n)

x =

50°,

hen

ce Â

= 5

0°


137CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.1

Geo

met

ry o

f 2d

sha

pes

sim

ilar a

nd c

ongr

uent

tria

ngle

s

• Th

roug

h in

vest

igat

ion,

est

ablis

h th

e m

inim

um c

ondi

tions

for c

ongr

uent

tri

angl

es

• Th

roug

h in

vest

igat

ion,

est

ablis

h th

e m

inim

um c

ondi

tions

for s

imila

r tri

angl

es

solv

ing

prob

lem

s

Sol

ve g

eom

etric

pro

blem

s in

volv

ing

unkn

own

side

s an

d an

gles

in

trian

gles

and

qua

drila

tera

ls, u

sing

kn

own

prop

ertie

s of

tria

ngle

s an

d qu

adril

ater

als,

as

wel

l as

prop

ertie

s of

co

ngru

ent a

nd s

imila

r tria

ngle

s.

• a

kite

has

two

pairs

of a

djac

ent s

ides

equ

al

• th

e di

agon

als

of a

squ

are,

rect

angl

e, p

aral

lelo

gram

and

rhom

bus

bise

ct e

ach

othe

r

• th

e di

agon

als

of a

squ

are,

rhom

bus

and

kite

are

per

pend

icul

ar

Con

grue

nt tr

iang

les

• C

onst

ruct

ions

are

a u

sefu

l con

text

for e

stab

lishi

ng th

e m

inim

um c

ondi

tions

for

two

trian

gles

to b

e co

ngru

ent.

See

not

es o

n C

onst

ruct

ions

abo

ve.

• C

ondi

tions

for t

wo

trian

gles

to b

e co

ngru

ent:

-th

ree

corr

espo

ndin

g si

des

are

equa

l (S

,S,S

)

-tw

o co

rres

pond

ing

side

s an

d th

e in

clud

ed a

ngle

are

equ

al (S

,A,S

)

-tw

o co

rres

pond

ing

angl

es a

nd a

cor

resp

ondi

ng s

ide

are

equa

l (A

,A,S

)

-rig

ht-a

ngle

, hyp

oten

use

and

one

othe

r cor

resp

ondi

ng s

ide

are

equa

l (R

,H,S

)

sim

ilar t

riang

les

• C

onst

ruct

ions

are

a u

sefu

l con

text

for e

stab

lishi

ng th

e m

inim

um c

ondi

tions

for

two

trian

gles

to b

e si

mila

r. S

ee n

otes

on

Con

stru

ctio

ns a

bove

.

• C

ondi

tion

for t

wo

trian

gles

to b

e si

mila

r: co

rres

pond

ing

angl

es a

re e

qual

and

co

rres

pond

ing

side

s ar

e pr

opor

tiona

l

solv

ing

prob

lem

s

• Le

arne

rs c

an s

olve

geo

met

ric p

robl

ems

to fi

nd u

nkno

wn

side

s an

d an

gles

in

tria

ngle

s an

d qu

adril

ater

als,

usi

ng k

now

n de

finiti

ons

as w

ell a

s an

gle

rela

tions

hips

on

stra

ight

line

s.

• Fo

r rig

ht-a

ngle

d tri

angl

es, l

earn

ers

can

also

use

the

Theo

rem

of P

ytha

gora

s to

fin

d un

know

n le

ngth

s.

• Le

arne

rs s

houl

d gi

ve re

ason

s an

d ju

stify

thei

r sol

utio

ns fo

r eve

ry w

ritte

n st

atem

ent.

• N

ote

that

sol

ving

geo

met

ric p

robl

ems

is a

n op

portu

nity

to p

ract

ise

solv

ing

equa

tions

.

exam

ple:

In ∆

AB

C, Â

= x

, ang

le B

= 5

0° a

nd a

ngle

Ĉ =

80°

. Wha

t is

the

size

of Â

?

Lear

ners

can

sol

ve x

in th

e fo

llow

ing

equa

tion:

x +

50°

+ 80

° =

180°

(bec

ause

the

sum

of t

he a

ngle

s in

a tr

iang

le =

180

°)

x =

180°

– 1

30°

(add

–50

° an

d –8

0° to

bot

h si

des

of th

e eq

uatio

n)

x =

50°,

hen

ce Â

= 5

0°

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

3.3

Geo

met

ry o

f st

raig

ht li

nes

ang

le re

latio

nshi

ps

• R

evis

e an

d w

rite

clea

r des

crip

tions

of

the

rela

tions

hip

betw

een

angl

es

form

ed b

y:

-pe

rpen

dicu

lar l

ines

-in

ters

ectin

g lin

es

-pa

ralle

l lin

es c

ut b

y a

trans

vers

al

solv

ing

prob

lem

s

Sol

ve g

eom

etric

pro

blem

s us

ing

the

rela

tions

hips

bet

wee

n pa

irs o

f ang

les

desc

ribed

abo

ve.

Wha

t is

diffe

rent

to G

rade

8?

Lear

ners

revi

se a

nd w

rite

clea

r des

crip

tions

of a

ngle

rela

tions

hips

on

stra

ight

line

s

ang

le re

latio

nshi

ps le

arne

rs s

houl

d kn

ow:

• th

e su

m o

f the

ang

les

on a

stra

ight

line

is 1

80°

• If

lines

are

per

pend

icul

ar li

nes,

then

adj

acen

t sup

plem

enta

ry a

ngle

s ar

e ea

ch

equa

l to

90°

.•

If lin

es in

ters

ect,

then

ver

tical

ly o

ppos

ite a

ngle

s ar

e eq

ual.

• if

para

llel l

ines

are

cut

by

a tra

nsve

rsal

, the

n co

rres

pond

ing

angl

es a

re e

qual

• if

para

llel l

ines

cut

by

a tra

nsve

rsal

, the

n al

tern

ate

angl

es a

re e

qual

• if

para

llel l

ines

cut

by

a tra

nsve

rsal

, the

n co

-inte

rior a

ngle

s ar

e su

pple

men

tary

The

abov

e an

gles

hav

e to

be

iden

tified

and

nam

ed b

y le

arne

rs.

solv

ing

prob

lem

s

• Le

arne

rs c

an s

olve

geo

met

ric p

robl

ems

to fi

nd u

nkno

wn

angl

es u

sing

the

angl

e re

latio

nshi

ps a

bove

, as

wel

l as

othe

r kno

wn

prop

ertie

s of

tria

ngle

s an

d qu

adril

ater

als.

• Le

arne

rs s

houl

d gi

ve re

ason

s an

d ju

stify

thei

r sol

utio

ns fo

r eve

ry w

ritte

n st

atem

ent.

• N

ote

that

sol

ving

geo

met

ric p

robl

ems

is a

n op

portu

nity

to p

ract

ise

solv

ing

equa

tions

.

exam

ple:

B1 a

nd B

2 are

two

angl

es o

n a

stra

ight

line

. B1 =

35°

. Wha

t is

the

size

of B

2 ?

Lear

ners

can

find

B2 b

y so

lvin

g th

e fo

llow

ing

equa

tion:

35°

+ B

2 =

180°

(bec

ause

the

sum

of a

ngle

s on

a s

traig

ht li

ne

180°

)

B2

= 18

0° –

35°

(add

–35

° to

bot

h si

des

of th

e eq

uatio

n)

B2 =

145

°

Tota

l tim

e fo

r ge

omet

ry o

f st

raig

ht li

nes

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

mea

sure

men

t4.

3

the

theo

rem

of

Py

thag

oras

solv

e pr

oble

ms

usin

g th

e th

eore

m

of P

ytha

gora

s

Use

the

Theo

rem

of P

ytha

gora

s to

so

lve

prob

lem

s in

volv

ing

unkn

own

leng

ths

in g

eom

etric

figu

res

that

con

-ta

in ri

ght-a

ngle

d tri

angl

es

• Th

e Th

eore

m o

f Pyt

hago

ras

was

intro

duce

d in

Gra

de 8

• It

is im

porta

nt th

at le

arne

rs u

nder

stan

d th

at th

e Th

eore

m o

f Pyt

hago

ras

appl

ies

only

to ri

ght-a

ngle

d tri

angl

es.

• Th

e Th

eore

m o

f Pyt

hago

ras

is b

asic

ally

a fo

rmul

a to

cal

cula

te u

nkno

wn

leng

th

of s

ides

in ri

ght-a

ngle

d tri

angl

es.

• In

par

ticul

ar, t

he T

heor

em o

f Pyt

hago

ras

can

be th

e fir

st s

tep

in c

alcu

latio

ns o

f pe

rimet

ers

or a

reas

of c

ompo

site

figu

res,

whe

n on

e of

the

figur

es is

a ri

ght-

angl

ed tr

iang

le w

ith a

n un

know

n le

ngth

. See

exa

mpl

e be

low

.

• In

the

FET

phas

e, th

e Th

eore

m o

f Pyt

hago

ras

is c

ruci

al to

the

furth

er s

tudy

of

Geo

met

ry a

nd T

rigon

omet

ry.

exam

ples

of s

olvi

ng p

robl

ems

usin

g th

e th

eore

m o

f Pyt

hago

ras:

AB

CD

is a

rect

angl

e w

here

BD

= 1

5 cm

and

BC

= 9

cm

.

