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IGCSE

Mathematics

Teaching the IGCSE in Mathematics (Specification A) alongside Core Mathematics 1

Scheme of work: Bridging to GCE

Edexcel IGCSE in Mathematics (Specification A) (4MA0)First examination 2012

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Acknowledgements

This document has been produced by Edexcel on the basis of consultation with

teachers, examiners, consultants and other interested parties. Edexcel would like to

thank all those who contributed their time and expertise to its development.

References to third-party material made in this document are made in good faith.

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websites.)

Authorised by Martin Stretton

Prepared by Sharon Wood

Publications Code UG029447

All the material in this publication is copyright

© Pearson Education Limited 2011

Introduction

The Edexcel IGCSE in Mathematics (Specification A) is designed for use in schools

and colleges. It is part of a suite of qualifications offered by Edexcel.

About this scheme of work

It provides opportunities for you to extend and enrich the mathematical learning of

your Higher Tier students. It is assumed that Higher Tier students have knowledge

of all Foundation Tier content. This scheme of work should be used together with

the higher tier course planner in the IGCSE in Mathematics (Specification A)

Teacher’s Guide.

Through teaching the Edexcel GCE Core Mathematics 1 unit alongside the Higher

Tier content, you will be able to prepare your Higher Tier students for the transition

from Level 2 Mathematics to AS Mathematics, and beyond. It also enables you to

extend several topic areas of the Mathematics IGCSE Higher Tier content.

This scheme of work introduces the following topics from Edexcel GCE Unit Core

Mathematics 1 in the first year of course:

Algebra and functions

Coordinate Geometry

The remaining topics from Edexcel GCE Unit Core Mathematics 1 could be taught

during the second year of the course:

Sequences and series

Differentiation

Integration.

This means that the scheme of work extends the following topic areas:

1.4 Powers and roots

2.2 Algebraic manipulation

2.7 Quadratic equations

2.8 Inequalities

3.3 Graphs.

It introduces new concepts within the following topic area:

3.4 Calculus – Differentiation.

It also introduces the following topic areas not included in the IGCSE:

Sequences and series

Integration.

Contents

Mapping of IGCSE Mathematics Higher tier content to GCE Core Mathematics 1 unit content 1

Higher tier IGCSE Mathematics/GCE Core Mathematics 1 unit content summary 33

Higher tier IGCSE Mathematics/GCE Core Mathematics 1 unit scheme of work 35

Mappin

g o

f IG

CSE M

ath

em

ati

cs

Hig

her

tier

conte

nt

to G

CE C

ore

Math

em

ati

cs

1 u

nit

conte

nt

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

1

Nu

mb

ers a

nd

th

e

nu

mb

er s

yste

m

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts n

eed

to learn

No

tes

1.1

In

teg

ers

See F

oundation T

ier

1.2

Fracti

on

s

See F

oundation T

ier

1.3

D

ecim

als

convert

recurr

ing

decim

als

into

fra

ctions

3.0

=

31,

0.2

33

3…

=

9021

1.4

P

ow

ers a

nd

ro

ots

unders

tand t

he

meanin

g o

f surd

s

m

anip

ula

te s

urd

s,

inclu

din

g r

ationalisin

g

the d

enom

inato

r w

here

the d

enom

inato

r is

a

pure

surd

Expre

ss in t

he f

orm

a2

: 82

, 1

8 +

32

Expre

ss in t

he f

orm

a +

b2

: (

3 +

52

)2

Exte

nsio

n t

op

ic

Use a

nd

man

ipu

lati

on

of

su

rd

s

Stu

den

ts s

ho

uld

be

ab

le t

o r

ati

on

alise

den

om

inato

rs

use index law

s t

o

sim

plify

and e

valu

ate

num

eri

cal expre

ssio

ns

involv

ing inte

ger,

fractional and n

egative

pow

ers

Evalu

ate

:

38

2,

21

62

5

,

23

251

Exte

nsio

n t

op

ic

Law

s o

f in

dic

es f

or

all r

ati

on

al

exp

on

en

ts

Th

e e

qu

ivale

nce o

f

nm

a a

nd

n

ma

sh

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ld

be k

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1

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un

it

AO

1 N

um

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nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

1

Nu

mb

ers a

nd

th

e

nu

mb

er s

yste

m

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts n

eed

to learn

No

tes

1.4

P

ow

ers a

nd

ro

ots

co

nti

nu

ed

evalu

ate

Hig

hest

Com

mon F

acto

rs (

HCF)

and L

ow

est

Com

mon

Multip

les (

LCM

)

1.5

S

et

lan

gu

ag

e

an

d n

ota

tio

n

unders

tand s

ets

defined in a

lgebra

ic

term

s

unders

tand a

nd u

se

subsets

unders

tand a

nd u

se

the c

om

ple

ment

of

a

set

If A

is a

subset

of

B,

then

A

B

Use t

he n

ota

tion A

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Schem

e o

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: Teachin

g I

GC

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ath

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atics a

longsid

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Core

Math

em

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– I

ssue 1

– J

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2011 ©

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011

2

IG

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un

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AO

1 N

um

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nd

alg

eb

ra

1

Nu

mb

ers a

nd

th

e

nu

mb

er s

yste

m

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1.5

Set

lan

gu

ag

e a

nd

no

tati

on

co

nti

nu

ed

use V

enn d

iagra

ms t

o

repre

sent

sets

and t

he

num

ber

of

ele

ments

in

sets

use t

he n

ota

tion n

(A)

for

the n

um

ber

of

ele

ments

in t

he s

et

A

use s

ets

in p

ractical

situations

1.6

Percen

tag

es

use r

evers

e

perc

enta

ges

repeate

d p

erc

enta

ge

change

In a

sale

, prices w

ere

reduced b

y 3

0%

. The

sale

pri

ce o

f an ite

m

was £

17.5

0.

Calc

ula

te

the o

rigin

al pri

ce o

f th

e

item

Calc

ula

te t

he t

ota

l

perc

enta

ge incre

ase

when a

n incre

ase o

f

30%

is f

ollow

ed b

y a

decre

ase o

f 20%

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Math

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– I

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2011 ©

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3

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un

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1 N

um

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nd

alg

eb

ra

1

Nu

mb

ers a

nd

th

e

nu

mb

er s

yste

m

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1.6

P

ercen

tag

es

co

nti

nu

ed

solv

e c

om

pound

inte

rest

pro

ble

ms

To inclu

de d

epre

cia

tion

1.7

R

ati

o a

nd

pro

po

rti

on

See F

oundation T

ier.

1.8

D

eg

ree o

f

accu

racy

solv

e p

roble

ms u

sin

g

upper

and low

er

bounds w

here

valu

es

are

giv

en t

o a

degre

e

of

accura

cy

The d

imensio

ns o

f a

recta

ngle

are

12cm

and 8

cm

to t

he

neare

st

cm

. Calc

ula

te,

to 3

sig

nific

ant

figure

s,

the s

mallest

possib

le

are

a a

s a

perc

enta

ge o

f

the larg

est

possib

le

are

a

1.9

Sta

nd

ard

fo

rm

expre

ss n

um

bers

in t

he

form

a

10

n w

here

n is

an inte

ger

and

1 ≤

a <

10

solv

e p

roble

ms

involv

ing s

tandard

form

15

0 0

00

00

0

= 1

.5

10

8

UG

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Math

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– I

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– J

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2011 ©

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011

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IG

CS

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un

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um

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nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

1 N

um

bers a

nd

th

e

nu

mb

er s

yste

m

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1.1

0 A

pp

lyin

g

nu

mb

er

See F

oundation T

ier

1.1

1 Ele

ctr

on

ic

calc

ula

tors

See F

oundation T

ier

2 E

qu

ati

on

s,

form

ula

e a

nd

iden

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts

need

to

learn

No

tes

2.1

U

se o

f sym

bo

ls

use index n

ota

tion

involv

ing f

ractional,

negative a

nd z

ero

pow

ers

Sim

plify

:

32

36

4t

, 31

4321

a

aa

Exte

nsio

n t

op

ic

law

s o

f in

dic

es f

or

all r

ati

on

al

exp

on

en

ts

Th

e e

qu

ivale

nce o

f

nm

a a

nd

n

ma

sh

ou

ld

be k

no

wn

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029447 –

Schem

e o

f W

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: Teachin

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GC

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ath

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Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

5

IG

CS

E M

ath

em

ati

cs (

Hig

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GC

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ore M

ath

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ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