Det

erm

ine

the:

a)

perim

eter

of A

BC

D

b)

area

of A

BC

D

15

cm

9

cm

A B C

D

Tota

l tim

e fo

r th

e Th

eore

m

of P

ytha

gora

s:

5 ho

urs


139CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

mea

sure

men

t4.

1

are

a an

d pe

rimet

er o

f 2d

sha

pes

are

a an

d pe

rimet

er

• U

se a

ppro

pria

te fo

rmul

ae a

nd

conv

ersi

ons

betw

een

SI u

nits

, to

sol

ve p

robl

ems

and

calc

ulat

e pe

rimet

er a

nd a

rea

of:

-po

lygo

ns

-ci

rcle

s

• In

vest

igat

e ho

w d

oubl

ing

any

or a

ll of

th

e di

men

sion

s of

a 2

D fi

gure

affe

cts

its p

erim

eter

and

its

area

Wha

t is

diffe

rent

to G

rade

8?

• Th

e ca

lcul

atio

ns a

re th

e sa

me

as in

Gra

de 8

, but

lear

ners

can

find

per

imet

ers

and

area

s of

mor

e co

mpo

site

and

com

plex

figu

res

• P

olyg

ons

can

incl

ude

trape

zium

s, p

aral

lelo

gram

s, rh

ombi

and

kite

s

•Fo

rmul

ae le

arne

rs s

houl

d kn

ow a

nd u

se:

Per

imet

er o

f a s

quar

e:P

= 4s

Per

imet

er o

f a re

ctan

gle:

P =

2(l +

b) o

r P =

2l +

2b

Are

a of

a s

quar

e:A

= l2

Are

a of

a re

ctan

gle:

A

= le

ngth

x b

read

th

Are

a of

a rh

ombu

s:A

= le

ngth

x h

eigh

t

Are

a of

a k

ite:

A =

1 2 (di

agon

al1 x

dia

gona

l 2

Are

a of

a p

aral

lelo

gram

:A

= ba

se x

hei

ght

Are

a of

a tr

apez

ium

A =

1 2 (su

m o

f par

alle

l sid

es) x

hei

ght

Are

a of

a tr

iang

le:

A =

1 2 (b

x h)

Dia

met

er o

f a c

ircle

: d

= 2r

Circ

umfe

renc

e of

circ

le:

c =

πd o

r c =

2πr

Are

a of

a c

ircle

: A

= πr

2

solv

ing

equa

tions

usi

ng fo

rmul

ae

The

use

of fo

rmul

ae p

rovi

des

a co

ntex

t to

prac

tise

solv

ing

equa

tions

by

insp

ectio

n or

usi

ng a

dditi

ve o

r mul

tiplic

ativ

e in

vers

es.

exam

ple:

a)

If th

e pe

rimet

er o

f a s

quar

e is

32

cm, w

hat i

s th

e le

ngth

of e

ach

side

? Le

arne

rs

shou

ld w

rite

this

as:

4s

= 3

2 an

d so

lve

by a

skin

g: 4

tim

es w

hat w

ill b

e 32

? or

say

ing

s= 32

4

.

b)

If th

e ar

ea o

f a re

ctan

gle

is 2

00 c

m2 ,

and

its le

ngth

is 5

0 cm

, wha

t is

its w

idth

? Le

arne

rs s

houl

d w

rite

this

as:

50

x b

= 2

00 a

nd s

olve

by

insp

ectio

n by

ask

ing:

50

times

wha

t will

be

200?

or

sayi

ng b

= 20

0 50

Tota

l tim

e fo

r Are

a an

d P

erim

eter

5 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

mea

sure

men

t4.

1

are

a an

d pe

rimet

er o

f 2d

sha

pes

For a

reas

of t

riang

les:

• m

ake

sure

lear

ners

kno

w th

at th

e he

ight

of a

tria

ngle

is a

line

seg

men

t dra

wn

from

any

ver

tex

perp

endi

cula

r to

the

oppo

site

sid

e.

exam

ple:

AD

is th

e he

ight

ont

o ba

se B

C o

f ∆A

BC

.

A D

B C

A D B

C

• P

oint

out

that

eve

ry tr

iang

le h

as 3

bas

es, e

ach

with

a re

late

d he

ight

or a

ltitu

de.

•Fo

r con

vers

ions

:

-If

1 cm

= 1

0 cm

then

1 c

m2 =

100

mm

2

-If

1 cm

= 1

00 c

m th

en 1

m2 =

10

000

cm2

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• co

nstru

ctin

g ge

omet

ric o

bjec

ts

• ge

omet

ry o

f 2D

sha

pes

• ge

omet

ry o

f stra

ight

line

s

• th

e Th

eore

m o

f Pyt

hago

ras

• ar

ea a

nd p

erim

eter

of 2

D s

hape

s

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

8 ho

urs


141CAPS

Gr

ad

e 9

– te

rm

3

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns &

al

gebr

a

2.2

Fu

nctio

ns

and

re

latio

nshi

ps

inpu

t and

out

put v

alue

s

• D

eter

min

e in

put v

alue

s, o

utpu

t va

lues

or r

ules

for p

atte

rns

and

rela

tions

hips

usi

ng:

-flo

w d

iagr

ams

-ta

bles

-fo

rmul

ae

-eq

uatio

ns

equi

vale

nt fo

rms

• D

eter

min

e, in

terp

ret a

nd ju

stify

eq

uiva

lenc

e of

diff

eren

t des

crip

tions

of

the

sam

e re

latio

nshi

p or

rule

pr

esen

ted:

-ve

rbal

ly

-in

flow

dia

gram

s

-in

tabl

es

-by

form

ulae

-by

equ

atio

ns

-by

gra

phs

on a

Car

tesi

an p

lane

Func

tions

and

rela

tions

hips

wer

e al

so d

one

in T

erm

1. T

he fo

cus

in th

is te

rm is

on

findi

ng o

utpu

t val

ues

for g

iven

equ

atio

ns, a

nd re

cogn

isin

g eq

uiva

lent

form

s be

-tw

een

diffe

rent

des

crip

tions

of t

he s

ame

rela

tions

hip.

see

addi

tiona

l not

es a

nd e

xam

ples

in t

erm

1.

Tim

e fo

r Fu

nctio

ns a

nd

rela

tions

hips

in

this

term

:

5 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns &

al

gebr

a

2.3

alg

ebra

icex

pres

sion

s

alg

ebra

ic la

ngua

ge

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-re

cogn

ize

and

iden

tify

conv

entio

ns

for w

ritin

g al

gebr

aic

expr

essi

ons

-id

entif

y an

d cl

assi

fy li

ke a

nd u

nlik

e te

rms

in a

lgeb

raic

exp

ress

ions

-re

cogn

ize

and

iden

tify

coef

ficie

nts

and

expo

nent

s in

alg

ebra

ic

expr

essi

ons

• R

ecog

nize

and

diff

eren

tiate

bet

wee

n m

onom

ials

, bin

omia

ls a

nd tr

inom

ials

expa

nd a

nd s

impl

ify a

lgeb

raic

ex

pres

sion

s

• R

evis

e th

e fo

llow

ing

done

in G

rade

8,

usi

ng th

e co

mm

utat

ive,

ass

ocia

tive

and

dstri

butiv

e la

ws

for r

atio

nal

num

bers

and

law

s of

exp

onen

ts to

:

-ad

d an

d su

btra

ct li

ke te

rms

in

alge

brai

c ex

pres

sion

s

-m

ultip

ly in

tege

rs a

nd m

onom

ials

by

: ♦m

onom

ials

♦bi

nom

ials

♦tri

nom

ials

-di

vide

the

follo

win

g by

inte

gers

or

mon

omia

ls:

♦m

onom

ials

♦bi

nom

ials

♦tri

nom

ials

-si

mpl

ify a

lgeb

raic

exp

ress

ions

in

volv

ing

the

abov

e op

erat

ions

-de

term

ine

the

squa

res,

cub

es,

squa

re ro

ots

and

cube

root

s of

si

ngle

alg

ebra

ic te

rms

or li

ke

alge

brai

c te

rms

Alg

ebra

ic e

xpre

ssio

ns w

ere

also

don

e in

Ter

m 1

. The

focu

s in

this

term

is o

n fa

ctor

izin

g ex

pres

sion

s.

see

addi

tiona

l not

es a

nd e

xam

ples

in t

erm

1.

Fact

oriz

ing

expr

essi

ons

• M

ake

sure

lear

ners

und

erst

and

that

fact

oriz

ing

is th

e re

vers

e of

exp

andi

ng a

n ex

pres

sion

thro

ugh

mul

tiplic

atio

n e.

g.

Exp

and:

2x

(x +

3) =

2x2 +

6

Fact

oriz

e:

2x2 +

6 =

2x(

x +

3)

• N

ote

that

1 a

nd –

1 ar

e co

mm

on fa

ctor

s of

eve

ry e

xpre

ssio

n e.

g.