2 Eq

uati

on

s,

form

ula

e a

nd

iden

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts

need

to

learn

No

tes

2.2

A

lgeb

raic

man

ipu

lati

on

expand t

he p

roduct

of

two lin

ear

expre

ssio

ns

unders

tand t

he c

oncept

of

a q

uadra

tic

expre

ssio

n a

nd b

e a

ble

to f

acto

rise s

uch

expre

ssio

ns

manip

ula

te a

lgebra

ic

fractions w

here

the

num

era

tor

and/o

r th

e

denom

inato

r can b

e

num

eri

c,

linear

or

quadra

tic

(2x

+ 3

)(3

x –

1)

(2x

– y

)(3

x +

y)

Facto

rise:

x2 +

12

x –

45

6x2

– 5

x –

4

Expre

ss a

s a

sin

gle

fraction: 4

3

3

1

xx

3

)3

5(2

2

)14(

3

xx

xx

34

23

, x

x

1

2

1

3

12

21

xx

xx

Exte

nsio

n t

op

ic

Alg

eb

raic

man

ipu

lati

on

of

po

lyn

om

ials

,

inclu

din

g

exp

an

din

g

brackets

an

d

co

llecti

ng

lik

e

term

s,

facto

ris

ati

on

Stu

den

ts s

ho

uld

be

ab

le t

o u

se b

rackets

.

Facto

ris

ati

on

of

po

lyn

om

ials

of

deg

ree n

, n

≤

3,

eg

. Th

e

no

tati

on

f(x

). (

Use o

f

the f

acto

r t

heo

rem

is

no

t req

uir

ed

)

xx

x3

42

3

Facto

rise a

nd s

implify

:

12

42

2

x

x

xx

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e o

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g I

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Math

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atics 1

– I

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– J

uly

2011 ©

Pears

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6

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CS

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ath

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ath

em

ati

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un

it

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1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

2 E

qu

ati

on

s,

form

ula

e a

nd

iden

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

2.3

Exp

ressio

ns a

nd

form

ula

e

unders

tand t

he p

rocess

of

manip

ula

ting

form

ula

e t

o c

hange t

he

subje

ct,

to inclu

de

cases w

here

the

subje

ct

may a

ppear

twic

e,

or

a p

ow

er

of

the s

ubje

ct

occurs

v2 =

u2 +

2g

s;

make s

the s

ubje

ct

m =

at

at

11

;

make t t

he s

ubje

ct

V =

34πr

3;

make r

the s

ubje

ct

glT

2

;

make

l the s

ubje

ct

2.4

Lin

ear e

qu

ati

on

s

See F

oundation T

ier.

4

17

x

= 2

– x

,

25

3

)2

(

6

)3

2(

x

x

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029447 –

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Math

em

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– I

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2011 ©

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on E

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011

7

IG

CS

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ath

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ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

2 E

qu

ati

on

s,

form

ula

e a

nd

iden

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

2.5

P

ro

po

rti

on

set

up p

roble

ms

involv

ing d

irect

or

invers

e p

roport

ion a

nd

rela

te a

lgebra

ic

solu

tions t

o g

raphic

al

repre

senta

tion o

f th

e

equations

To inclu

de o

nly

the

follow

ing:

y

x,

y

x1

,

y

x2,

y

21 x

,

y

x3,

y

x

2.6

S

imu

ltan

eo

us

lin

ear e

qu

ati

on

s

calc

ula

te t

he e

xact

solu

tion o

f tw

o

sim

ultaneous e

quations

in t

wo u

nknow

ns

inte

rpre

t th

e e

quations

as lin

es a

nd t

he

com

mon s

olu

tion a

s

the p

oin

t of

inte

rsection

3x

– 4

y =

7

2x

– y

= 8

2x

+ 3

y =

17

3x

– 5

y =

35

UG

029447 –

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e o

f W

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: Teachin

g I

GC

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ath

em

atics a

longsid

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Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

8

IG

CS

E M

ath

em

ati

cs (

Hig

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GC

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ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

2 E

qu

ati

on

s,

form

ula

e a

nd

iden

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts n

eed

to learn

No

tes

2.7

Q

uad

rati

c

eq

uati

on

s

solv

e q

uadra

tic

equations b

y

facto

risation

solv

e q

uadra

tic

equations b

y u

sin

g t

he

quadra

tic f

orm

ula

form

and s

olv

e

quadra

tic e

quations

from

data

giv

en in a

conte

xt

2x2

– 3

x +

1 =

0,

x(3

x – 2

) =

5

New

to

pic

Co

mp

leti

ng

th

e

sq

uare.