Exp

and:

a

– 4b

= 1

(a –

4b)

Fact

oriz

e:

4b –

a =

1(a

– 4

b)

exam

ples

of e

xpre

ssio

ns w

ith c

omm

on fa

ctor

s th

at c

an b

e fa

ctor

ized

a)

6a4 –

4a2

b)

ax –

bx

+ 2a

– 2

b

c)

2x(a

– b

) – 3

(a –

b)

d)

2x(a

– b

) – 3

(b –

a)

e)

(a +

b)2 –

5(a

+ b

)

exam

ples

of e

xpre

ssio

ns w

ith a

diff

eren

ce o

f tw

o sq

uare

s th

at c

an b

e fa

ctor

ized

a)

25a2 =

1

b)

a4 – b

4

c)

9(a

+ b)

2 – 1

d)

3x 3 –

27

exam

ples

of a

lgeb

raic

frac

tions

that

can

be

fact

oriz

ed

a)

2x +

6y

x +

3y

b)

3x –

3y

6x –

6y

c)

9a2 –

13a

+ 1

Tota

l tim

e fo

r alg

ebra

ic

expr

essi

ons:

9 ho

urs


143CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

Patte

rns,

fu

nctio

ns &

al

gebr

a

2.3

alg

ebra

icex

pres

sion

s

-de

term

ine

the

num

eric

al v

alue

of

alg

ebra

ic e

xpre

ssio

ns b

y su

bstit

utio

n

• E

xten

d th

e ab

ove

alge

brai

c m

anip

ulat

ions

to in

clud

e:

-m

ultip

ly in

tege

rs a

nd m

onom

ials

by

poly

nom

ials

-di

vide

pol

ynom

ials

by

inte

gers

or

mon

omia

ls

-th

e pr

oduc

t of t

wo

bino

mia

ls

-th

e sq

uare

of a

bin

omia

l

Fact

oriz

e al

gebr

aic

expr

essi

ons

• Fa

ctor

ize

alge

brai

c ex

pres

sion

s th

at

invo

lve:

-co

mm

on fa

ctor

s

-di

ffere

nce

of tw

o sq

uare

s

-tri

nom

ials

of t

he fo

rm:

♦x2 +

bx

+ c

♦ax

2 + b

x +

c, w

here

a is

a c

omm

on

fact

or

• S

impl

ify a

lgeb

raic

exp

ress

ions

that

in

volv

e th

e ab

ove

fact

oris

atio

n pr

oces

ses

• S

impl

ify a

lgeb

raic

frac

tions

usi

ng

fact

oris

atio

n

exam

ples

of t

rinom

ials

that

can

be

fact

oriz

ed

a)

x2 + 5

x +

6

b)

x2 – 5

x +

6

c)

x2 – x

– 6

d)

x2 – 6

x +

9

e)

2x2 +

10x

+ 1

2



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.4

alg

ebra

iceq

uatio

ns

equa

tions

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-se

t up

equa

tions

to d

escr

ibe

prob

lem

situ

atio

ns

-an

alys

e an

d in

terp

ret e

quat

ions

th

at d

escr

ibe

a gi

ven

situ

atio

n

-so

lve

equa

tions

by:

♦in

spec

tion

♦us

ing

addi

tive

and

mul

tiplic

ativ

e in

vers

es

♦us

ing

law

s of

exp

onen

ts

-de

term

ine

the

num

eric

al v

alue

of

an e

xpre

ssio

n by

sub

stitu

tion.

-us

e su

bstit

utio

n in

equ

atio

ns to

ge

nera

te ta

bles

of o

rder

ed p

airs

• E

xten

d so

lvin

g eq

uatio

ns to

incl

ude:

-us

ing

fact

oris

atio

n

-eq

uatio

ns o

f the

form

: a p

rodu

ct o

f fa

ctor

s =

0

Alg

ebra

ic e

quat

ions

wer

e do

ne in

Ter

m 1

. The

focu

s in

this

term

is o

n so

lvin

g eq

uatio

ns u

sing

fact

oriz

atio

n, a

nd e

quat

ions

with

a p

rodu

ct o

f fac

tors

. Thi

s te

rm

also

focu

ses

on u

sing

equ

atio

ns to

gen

erat

e ta

bles

of o

rder

ed p

airs

.

see

addi

tiona

l not

es a

nd e

xam

ples

in t

erm

1.

exam

ples

of e

quat

ions

a)

Sol

ve x

if

x2 – 3

x =

0

x(

x –

3) =

0 (f

acto

rize

LHS

)

x

= 0

or

x –

3 =

0 (a

t lea

st o

ne fa

ctor

= 0

)

Th

eref

ore,

x =

0 o

r x

= 3

(add

3 to

bot

h si

des

of th

e se

cond

equ

atio

n)

b)

Sol

ve x

if

x2 – 2

5 =

0

(x

+ 5

)(x

– 5)

= 0

(fac

toriz

e th

e di

ffere

nce

of tw

o sq

uare

s on

LH

S)

x +

5 =

0 or

x –

5 =

0 (a

t lea

st o

ne fa

ctor

= 0

)

Th

eref

ore,

x =

– 5

(add

– 5

to b

oth

side

s of

the

equa

tion)

or

x =

5 (a

dd 5

to

both

sid

es o

f the

equ

atio

n)

c)

Writ

e an

equ

atio

n to

find

the

volu

me

of a

rect

angu

lar p

rism

with

leng

th 2

x cm

; w

idth

(2x

+ 1)

cm

and

hei

ght (

2x +

3) c

m

d)

If y

= 2x

2 +

4x +

3, c

alcu

late

y w

hen

x =

–2

e)

Than

di is

6 y

ears

old

er th

an S

ophi

e. In

3 y

ears

' tim

e Th

andi

will

be

twic

e as

old

as

Sop

hie.

How

old

is T

hand

i now

?

exam

ples

of g

ener

atin

g or

dere

d pa

irs

a)

Com

plet

e th

e ta

ble

belo

w fo

r x a

nd y

val

ues

for t

he e

quat

ion:

y =

2x2

– 3

x–2

–10

12

y

b)

Com

plet

e th

e ta

ble

belo

w fo

r x a

nd y

val

ues

for t

he e

quat

ion:

y =

x2 –

2

x–3

–20

y–2

2

Tim

e fo

r al

gebr

aic

equa

tions

in

this

term

:

9 ho

urs


145CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

2.5

Gra

phs

inte

rpre

ting

grap

hs

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-an

alys

e an

d in

terp

ret g

loba

l gr

aphs

of p

robl

em s

ituat

ions

, with

a

spec

ial f

ocus

on

the

follo

win

g tre

nds

and

feat

ures

:

♦lin

ear o

r non

-line

ar

♦co

nsta

nt, i

ncre

asin

g or

de

crea

sing

♦m

axim

um o

r min

imum

♦di

scre

te o

r con

tinuo

us

• E

xten

d th

e ab

ove

with

spe

cial

focu

s on

the

follo

win

g fe

atur

es o

f lin

ear

grap

hs:

-x-

inte

rcep

t and

y-in

terc

ept

-gr

adie

nt

dra

win

g gr

aphs

• R

evis

e th

e fo

llow

ing

done

in G

rade

8:

-dr

aw g

loba

l gra

phs

from

giv

en

desc

riptio

ns o

f a p

robl

em s

ituat

ion,

id

entif

ying

feat

ures

list

ed a

bove

.

-us

e ta

bles

of o

rder

ed p

airs

to p

lot

poin

ts a

nd d

raw

gra

phs

on th

e C

arte

sian

pla

ne

• E

xten

d th

e ab

ove

with

a s

peci

al

focu

s on

:

-dr

awin

g lin

ear g

raph

s fro

m g

iven

eq

uatio

ns

-de

term

ine

equa

tions

from

giv

en

linea

r gra

phs

Wha

t is

diffe

rent

to G

rade

8?

• x-

inte

rcep

t, y-

inte

rcep

t and

gra

dien

t of l

inea

r gra

phs

• D

raw

line

ar g

raph

s fro

m g

iven

equ

atio

ns

• D

eter

min

e eq

uatio

ns o

f lin

ear g

raph

s

Lear

ners

sho

uld

cont

inue

to a

naly

se a

nd in

terp

ret g

raph

s of

pro

blem

situ

atio

ns.

inve

stig

atin

g lin

ear g

raph

s

• To

ske

tch

linea

r gra

phs

from

giv

en e

quat

ions

, lea

rner

s sh

ould

firs

t dra

w u

p a

tabl

e of

ord

ered

pai

rs, t

hat i

nclu

des

the

inte

rcep

t poi

nts

(x; 0

) and

(0; y

), an

d th

en p

lotti

ng th

e po

ints

.

• Le

arne

rs s

houl

d in

vest

igat

e gr

adie

nts

by c

ompa

ring

vert

ical

cha

nge

hori

zont

al c

hang

e b

etw

een

any

two

poin

ts o

n a

stra

ight

line

gra

ph.

• Le

arne

rs s

houl

d al

so in

vest

igat

e th

e re

latio

nshi

p be

twee

n th

e va

lue

of th

e gr

adie

nt a

nd th

e co

effic

ient

of x

in th

e eq

uatio

n of

a s

traig

ht li

ne g

raph

.

• Le

arne

rs s

houl

d co

mpa

re y

-inte

rcep

ts o

f lin

ear g

raph

s to

the

valu

e of

the

cons

tant

in th

e eq

uatio

n of

the

stra

ight

line

gra

ph.

exam

ples

of l

inea

r gra

phs

a)

Ske

tch

and

com

pare

the

grap

hs o

f: y

= 4

and

x =

4

b)

Ske

tch

and

com

pare

the

grap

hs o

f: y

= x

and

y =

–x

c)

Ske

tch

and

com

pare

the

grap

hs o

f: y

= 2x

; y =

2x

+ 1;

y =

2x

– 1

d)

Ske

tch

and

com

pare

the

grap

hs o

f: y

= 3x

; y =

4x;

y =

5x

e)

Ske

tch

the

grap

hs o

f: y

= –3

x +

2; u

sing

the

tabl

e m

etho

d

f) D

eter

min

e th

e eq

uatio

n of

the

stra

ight

line

pas

sing

thro

ugh

the

follo

win

g po

ints

:

x–4

–3–2

–10

12

34

y–1

01

23

45

67

Tota

l tim

e fo

r gr

aphs

:

12 h

ours



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

mea

sure

men

t4.

2

surf

ace

area

an

d vo

lum

e of

3d

obj

ects

surf

ace

area

and

vol

ume

• U

se a

ppro

pria

te fo

rmul

ae a

nd

conv

ersi

ons

betw

een

SI u

nits

to

solv

e pr

oble

ms

and

calc

ulat

e th

e su

rface

are

a, v

olum

e an

d ca

paci

ty o

f:

-cu

bes

-re

ctan

gula

r pris

ms

-tri

angu

lar p

rism

s

-cy

linde

rs

• In

vest

igat

e ho

w d

oubl

ing

any

or a

ll th

e di

men

sion

s of

righ

t pris

ms

and

cylin

ders

affe

cts

the

volu

me

Wha

t is

diffe

rent

to G

rade

8?

• S

urfa

ce a

rea

and

volu

me

of c

ylin

ders

•Fo

rmul

ae le

arne

rs s

houl

d kn

ow a

nd u

se:

-th

e vo

lum

e of

a p

rism

= th

e ar

ea o

f the

bas

e x

the

heig

ht

-th

e su

rface

are

a of

a p

rism

= th

e su

m o

f the

are

a of

all

its fa

ces

-th

e vo

lum

e of

a c

ube

= l3

-th

e vo

lum

e of

a re

ctan

gula

r pris

m =

l x

b x

h

-th

e vo

lum

e of

a tr

iang

ular

pris

m =

(1 2

b x

h) x

hei

ght o

f the

pris

m

-th

e vo

lum

e of

a c

ylin

der =

(πr2

) x th

e he

ight

of t

he c

ylin

der

•Fo

r con

vers

ions

, not

e:

-if

1 cm

= 1

0 m

m th

en 1

cm

3 = 1

000

mm

3 ; an

d

- if

1 m

= 1

00 c

m th

en 1

m3 =

1 0

00 0

00 o

r 106

cm3

-an

obj

ect w

ith a

vol

ume

of 1

cm

3 will

dis

plac

e ex

actly

1 m

l of w

ater

; and

-an

obj

ect w

ith a

vol

ume

of 1

m3 w

ill d

ispl

ace

exac

tly 1

kl o

f wat

er.

exam

ples

of s

olvi

ng p

robl

ems

invo

lvin

g su

rfac

e ar

ea a

nd v

olum

e of

cy

linde

rs

• C

alcu

late

the

volu

me

of a

cyl

inde

r, w

ithou

t usi

ng a

cal

cula

tor i

f its

dia

met

er is

28

cm

, its

hei

ght i

s 30

cm

andπ=

22 7

• C

alcu

late

the

surfa

ce a

rea

of a

cyl

inde

r, co

rrec

t to

2 de

cim

al p

lace

s, if

its

heig

ht

is 6

5 cm

and

the

circ

umfe

renc

e of

its

base

is 4

7,6

cm.

Tota

l tim

e fo

r su

rface

are

a an

d vo

lum

e:

5 ho

urs

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• fu

nctio

ns a

nd re

latio

nshi

ps

• al

gebr

aic

expr

essi

ons

• A

lgeb

raic

equ

atio

ns

• gr

aphs

• vo

lum

e an

d su

rface

are

a

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

5 ho

urs


147CAPS

Gr

ad

e 9

– te

rm

4

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

spac

e an

d sh

ape

(geo

met

ry)

3.4

tran

sfor

ma-

tion

G

eom

etry

tran

sfor

mat

ions

• R

ecog

nize

, des

crib

e an

d pe

rform

tra

nsfo

rmat

ions

with

poi

nts,

line

se

gmen

ts a

nd s

impl

e ge

omet

ric

figur

es o

n a

co-o

rdin

ate

plan

e,

focu

sing

on:

-re

flect

ion

in th

e y-

axis

or x

-axi

s

-tra

nsla

tion

with

in a

nd a

cros

s qu

adra

nts

-re

flect

ion

in th

e lin

e y

= x

• Id

entif

y w

hat t

he tr

ansf

orm

atio

n of

a

poin

t is,

if g

iven

the

co-o

rdin

ates

of

its im

age

enla

rgem

ents

and

redu

ctio

ns•

Use

pro

porti

on to

des

crib

e th

e ef

fect

of

enl

arge

men

t or r

educ

tion

on a

rea

and

perim

eter

of g

eom

etric

figu

res

• In

vest

igat

e th

e co

-ord

inat

es o

f the

ve

rtice

s of

figu

res

that

hav

e be

en

enla

rged

or r

educ

ed b

y a

give

n sc

ale

fact

or

Wha

t is

diffe

rent

to G

rade

8?

• R

eflec

tion

in th

e lin

e y

= x

• Id

entif

y tra

nsfo

rmat

ions

from

co-

ordi

nate

poi

nts

of th

e im

age

• C

o-or

dina

tes

of v

ertic

es

Co-

ordi

nate

pla

ne•

Doi

ng tr

ansf

orm

atio

ns o

n th

e co

-ord

inat

e pl

ane

is a

n op

portu

nity

to p

ract

ise

read

ing

and

plot

ting

poin

ts w

ith o

rder

ed p

airs

, and

link

s to

dra

win

g al

gebr

aic

grap

hs.

• m

ake

sure

lear

ners

kno

w h

ow to

plo

t poi

nts

on th

e co

-ord

inat

e pl

ane

and

can

read

the

co-o

rdin

ates

of p

oint

s of

f the

x-a

xis

and

y-ax

is.

• M

ake

sure

lear

ners

kno

w th

e co

nven

tion

for w

ritin

g or

dere

d pa

irs (x

; y)

• P

oint

out

the

diffe

renc

es b

etw

een

the

axes

in th

e fo

ur q

uadr

ants

.

Focu

s of

tran

sfor

mat

ions

• D

oing

tran

sfor

mat

ions

on

a co

-ord

inat

e pl

ane

focu

ses

atte

ntio

n on

the

co-

ordi

nate

s of

poi

nts

and

verti

ces

of s

hape

s.

• Le

arne

rs s

houl

d re

cogn

ize

that

tran

slat

ions

, refl

ectio

ns a

nd ro

tatio

ns o

nly

chan

ge th

e po

sitio

n of

the

figur

e, a

nd n

ot it

s sh

ape

or s

ize.

• Le

arne

rs s

houl

d re

cogn

ize

that

the

abov

e tra

nsfo

rmat

ions

pro

duce

con

grue

nt

figur

es.

• Le

arne

rs s

houl

d be

gin

to s

ee p

atte

rns

in te

rms

of th

e co

-ord

inat

e po

ints

, for

the

diffe

rent

tran

sfor

mat

ions

, suc

h as

:

-fo

r tra

nsla

tions

to th

e rig

ht o

r lef

t, th

e x–

valu

e ch

ange

s an

d y–

valu

e st

ays

the

sam

e

-fo

r tra

nsla

tions

up

or d

own,

the

y–va

lue

chan

ges

and

the

x–va

lue

stay

s th

e sa

me

-fo

r refl

ectio

ns in

the

y–a

xis,

the

x–va

lue

chan

ges

sign

and

the

y–va

lue

stay

s th

e sa

me

-fo

r refl

ectio

ns in

the

x–a

xis,

the

y–va

lue

chan

ges

sign

and

the

x–va

lue

stay

s th

e sa

me

-fo

r refl

ectio

ns in

the

line

y =

x, th

e x–

valu

e an

d y–

valu

e ar

e in

terc

hang

ed.

• Le

arne

rs s

houl

d re

cogn

ize

that

enl

arge

men

ts a

nd re

duct

ions

cha

nge

the

size

of

figur

es b

y in

crea

sing

or d

ecre

asin

g th

e le

ngth

of s

ides

, but

kee

ping

the

angl

es

the

sam

e, p

rodu

ces

sim

ilar r

athe

r tha

n co

ngru

ent fi

gure

s.

• Le

arne

rs s

houl

d al

so b

e ab

le to

wor

k ou

t the

fact

or o

f enl

arge

men

t or r

educ

tion

of a

figu

re.