So

lve

qu

ad

rati

c e

qu

ati

on

s

So

lve q

uad

rati

c

eq

uati

on

s b

y

co

mp

leti

ng

th

e

sq

uare

solv

e s

imultaneous

equations in t

wo

unknow

ns,

one

equation b

ein

g lin

ear

and t

he o

ther

bein

g

quadra

tic

y =

2x

– 1

1 a

nd

x2 +

y2 =

25

y =

11

x –

2 a

nd

y =

5x2

New

to

pic

Sim

ult

an

eo

us

eq

uati

on

s:

an

aly

tical

so

luti

on

by

su

bsti

tuti

on

Fo

r e

xam

ple

, w

here

on

e e

qu

ati

on

is

lin

ear a

nd

on

e

eq

uati

on

is q

uad

rati

c

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

9

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

2 E

qu

ati

on

s,

form

ula

e a

nd

iden

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts n

eed

to learn

No

tes

2.8

In

eq

ualiti

es

solv

e q

uadra

tic

inequalities in o

ne

unknow

n a

nd r

epre

sent

the s

olu

tion s

et

on a

num

ber

line

identify

hard

er

exam

ple

s o

f re

gio

ns

defined b

y lin

ear

inequalities

For

exam

ple

,

ba

x

>d

cx

,

x2 ≤

25,

4x2

> 2

5

Shade t

he r

egio

n

defined b

y t

he

inequalities

x ≤

4,

y ≤

2x

+ 1

,

5x

+ 2

y ≤

20

New

to

pic

So

luti

on

of

lin

ear

an

d q

uad

rati

c

ineq

ualiti

es

Fo

r e

xam

ple

,

ax

+ b

, cx

+ d

,

,

02

r

qx

px

ax

rqx

px

2

b

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

10

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Seq

uen

ces a

nd

serie

s

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

3

Seq

uen

ces

an

d s

erie

s

Wh

at

stu

den

ts n

eed

to

learn

No

tes

3.1

S

eq

uen

ces

use lin

ear

expre

ssio

ns

to d

escri

be t

he n

th

term

of

an a

rith

metic

sequence

1,

3,

5,

7,

9,

…

nth

term

= 2

n –

1

New

to

pic

New

to

pic

Seq

uen

ces,

inclu

din

g

tho

se g

iven

by a

fo

rm

ula

for t

he n

th t

erm

an

d t

ho

se

gen

erate

d b

y a

sim

ple

rela

tio

n o

f th

e f

orm

1n

x =

f(

) n

x

Arit

hm

eti

c s

erie

s,

inclu

din

g t

he f

orm

ula

fo

r

the s

um

of

the f

irst

n

natu

ral n

um

bers

Th

e g

en

eral te

rm

an

d t

he s

um

of

n

term

s o

f th

e s

erie

s

are r

eq

uir

ed

. Th

e

pro

of

of

the s

um

form

ula

sh

ou

ld b

e

kn

ow

n

Un

dersta

nd

ing

of

no

tati

on

wil

l b

e

exp

ecte

d

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

11

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts n

eed

to

learn

No

tes

3.2

Fu

ncti

on

no

tati

on

unders

tand t

he c

oncept

that

a f

unction is a

mappin

g b

etw

een

ele

ments

of

two s

ets

use f

unction n

ota

tions

of

the f

orm

f(x)

= …

and

f :

x

…

unders

tand t

he t

erm

s

dom

ain

and r

ange a

nd

whic

h v

alu

es m

ay n

eed

to b

e e

xclu

ded f

rom

the

dom

ain

unders

tand a

nd f

ind t

he

com

posite f

unction f

g

and t

he invers

e f

unction

f 1

f(x)

=

x1,

exclu

de x

= 0

f(x)

=

3

x,

exclu

de x

< –

3

‘ fg’ w

ill m

ean ‘do

g

firs

t, t

hen

f’

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

12

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

3.3

G

rap

hs

plo

t and d

raw

gra

phs

with e

quation:

y =

Ax3

+ B

x2 +

Cx

+ D

in w

hic

h:

(i)

the c

onsta

nts

are

inte

gers

and s

om

e

could

be z

ero

(ii)

th

e lett

ers

x a

nd y

can b

e r

epla

ced

with a

ny o

ther

two

lett

ers

y =

x3,

y =

3x3

– 2

x2 +

5x

– 4

,

y =

2x3

– 6

x +

2,

V =

60

w(6

0 –

w)