Tota

l tim

e fo

r tra

nsfo

rma-

tions

:

9 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

3.2

Geo

met

ry o

f 3d

obj

ects

Cla

ssify

ing

3d o

bjec

ts

• R

evis

e pr

oper

ties

and

defin

ition

s of

the

5 P

lato

nic

solid

s in

term

s of

th

e sh

ape

and

num

ber o

f fac

es, t

he

num

ber o

f ver

tices

and

the

num

ber

of e

dges

• R

ecog

nize

and

des

crib

e th

e pr

oper

ties

of:

-sp

here

s

-cy

linde

rs

Bui

ldin

g 3d

mod

els

• U

se n

ets

to c

reat

e m

odel

s of

ge

omet

ric s

olid

s, in

clud

ing:

-cu

bes

-pr

ism

s

-py

ram

ids

-cy

linde

rs

Wha

t is

diffe

rent

to G

rade

8?

• P

rope

rties

of s

pher

es a

nd c

ylin

ders

• N

ets

of c

ylin

ders

Plat

onic

sol

ids

• P

rope

rties

of P

lato

nic

solid

s sh

ould

be

revi

sed

• P

lato

nic

solid

s ar

e a

spec

ial g

roup

of p

olyh

edra

that

hav

e fa

ces

that

are

co

ngru

ent r

egul

ar p

olyg

ons.

• Th

ere

are

only

5 P

lato

nic

solid

s:

-te

trahe

dron

-he

xahe

dron

(cub

e)

-oc

tahe

dron

-do

deca

hedr

on

-ic

osah

edro

ns

• Th

e na

me

of e

ach

Pla

toni

c so

lid is

der

ived

from

its

num

ber o

f fac

es.

• P

lato

nic

solid

s pr

ovid

e an

inte

rest

ing

cont

ext i

n w

hich

to in

vest

igat

e th

e re

latio

nshi

p be

twee

n th

e nu

mbe

r of f

aces

, ver

tices

and

edg

es. B

y lis

ting

thes

e pr

oper

ties

for a

ll th

e P

lato

nic

solid

s le

arne

rs c

an in

vest

igat

e th

e pa

ttern

that

em

erge

s, to

com

e up

with

the

gene

ral r

ule:

V

– E

+ F

= 2,

whe

re V

= n

umbe

r of v

ertic

es; E

= n

umbe

r of e

dges

; F =

num

ber o

f fa

ces

usi

ng a

nd c

onst

ruct

ing

nets

• U

sing

and

con

stru

ctin

g ne

ts a

re u

sefu

l con

text

s fo

r exp

lorin

g or

con

solid

atin

g pr

oper

ties

of p

olyh

edra

.

• Le

arne

rs s

houl

d re

cogn

ize

the

nets

of d

iffer

ent s

olid

s.

• Le

arne

rs s

houl

d m

ake

sket

ches

of t

he n

ets

usin

g th

eir k

now

ledg

e of

the

the

shap

e an

d nu

mbe

r of f

aces

of t

he s

olid

s, b

efor

e dr

awin

g an

d cu

tting

out

the

nets

to

bui

ld m

odel

s.

• S

ince

lear

ners

hav

e m

ore

know

ledg

e ab

out t

he s

ize

of th

e in

tern

al a

ngle

s of

po

lygo

ns, a

nd c

an m

easu

re a

ngle

s, th

eir c

onst

ruct

ions

of n

ets

shou

ld b

e m

ore

accu

rate

.

• Le

arne

rs h

ave

to w

ork

out t

he re

lativ

e po

sitio

n of

face

s of

the

nets

in o

rder

to

build

the

3D m

odel

.

Tota

l tim

e fo

r ge

omet

ry o

f 3D

obj

ects

:

9 ho

urs


149CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

dat

a ha

ndlin

g5.

1 C

olle

ct,

orga

nize

and

su

mm

ariz

e da

ta

Col

lect

dat

a

• P

ose

ques

tions

rela

ting

to s

ocia

l, ec

onom

ic, a

nd e

nviro

nmen

tal i

ssue

s

• S

elec

t and

just

ify a

ppro

pria

te s

ourc

es

for t

he c

olle

ctio

n of

dat

a

• D

istin

guis

h be

twee

n sa

mpl

es a

nd

popu

latio

ns, a

nd s

ugge

st a

ppro

pria

te

sam

ples

for i

nves

tigat

ion

• S

elec

t and

just

ify a

ppro

pria

te

met

hods

for c

olle

ctin

g da

ta

org

aniz

e an

d su

mm

ariz

e da

ta

• O

rgan

ize

num

eric

al d

ata

in d

iffer

ent

way

s in

ord

er to

sum

mar

ize

by

dete

rmin

ing:

-m

easu

res

of c

entra

l ten

denc

y

-m

easu

res

of

disp

ersi

on

incl

udin

g ex

trem

es a

nd o

utlie

rs

• O

rgan

ize

data

acc

ordi

ng to

mor

e th

an o

ne c

riter

ia

Wha

t is

diffe

rent

to G

rade

8?

• O

rgan

izin

g da

ta a

ccor

ding

to m

ore

than

one

crit

eria

• O

utlie

rs

• S

catte

r plo

ts

dat

a se

ts a

nd c

onte

xts

Lear

ners

sho

uld

be e

xpos

ed to

a v

arie

ty o

f con

text

s th

at d

eal w

ith s

ocia

l and

en

viro

nmen

tal i

ssue

s, a

nd s

houl

d w

ork

with

giv

en d

ata

sets

, rep

rese

nted

in

a va

riety

of w

ays,

that

incl

ude

big

num

ber r

ange

s, p

erce

ntag

es a

nd d

ecim

al

fract

ions

. Lea

rner

s sh

ould

then

pra

ctis

e or

gani

zing

and

sum

mar

izin

g th

is d

ata,

an

alys

ing

and

inte

rpre

ting

the

data

, and

writ

ing

a re

port

abou

t the

dat

a.

Com

plet

e a

data

cyc

le

Lear

ners

sho

uld

com

plet

e at

leas

t one

dat

a cy

cle

for t

he y

ear,

star

ting

with

pos

ing

thei

r ow

n qu

estio

ns, s

elec

ting

the

sour

ces

and

met

hod

for c

olle

ctin

g, re

cord

ing,

or

gani

zing

, rep

rese

ntin

g, th

en a

naly

sing

, sum

mar

izin

g, in

terp

retin

g an

d re

porti

ng

on th

e da

ta. C

halle

nge

lear

ners

to th

ink

abou

t wha

t kin

ds o

f que

stio

ns a

nd d

ata

need

to b

e co

llect

ed to

be

repr

esen

ted

on a

his

togr

am, a

pie

cha

rt, a

bar

gra

ph, a

lin

e gr

aph

or a

sca

tter p

lot.

Tota

l tim

e fo

r co

llect

ing

and

orga

nizi

ng

data

:

4 ho

urs



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.2

rep

rese

nt

data

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy to

dis

play

and

inte

rpre

t da

ta in

clud

ing:

-ba

r gra

phs

and

doub

le b

ar g

raph

s

-hi

stog

ram

s w

ith

give

n an

d ow

n in

terv

als

-pi

e ch

arts

-br

oken

-line

gra

phs

-sc

atte

r plo

ts

rep

rese

ntin

g da

ta

• P

ie c

harts

to re

pres

ent d

ata

do n

ot h

ave

to b

e ac

cura

tely

dra

wn

with

a c

ompa

ss

and

prot

ract

or, e

tc. L

earn

ers

can

use

any

roun

d ob

ject

to d

raw

a c

ircle

, the

n di

vide

the

circ

le in

to h

alve

s an

d qu

arte

rs a

nd e

ight

hs if

nee

ded,

as

a gu

ide

to

estim

ate

the

prop

ortio

ns o

f the

circ

le th

at n

eed

to b

e sh

own

to re

pres

ent t

he

data

. Wha

t is

impo

rtant

is th

at th

e va

lues

or p

erce

ntag

es a

ssoc

iate

d w

ith th

e da

ta, a

re s

how

n pr

opor

tiona

lly o

n th

e pi

e ch

art.

• D

raw

ing,

read

ing

and

inte

rpre

ting

pie

char

ts is

a u

sefu

l con

text

to re

-vis

it eq

uiva

lenc

e be

twee

n fra

ctio

ns a

nd p

erce

ntag

es, e

.g. 2

5% o

f the

dat

a is

re

pres

ente

d by

a 1 4 s

ecto

r of t

he c

ircle

.

• It

is a

lso

a co

ntex

t in

whi

ch le

arne

rs c

an fi

nd p

erce

ntag

es o

f who

le n

umbe

rs e

.g.

if 25

% o

f 300

lear

ners

like

rugb

y, h

ow m

any

(act

ual n

umbe

r) le

arne

rs li

ke ru

gby?

• H

isto

gram

s ar

e us

ed to

repr

esen

t gro

uped

dat

a sh

own

in in

terv

als

on th

e ho

rizon

tal a

xis

of th

e gr

aph.

Poi

nt o

ut th

e di

ffere

nces

bet

wee

n hi

stog

ram

s an

d ba

r gra

phs,

in p

artic

ular

bar

gra

phs

that

repr

esen

t dis

cret

e da

ta (e

.g. f

avou

rite

spor

ts) c

ompa

red

to h

isto

gram

s th

at s

how

cat

egor

ies

in c

onse

cutiv

e, n

on-

over

lapp

ing

inte

rval

, (e.

g. te

st s

core

s ou

t of 1

00 s

how

n in

inte

rval

s of

10)

. The

ba

rs o

n ba

r gra

phs

do n

ot h

ave

to to

uch

each

oth

er, w

hile

in a

his

togr

am th

ey

have

to to

uch

sinc

e th

ey s

how

con

secu

tive

inte

rval

s.

• B

roke

n-lin

e gr

aphs

refe

r to

data

gra

phs

that

repr

esen

t dat

a po

ints

join

ed b

y a

line

and

are

not t

he s

ame

as s

traig

ht li

ne g

raph

s th

at a

re d

raw

n us

ing

the

equa

tion

of th

e lin

e.

• B

roke

n-lin

e gr

aphs

are

use

d to

repr

esen

t dat

a th

at c

hang

es c

ontin

uous

ly o

ver

time,

e.g

. ave

rage

dai

ly te

mpe

ratu

re fo

r a m

onth

. Eac

h da

y’s

tem

pera

ture

is

repr

esen

ted

by a

poi

nt o

n th

e gr

aph,

and

onc

e th

e w

hole

mon

th h

as b

een

plot

ted,

the

poin

ts a

re jo

ined

to s

how

a b

roke

- lin

e gr

aph.

• B

roke

n-lin

e gr

aphs

are

use

ful t

o re

ad ‘t

rend

s’ a

nd p

atte

rns

in th

e da

ta, f

or

pred

ictiv

e pu

rpos

es e

.g. W

ill th

e te

mpe

ratu

res

go u

p or

dow

n in

the

next

mon

th?

• A

scat

ter p

lot i

s us

ed to

repr

esen

t dat

a th

at in

volv

es tw

o di

ffere

nt c

riter

ia a

nd

the

grap

h is

use

d to

look

at t

he re

latio

nshi

p be

twee

n th

e tw

o cr

iteria

. e.g

. How

do

es th

e pe

rform

ance

of l

earn

ers

in m

athe

mat

ics

com

pare

to th

eir p

erfo

rman

ce

in E

nglis

h? E

ach

poin

t on

the

grap

h re

pres

ents

the

resu

lts o

f one

lear

ner i

n M

athe

mat

ics

and

Eng

lish.

Afte

r all

the

resu

lts h

ave

been

plo

tted,

you

can

co

mpa

re th

e re

latio

nshi

p be

twee

n pe

rform

ance

in E

nglis

h an

d m

athe

mat

ics

for

all t

he le

arne

rs i.

e. if

they

sco

re h

igh

in M

athe

mat

ics,

do

they

als

o sc

ore

high

in

Eng

lish?

or,

if th

ey s

core

hig

h in

mat

hem

atic

s do

they

sco

re lo

w in

Eng

lish,

or

is th

ere

no re

latio

nshi

p be

twee

n w

hat t

hey

scor

e on

mat

hem

atic

s to

wha

t the

y sc

ore

in E

nglis

h?

• Th

e sc

atte

r plo

t allo

ws

one

to s

ee tr

ends

and

mak

e pr

edic

tions

, as

wel

l as

iden

tify

outli

ers

in th

e da

ta.

Tota

l tim

e fo

r re

pres

entin

g da

ta:

3 ho

urs


151CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.2

rep

rese

nt

data

rep

rese

nt d

ata

• D

raw

a v

arie

ty o

f gra

phs

by h

and/

tech

nolo

gy to

dis

play

and

inte

rpre

t da

ta in

clud

ing:

-ba

r gra

phs

and

doub

le b

ar g

raph

s

-hi

stog

ram

s w

ith

give

n an

d ow

n in

terv

als

-pi

e ch

arts

-br

oken

-line

gra

phs

-sc

atte

r plo

ts

rep

rese

ntin

g da

ta

• P

ie c

harts

to re

pres

ent d

ata

do n

ot h

ave

to b

e ac

cura

tely

dra

wn

with

a c

ompa

ss

and

prot

ract

or, e

tc. L

earn

ers

can

use

any

roun

d ob

ject

to d

raw

a c

ircle

, the

n di

vide

the

circ

le in

to h

alve

s an

d qu

arte

rs a

nd e

ight

hs if

nee

ded,

as

a gu

ide

to

estim

ate

the

prop

ortio

ns o

f the

circ

le th

at n

eed

to b

e sh

own

to re

pres

ent t

he

data

. Wha

t is

impo

rtant

is th

at th

e va

lues

or p

erce

ntag

es a

ssoc

iate

d w

ith th

e da

ta, a

re s

how

n pr

opor

tiona

lly o

n th

e pi

e ch

art.

• D

raw

ing,

read

ing

and

inte

rpre

ting

pie

char

ts is

a u

sefu

l con

text

to re

-vis

it eq

uiva

lenc

e be

twee

n fra

ctio

ns a

nd p

erce

ntag

es, e

.g. 2

5% o

f the

dat

a is

re

pres

ente

d by

a 1 4 s

ecto

r of t

he c

ircle

.

• It

is a

lso

a co

ntex

t in

whi

ch le

arne

rs c

an fi

nd p

erce

ntag

es o

f who

le n

umbe

rs e

.g.

if 25

% o

f 300

lear

ners

like

rugb

y, h

ow m

any

(act

ual n

umbe

r) le

arne

rs li

ke ru

gby?

• H

isto

gram

s ar

e us

ed to

repr

esen

t gro

uped

dat

a sh

own

in in

terv

als

on th

e ho

rizon

tal a

xis

of th

e gr

aph.

Poi

nt o

ut th

e di

ffere

nces

bet

wee

n hi

stog

ram

s an

d ba

r gra

phs,

in p

artic

ular

bar

gra

phs

that

repr

esen

t dis

cret

e da

ta (e

.g. f

avou

rite

spor

ts) c

ompa

red

to h

isto

gram

s th

at s

how

cat

egor

ies

in c

onse

cutiv

e, n

on-

over

lapp

ing

inte

rval

, (e.

g. te

st s

core

s ou

t of 1

00 s

how

n in

inte

rval

s of

10)

. The

ba

rs o

n ba

r gra

phs

do n

ot h

ave

to to

uch

each

oth

er, w

hile

in a

his

togr

am th

ey

have

to to

uch

sinc

e th

ey s

how

con

secu

tive

inte

rval

s.

• B

roke

n-lin

e gr

aphs

refe

r to

data

gra

phs

that

repr

esen

t dat

a po

ints

join

ed b

y a

line

and

are

not t

he s

ame

as s

traig

ht li

ne g

raph

s th

at a

re d

raw

n us

ing

the

equa

tion

of th

e lin

e.

• B

roke

n-lin

e gr

aphs

are

use

d to

repr

esen

t dat

a th

at c

hang

es c

ontin

uous

ly o

ver

time,

e.g

. ave

rage

dai

ly te

mpe

ratu

re fo

r a m

onth

. Eac

h da

y’s

tem

pera

ture

is

repr

esen

ted

by a

poi

nt o

n th

e gr

aph,

and

onc

e th

e w

hole

mon

th h

as b

een

plot

ted,

the

poin

ts a

re jo

ined

to s

how

a b

roke

- lin

e gr

aph.

• B

roke

n-lin

e gr

aphs

are

use

ful t

o re

ad ‘t

rend

s’ a

nd p

atte

rns

in th

e da

ta, f

or

pred

ictiv

e pu

rpos

es e

.g. W

ill th

e te

mpe

ratu

res

go u

p or

dow

n in

the

next

mon

th?

• A

scat

ter p

lot i

s us

ed to

repr

esen

t dat

a th

at in

volv

es tw

o di

ffere

nt c

riter

ia a

nd

the

grap

h is

use

d to

look

at t

he re

latio

nshi

p be

twee

n th

e tw

o cr

iteria

. e.g

. How

do

es th

e pe

rform

ance

of l

earn

ers

in m

athe

mat

ics

com

pare

to th

eir p

erfo

rman

ce

in E

nglis

h? E

ach

poin

t on

the

grap

h re

pres

ents

the

resu

lts o

f one

lear

ner i

n M

athe

mat

ics

and

Eng

lish.

Afte

r all

the

resu

lts h

ave

been

plo

tted,

you

can

co

mpa

re th

e re

latio

nshi

p be

twee

n pe

rform

ance

in E

nglis

h an

d m

athe

mat

ics

for

all t

he le

arne

rs i.

e. if

they

sco

re h

igh

in M

athe

mat

ics,

do

they

als

o sc

ore

high

in

Eng

lish?

or,

if th

ey s

core

hig

h in

mat

hem

atic

s do

they

sco

re lo

w in

Eng

lish,

or

is th

ere

no re

latio

nshi

p be

twee

n w

hat t

hey

scor

e on

mat

hem

atic

s to

wha

t the

y sc

ore

in E

nglis

h?

• Th

e sc

atte

r plo

t allo

ws

one

to s

ee tr

ends

and

mak

e pr

edic

tions

, as

wel

l as

iden

tify

outli

ers

in th

e da

ta.