New

to

pic

Qu

ad

rati

c f

un

cti

on

s a

nd

their

grap

hs

Th

e d

iscrim

inan

t o

f a

qu

ad

rati

c f

un

cti

on

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

13

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

3.3

G

rap

hs

co

nti

nu

ed

or:

y =

Ax3

+ B

x2 +

Cx

+ D

+ E

/x +

F/x

2

in w

hic

h:

(i)

the c

onsta

nts

are

num

eri

cal and a

t

least

thre

e o

f

them

are

zero

(ii)

th

e lett

ers

x a

nd y

can b

e r

epla

ced

with a

ny o

ther

two

lett

ers

find t

he g

radie

nts

of

non-l

inear

gra

phs

y =

x1

, x

0,

y =

2x2

+ 3

x +

1/x

,

x

0,

y =

x1

(3x2

– 5

),

x

0,

W =

25 d

, d

0

By d

raw

ing a

tangent

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

14

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fun

cti

on

s

Wh

at

stu

den

ts n

eed

to

learn

No

tes

3.3

Grap

hs

co

nti

nu

ed

find t

he inte

rsection

poin

ts o

f tw

o g

raphs,

one lin

ear

(y1)

and o

ne

non-l

inear

(y2),

and

recognis

e t

hat

the

solu

tions c

orr

espond

to t

he s

olu

tions o

f

y 2 –

y1 =

0

The

x-v

alu

es o

f th

e

inte

rsection o

f th

e t

wo

gra

phs:

y =

2x

+ 1

y =

x2 +

3x

– 2

are

the s

olu

tions o

f:

x2 +

x –

3 =

0

Exte

nsio

n t

op

ic

Grap

hs o

f fu

ncti

on

s;

sketc

hin

g c

urves d

efi

ned

by s

imp

le e

qu

ati

on

s.

Geo

metr

ical

inte

rp

reta

tio

n o

f alg

eb

raic

so

luti

on

of

eq

uati

on

s.

Use

inte

rsecti

on

po

ints

of

grap

hs o

f fu

ncti

on

s t

o

so

lve e

qu

ati

on

s

Fu

ncti

on

s t

o in

clu

de

sim

ple

cu

bic

fun

cti

on

s a

nd

th

e

recip

ro

cal fu

ncti

on

xky

, w

ith

0

x

Kn

ow

led

ge o

f th

e

term

asym

pto

te is

exp

ecte

d

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

15

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

3.3

G

rap

hs

co

nti

nu

ed

calc

ula

te t

he g

radie

nt

of

a s

traig

ht

line g

iven

the C

art

esia

n

coord

inate

s o

f tw

o

poin

ts

Sim

ilarly,

the x

-valu

es

of

the inte

rsection o

f

the t

wo g

raphs:

y =

5

y =

x3 –

3x2

+ 7

are

the s

olu

tions o

f:

x3 –

3x2

+ 2

= 0

re

cognis

e t

hat

equations o

f th

e f

orm

y =

mx

+ c

are

str

aig

ht

line g

raphs w

ith

gra

die

nt

m a

nd

inte

rcept

on t

he

y a

xis

at

the p

oin

t ( 0

, c)

find t

he e

quation o

f a

str

aig

ht

line p

ara

llel to

a g

iven lin

e

Fin

d t

he e

quation o

f

the s

traig

ht

line

thro

ugh

(1,

7)

and (

2,

9)

Exte

nsio

n t

op

ic

Eq

uati

on

s o

f a s

traig

ht

lin

e,

inclu

din

g t

he

form

s)

(2

12

1x

xm

yy

an

d

0

cby

ax

To

in

clu

de:

1) t

he e

qu

ati

on

of

a

lin

e t

hro

ug

h t

wo

g

iven

po

ints

2) t

he e

qu

ati

on

of

a

lin

e p

arall

el (o

r

perp

en

dic

ula

r) t

o a

g

iven

lin

e t

hro

ug

h

a g

iven

po

int.