Tota

l tim

e fo

r re

pres

entin

g da

ta:

3 ho

urs

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.3

inte

rpre

t, an

alys

e an

d re

port

dat

a

inte

rpre

t dat

a

• C

ritic

ally

read

and

inte

rpre

t dat

a re

pres

ente

d in

a v

arie

ty o

f way

s.

• C

ritic

ally

com

pare

two

sets

of d

ata

rela

ted

to th

e sa

me

issu

e

ana

lyse

dat

a

• C

ritic

ally

ana

lyse

dat

a by

ans

wer

ing

ques

tions

rela

ted

to:

-D

ata

colle

ctio

n m

etho

ds

-S

umm

ary

stat

istic

s of

dat

a

-S

ourc

es o

f err

or a

nd b

ias

in th

e da

ta

rep

ort d

ata

• S

umm

ariz

e da

ta in

sho

rt pa

ragr

aphs

th

at in

clud

e

-dr

awin

g co

nclu

sion

s ab

out t

he d

ata

-m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

-m

akin

g co

mpa

rison

s be

twee

n tw

o se

ts o

f dat

a

-id

entif

ying

sou

rces

of e

rror

and

bi

as in

the

data

-ch

oosi

ng a

ppro

pria

te s

umm

ary

stat

istic

s fo

r the

dat

a (m

ean,

m

edia

n, m

ode,

rang

e)

-th

e ro

le o

f ext

rem

es a

nd o

utlie

rs in

th

e da

ta

dev

elop

ing

criti

cal a

naly

sis

skill

s

• Le

arne

rs s

houl

d co

mpa

re th

e sa

me

data

repr

esen

ted

in d

iffer

ent w

ays

e.g.

in a

pi

e ch

art o

r a b

ar g

raph

or a

tabl

e, a

nd d

iscu

ss w

hat i

nfor

mat

ion

is s

how

n an

d w

hat i

s hi

dden

; the

y sh

ould

eva

luat

e w

hich

form

of r

epre

sent

atio

n w

orks

bes

t for

th

e gi

ven

data

.

• Le

arne

rs s

houl

d co

mpa

re g

raph

s on

the

sam

e to

pic

but w

here

dat

a ha

s be

en

colle

cted

from

diff

eren

t gro

ups

of p

eopl

e, a

t diff

eren

t tim

es, i

n di

ffere

nt p

lace

s or

in d

iffer

ent w

ays.

Her

e le

arne

rs s

houl

d di

scus

s di

ffere

nces

bet

wee

n th

e da

ta

with

an

awar

enes

s of

bia

s re

late

d to

the

impa

ct o

f dat

a so

urce

s an

d m

etho

ds o

f da

ta c

olle

ctio

n on

the

inte

rpre

tatio

n of

the

data

.

• Le

arne

rs s

houl

d co

mpa

re d

iffer

ent w

ays

of s

umm

ariz

ing

the

sam

e da

ta s

ets,

de

velo

ping

an

awar

enes

s of

how

dat

a re

porti

ng c

an b

e m

anip

ulat

ed; t

hey

shou

ld e

valu

ate

whi

ch s

umm

ary

stat

istic

s be

st re

pres

ent t

he d

ata.

• Le

arne

rs s

houl

d co

mpa

re g

raph

s of

the

sam

e da

ta, w

here

the

scal

es o

f th

e gr

aphs

are

diff

eren

t. H

ere

lear

ners

sho

uld

disc

uss

diffe

renc

es w

ith a

n aw

aren

ess

of h

ow re

pres

enta

tion

of d

ata

can

be m

anip

ulat

ed; t

hey

shou

ld

eval

uate

whi

ch fo

rm o

f rep

rese

ntat

ion

wor

ks b

est f

or th

e gi

ven

data

.

• Le

arne

rs s

houl

d co

mpa

re d

ata

on th

e sa

me

topi

c, w

here

one

set

of d

ata

has

extre

mes

or o

utlie

rs, a

nd d

iscu

ss d

iffer

ence

s w

ith a

n aw

aren

ess

of th

e ef

fect

of

the

extre

mes

or o

utlie

rs o

n th

e in

terp

reta

tion

of th

e da

ta. I

n pa

rticu

lar,

extre

mes

af

fect

the

rang

e an

d ou

tlier

s w

hich

are

iden

tified

on

scat

ter p

lots

.

• Le

arne

rs s

houl

d w

rite

repo

rts o

n th

e da

ta in

sho

rt pa

ragr

aphs

.

Tota

l tim

e fo

r an

alys

ing,

in

terp

retin

g,

sum

mar

izin

g an

d re

porti

ng

data

:

3,5

hour

s



Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.4

Prob

abili

tyPr

obab

ility

• C

onsi

der s

ituat

ions

with

equ

ally

pr

obab

le o

utco

mes

, and

:

-de

term

ine

prob

abili

ties

of

com

poun

d ev

ents

usi

ng tw

o-w

ay

tabl

es a

nd tr

ee d

iagr

ams

-de

term

ine

the

prob

abili

ties

of

outc

omes

of e

vent

s an

d pr

edic

t th

eir r

elat

ive

frequ

ency

in s

impl

e ex

perim

ents

-co

mpa

re re

lativ

e fre

quen

cy w

ith

prob

abili

ty a

nd e

xpla

in p

ossi

ble

diffe

renc

es

Prob

abili

ty e

xper

imen

tsIn

Gra

des

8 an

d 9

prob

abili

ty e

xper

imen

ts a

re le

ss im

porta

nt, a

nd le

arne

rs s

houl

d co

nsid

er p

roba

bilit

y fo

r hyp

othe

tical

eve

nts.

In G

rade

9, l

earn

ers

have

to c

onsi

der

outc

omes

of c

ompo

und

even

ts a

nd u

se tw

o-w

ay ta

bles

and

tree

dia

gram

s to

wor

k ou

t the

pro

babi

lity

of a

n ou

tcom

e.

Prob

abili

ty o

f com

poun

d ev

ents

For e

xam

ple,

wha

t is

the

prob

abili

ty o

f a w

oman

giv

ing

birth

to tw

o bo

ys a

fter e

ach

othe

r? A

two-

way

tabl

e ca

n be

use

d:

1st b

irth

2nd

birth

Boy

Girl

Boy

BB

BG

Girl

GB

GG

Two

boys

afte

r eac

h ot

her (

BB

) is

1 of

4 p

ossi

ble

outc

omes

, so

the

prob

abili

ty o

f tw

o bo

ys is

1 o

ut o

f 4, o

r 25%

.

How

doe

s th

is p

roba

bilit

y ch

ange

if y

ou a

sk w

hat i

s th

e pr

obab

ility

of g

ivin

g bi

rth to

th

ree

boys

afte

r eac

h ot

her?

A tr

ee d

iagr

am c

an th

en b

e us

ed:

Girl

[GG

G]

Girl

Boy

Girl

[BB

G]

Boy

[BBB

]

Girl

[BG

G]

Boy

[BGB

]

Girl

[GB

G]

Boy

[GBB

] Bo

y [G

GB]

Boy

Girl

Boy

Girl

2nd b

irth

3rd b

irth

1st b

irth

Tota

l tim

e fo

r pr

obab

ility

:

4,5

hour

s


153CAPS

Co

nte

nt

ar

eato

PiC

sC

on

CeP

ts a

nd

sK

ills

som

e C

lar

iFiC

atio

n n

ote

s o

r t

eaC

Hin

G G

uid

elin

esd

ur

atio

n

(in h

ours

)

5.4

Prob

abili

tyto

tal o

utco

mes

8

Thre

e bo

ys a

fter e

ach

othe

r, [B

BB

], is

1 o

ut 8

of p

ossi

ble

outc

omes

, hen

ce th

e pr

obab

ility

of t

hree

boy

s af

ter e

ach

othe

r is

1 8 or 1

2,5%

.

rev

isio

n/a

sses

smen

t:A

t thi

s st

age

lear

ners

sho

uld

have

bee

n as

sess

ed o

n:

• tra

nsfo

rmat

ion

geom

etry

• ge

omet

ry o

f 3D

obj

ects

• co

llect

ing,

org

aniz

ing,

repr

esen

ting,

ana

lysi

ng, s

umm

ariz

ing,

inte

rpre

ting

and

repo

rting

dat

a

• pr

obab

ility

Tota

l tim

e fo

r rev

isio

n/as

sess

men

t fo

r the

term

12 h

ours



SECTION 4: ASSESSMENT

4.1 introduCtion

Assessment is a continuous planned process of identifying, gathering and interpreting information regarding the per-formance 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 thereby assist the learner’s development in order to improve the process of learning and teaching. Assessment should be both informal and formal. In both cases regular feedback should be provided to learners to enhance their learning experience. This will assist the learner to achieve the minimum performance level of 40% to 49% required in Math-ematics for promotion purposes.

4.2 tyPes oF assessment

The following types of assessment are very useful in mathematics; as a result teachers are encouraged to use them to serve the purpose associated with each.