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

16

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Co

ord

inate

geo

metr

y in

th

e (

x, y

)

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

2

Co

ord

inate

geo

metr

y in

the (

x, y

)

Wh

at

stu

den

ts n

eed

to

learn

No

tes

Exte

nsio

n t

op

ic

Co

nd

itio

ns f

or t

wo

str

aig

ht

lin

es t

o b

e

parall

el o

r p

erp

en

dic

ula

r

to e

ach

oth

er

Fo

r e

xam

ple

, th

e

lin

e p

erp

en

dic

ula

r

to t

he lin

e

18

43

yx

,

thro

ug

h t

he p

oin

t

,2(

)3

, h

as t

he

eq

uati

on

)2

(34

3

xy

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

17

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Co

ord

inate

geo

metr

y in

th

e (

x, y

)

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

2

Co

ord

inate

geo

metr

y in

the (

x, y

)

Wh

at

stu

den

ts n

eed

to

learn

No

tes

3.3

Grap

hs

co

nti

nu

ed

Exte

nsio

n t

op

ic

Kn

ow

led

ge o

f th

e e

ffect

of

sim

ple

tran

sfo

rm

ati

on

s

on

th

e g

rap

h o

f y

= f

(x)

as

rep

resen

ted

by y

= a

f(x)

,

y =

f (

x) +

a,

y =

f (

x +

a),

y =

f (

ax)

Stu

den

ts s

ho

uld

be a

ble

to

ap

ply

on

e o

f th

ese

tran

sfo

rm

ati

on

s

to a

ny f

un

cti

on

(q

uad

rati

cs,

cu

bic

s,

recip

ro

cal)

an

d

sketc

h t

he r

esu

ltin

g

grap

h.

Giv

en

th

e g

rap

h o

f

an

y f

un

cti

on

y =

f(x

)

stu

den

ts s

ho

uld

be

ab

le t

o s

ketc

h t

he

grap

h r

esu

ltin

g

fro

m o

ne o

f th

ese

tran

sfo

rm

ati

on

s.

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

18

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Dif

feren

tiati

on

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4

Dif

feren

tiati

on

W

hat

stu

den

ts n

eed

to

learn

No

tes

3.4

C

alc

ulu

s

Dif

feren

tiati

on

unders

tand t

he

concept

of

a v

ari

able

rate

of

change

diffe

rentiate

inte

ger

pow

ers

of x

dete

rmin

e g

radie

nts

,

rate

s o

f change,

turn

ing p

oin

ts

(maxim

a a

nd m

inim

a)

by d

iffe

rentiation a

nd

rela

te t

hese t

o g

raphs

dis

tinguis

h b

etw

een

maxim

a a

nd m

inim

a

by c

onsid

eri

ng t

he

genera

l shape o

f th

e

gra

ph

y =

x +

x9

Fin

d t

he C

art

esia

n

coord

inate

s o

f th

e

maxim

um

and

min

imum

poin

ts

Exte

nsio

n t

op

ic

Th

e d

eriv

ati

ve o

f f(

x) a

s

the g

rad

ien

t o

f th

e

tan

gen

t to

th

e g

rap

h o

f

y =

f(x

) at

a p

oin

t; t

he

grad

ien

t o

f th

e t

an

gen

t as

a lim

it;

inte

rp

reta

tio

n a

s a

rate

of

ch

an

ge;

seco

nd

ord

er d

eriv

ati

ves

Fo

r e

xam

ple

,

kn

ow

led

ge t

hat

dx

dy

is t

he r

ate

of

ch

an

ge

of

y w

ith

resp

ect

to

x

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

19

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Dif

feren

tiati

on

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4

Dif

feren

tiati

on

co

nti

nu

ed

Wh

at

stu

den

ts n

eed

to

learn

No

tes

3.4

C

alc

ulu

s

Dif

feren

tiati

on

co

nti

nu

ed

apply

calc

ulu

s t

o

linear

kin

em

atics a

nd

to o

ther

sim

ple

pra

ctical pro

ble

ms

The d

ispla

cem

ent,

s

metr

es,

of

a p

art

icle

from

a f

ixed p

oin

t 0

aft

er

t seconds is g

iven

by:

s =

24

t2 –

t3,

0 ≤

t ≤

20

Fin

d e

xpre

ssio

ns f

or

the v

elo

city a

nd t

he

accele

ration

Exte

nsio

n t

op

ic

K

no

wle

dg

e o

f th

e

ch

ain

ru

le i

s n

ot

req

uir

ed

Th

e n

ota

tio

n f

/ (x)