Baseline assessment: mathematics teachers who might want to establish whether their learners meet the basic skills and knowledge levels required to learn a specific Mathematics topic will use baseline assessment. Knowing learners’ level of proficiency in a particular Mathematics topic enables the teacher to plan her/his Mathematics lesson appropriately and to pitch it at the appropriate level. Baseline assessment, as the name suggests, should therefore be administered prior to teaching a particular Mathematics topic. The results of the baseline assessment should not be used for promotion purposes.

diagnostic assessment: It is not intended for promotion purposes but to inform the teacher about the learner’s Mathematics problem areas that have the potential to hinder performance. Two broad areas form the basis of diag-nostic assessment: content-related challenges where learners find certain difficulties to comprehend, and psycho-social factors such as negative attitudes, Mathematics anxiety, poor study habits, poor problem-solving behaviour, etc. Appropriate interventions should be implemented to assist learners in overcoming these challenges early in their school careers.

Formative assessment: Formative assessment is used to aid the teaching and learning processes, hence assess-ment for learning. It is the most commonly used type of assessment because it can be used in different forms at any time during a Mathematics lesson, e.g. short class works during or at the end of each lesson, verbal questioning dur-ing the lesson. It is mainly informal and should not be used for promotion purposes. The fundamental distinguishing characteristic of formative assessment is constant feedback to learners, particularly with regard to learners’ learning processes. The information provided by formative assessment can also be used by teachers to inform their methods of teaching.

summative assessment: Contrary to the character of formative assessment, summative assessment is carried out after the completion of a Mathematics topic or a cluster of related topics. It is therefore referred to as assessment of learning since it is mainly focusing on the product of learning. The results of summative assessment are recorded and used for promotion purposes. The forms of assessment presented in Table 4.1 are examples of summative as-sessment.


155CAPS

4.3 inFormal or daily assessment

Assessment for learning has the purpose of continuously collecting information on learner performance that can be used to improve their learning.

Informal assessment is a daily monitoring of learners’ progress. This is done through observations, discussions, prac-tical demonstrations, learner-teacher conferences, informal classroom interactions, etc. 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 learners and to inform planning for teaching, but need not be recorded. It should not be seen as separate from the learning activities taking place in the classroom.

Self-assessment and peer assessment actively allow learners to assess themselves. This is important as it allows learners to learn from, and reflect on their own performance. The results of the informal daily assessment tasks are not formally recorded unless the teacher wishes to do so. The results of daily assessment tasks are not taken into account for promotion purposes.

4.4 Formal assessment

Formal assessment comprises School-Based Assessment (SBA) and End of the Year Examination. Formal assessment tasks are marked and formally recorded by the teacher for promotion purposes. All Formal Assessment tasks are subject to moderation for the purpose of quality assurance and to ensure that appropriate standards are maintained. The SBA component may take various forms. However, tests, examinations, projects, assignments and investigations are recommended for Mathematics. The Senior Phase Mathematics minimum formal programme of assessment tasks are outlined in Table 4.1

table 4.1: minimum requirements for formal assessment: senior Phase mathematics

Forms of assessment

minimum requirements per term number of tasks per

yearWeighting

Term 1 Term 2 Term 3 Term 4

sBa

Test 1 1 1 3

40%

Examination 1 1

Assignment 1 1 1 3

Investigation 1 1 2

Project 1 1

total 2 3 3 2 10*

end of the year examination 1 60%

*To be completed before the End of the year Examination

tests and examinations are individualised assessment tasks and should be carefully designed to ensure that learners demonstrate their full potential in Mathematics content. The questions should be carefully spread to cater for different cognitive levels of learners. Tests and examinations are predominantly assessed using a memorandum.

the assignment, as is the case with tests and examinations, is mainly an individualised task. It can be a collection of past questions, but should focus on more demanding work as any resource material can be used, which is not the case in a task that is done in class under supervision.



Projects are used to assess a range of skills and competencies. Through projects, learners are able to demonstrate their understanding of different Mathematics concepts and apply them in real-life situations. Caution should, how-ever, be exercised not to give projects that are above learners’ cognitive levels. The assessment criteria should be clearly indicated on the project specification and should focus on the Mathematics involved and not on duplicated pictures and facts copied from reference material. Good projects contain the collection and display of real data, fol-lowed by deductions that can be substantiated.

Investigation promotes critical and creative thinking. It can be used to discover rules or concepts and may involve inductive reasoning, identifying or testing patterns or relationships, drawing conclusions, and establishing general trends. To avoid having to assess work which is copied without understanding, it is recommended that whilst initial investigation could be done at home, the final write-up should be done in class, under supervision, without access to any notes. Investigations are assessed with rubrics, which can be specific to the task, or generic, listing the number of marks awarded for each skill. These skills include:

• organizing and recording ideas and discoveries using, for example, diagrams and tables.

• communicating ideas with appropriate explanations

• calculations showing clear understanding of mathematical concepts and procedures.

• generalizing and drawing conclusions,

The forms of assessment used should be appropriate to the age and cognitive level of learners. The design of these tasks should cover the content of the subject and designed to achieve the broad aims of the subject. Appropriate instruments, such as rubrics and memoranda, should be used for marking. Formal assessments should cater for a range of cognitive levels and abilities of learners as shown in Table 4.2:


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table 4.2: Cognitive levels

desCriPtion and eXamPles oF CoGnitive levelsCognitive levels description of skills to be demonstrated examples

Knowledge(≈25%)

• Estimation and appropriate rounding of numbers

• Straight recall

• Identification and direct use of correct formula

• Use of mathematical facts

• Appropriate use of mathematical vocabulary

1. Estimate the answer and then calculate with a calculator:

62 816325 + 279 [Grade 7]

2. Use the formula A = πr2 to calculate the area of a circle if the diameter is equal to 10 cm. [Grade 8]

3. Write down the y-intercept of the function y = 2x + 1 [Grade 9]

routine procedures

(≈45%)

• Perform well-known procedures

• Simple applications and calculations which might involve many 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. Determine the mean of 5 Grade 7 learners’ marks if they have respectively achieved 25; 40; 21; 85; 14 out of 50. [Grade 7]

2. Solve x in x – 6 = 9 [Grade 8]

3. R600 invested at r% per annum for a period of 3 years yields R150 interest. Calculate the value of r if SI = P.n.r

100 . [Grade 9]

Complex procedures

(≈20%)

• Problems involving complex calculations and/or higher order reasoning

• Investigate elementary axioms to generalize them into proofs for straight line geometry, congruence and similarity

• No obvious route to the solution

• Problems not necessarily based on real world contexts

• Making significant connections between different representations

• Require conceptual understanding

1. Mr Mnisi pays R75 for a book which he marks up to provide 20% profit. He then sells it for cash at 4% discount. Calculate the selling price. [Grade 7]

2. A car travelling at a constant speed travels 60 km in 18 minutes. How far, travelling at the same constant speed, will the car travel in 1 hour 12 minutes? [Grade 8]

3. Use investigation skills to prove that the angles on a straight line are supplementary. [Grade 9]

Problem solving(≈10%)

• Unseen, non-routine problems (which are not necessarily difficult)

• Higher order understanding and processes are often involved

• might require the ability to break the problem down into its constituent parts

1. The sum of three consecutive numbers is 87. Find the numbers. [Grade 7]

2. Mary travels a distance of km in 6 hours if she travels at an average speed of 20 km/h on her bicycle. What should be her average speed if she wants to cover the same distance in 5 hours? [Grade 8]

3. The combined age of a father and son is 84 years old. In 6 years time the father will be twice as old as the son was 3 years ago. How old are they now? [Grade 9]

4.5 reCordinG and rePortinG

Recording is a process in which the teacher documents the level of a learner’s performance in a specific assessment task. It indicates the learner’s progress towards the achievement of the knowledge as prescribed in the National 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 be promoted to the next grade. Records of learner performance should also be used to verify 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. Primary schooling is a critical period for the acquisition of foundational Mathematics skills and conceptual knowledge. Reporting of learner performance is therefore essential and should not be limited to the quarterly report card. Other methods of reporting should be explored, e.g. parents’ meetings, school visitation days, parent-teacher conferences, phone calls, letters. These extreme, but worthwhile modalities will ensure that any underperformance is communi-cated promptly and appropriate measures of intervention are implemented collaboratively by teachers and parents. Formal reporting is done on a 7-point rating scale (see Table 4.3)

table 4.3: scale of achievement for the national Curriculum statement Grades 7 - 9

ratinG Code desCriPtion oF ComPetenCe PerCentaGe

7 Outstanding achievement 80 – 100

6 meritorious achievement 70 – 79

5 Substantial achievement 60 – 69

4 Adequate achievement 50 – 59

3 moderate achievement 40 – 49

2 Elementary achievement 30 – 39

1 Not achieved 0 – 29

4.6 moderation oF assessment

Moderation refers to the process that ensures that the assessment tasks are fair, valid and reliable. Moderation should be carried out internally at school and/or externally at district, provincial and national levels. Given that the promotion of learners in the Senior Phase is largely dependent upon the SBA (which contributes 40%); the moderation process should be intensified to ensure that:

• learners are not disadvantaged by the invalid and unreliable assessment tasks,

• quality assessment is given and high but achievable standards are maintained.

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 National Protocol for Assessment Grades R-12.


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senior phase - department of basic education

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