may b

e u

sed

Fo

r e

xam

ple

, fo

r

1

n,

the a

bilit

y t

o

dif

feren

tiate

exp

ressio

ns s

uch

as

)1)(

52(

xx

an

d

21

2

3

5 x

xx

3

is

exp

ecte

d

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

20

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Dif

feren

tiati

on

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4

Dif

feren

tiati

on

co

nti

nu

ed

Wh

at

stu

den

ts n

eed

to

learn

No

tes

New

co

ncep

ts

Ap

plicati

on

s o

f

dif

feren

tiati

on

to

grad

ien

ts,

tan

gen

ts a

nd

no

rm

als

Use o

f

dif

feren

tiati

on

to

fin

d e

qu

ati

on

s o

f

tan

gen

ts a

nd

no

rm

als

at

sp

ecif

ic

po

ints

on

a c

urve

In

teg

rati

on

5

In

teg

rati

on

W

hat

stu

den

ts n

eed

to

learn

No

tes

3.4

C

alc

ulu

s

Dif

feren

tiati

on

co

nti

nu

ed

New

to

pic

In

defi

nit

e in

teg

rati

on

as

the r

everse o

f

dif

feren

tiati

on

inte

grati

on

of

n

x

Stu

den

ts s

ho

uld

kn

ow

th

at

a

co

nsta

nt

of

inte

grati

on

is

req

uir

ed

.

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

21

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

1 N

um

ber a

nd

alg

eb

ra

Dif

feren

tiati

on

3 S

eq

uen

ces,

fun

cti

on

s a

nd

grap

hs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

5

In

teg

rati

on

co

nti

nu

ed

Wh

at

stu

den

ts n

eed

to

learn

No

tes

3.4

Calc

ulu

s

Dif

feren

tiati

on

co

nti

nu

ed

New

to

pic

Fo

r e

xam

ple

, th

e

ab

ilit

y t

o in

teg

rate

exp

ressio

ns s

uch

as:

21

23

21

x

x a

nd

21

2 )2

(

x

x

is e

xp

ecte

d

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

22

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 N

um

ber a

nd

alg

eb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4.1

Lin

es a

nd

tria

ng

les

See F

oundation T

ier.

4.2

P

oly

go

ns

See F

oundation T

ier.

4.3

S

ym

metr

y

See F

oundation T

ier.

4.4

M

easu

res

See F

oundation T

ier.

4.5

C

on

str

ucti

on

See F

oundation T

ier.

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

23

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 N

um

ber a

nd

alg

eb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4.6

C

ircle

pro

perti

es

unders

tand a

nd u

se

the inte

rnal and

exte

rnal in

ters

ecting

chord

pro

pert

ies

recognis

e t

he t

erm

cyclic q

uadrila

tera

l

unders

tand a

nd u

se

angle

pro

pert

ies o

f th

e

cir

cle

inclu

din

g:

angle

subte

nded

by a

n a

rc a

t th

e

centr

e o

f a c

ircle

is

twic

e t

he a

ngle

subte

nded a

t any

poin

t on t

he

rem

ain

ing p

art

of

the c

ircum

fere

nce

angle

subte

nded

at

the

cir

cum

fere

nce b

y

a d

iam

ete

r is

a

right

angle

Form

al pro

of of

these

theore

ms is n

ot

requir

ed

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

24

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 N

um

ber a

nd

alg

eb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4.6

C

ircle

pro

perti

es

co

nti

nu

ed

angle

s in t

he s

am

e

segm

ent

are

equal

th

e s

um

of

the

opposite a

ngle

s o

f

a c

yclic

quadri

late

ral is

180

th

e a

ltern

ate

segm

ent

theore

m

4.7

G

eo

metr

ical

reaso

nin

g

pro

vid

e r

easons,

usin

g

sta

ndard

geom

etr

ical

sta

tem

ents

, to

support

num

eri

cal valu

es f

or

angle

s o

bta

ined in a

ny

geom

etr

ical conte

xt

involv

ing lin

es,

poly

gons a

nd c

ircle

s

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

25

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 N

um

ber a

nd

alg

eb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4.8

Trig

on

om

etr

y

an

d P

yth

ag

oras’

Th

eo

rem

unders

tand a

nd u

se

sin

e,

cosin

e a

nd

tangent

of

obtu

se

angle

s

unders

tand a

nd u

se

angle

s o

f ele

vation

and d

epre

ssio

n

unders

tand a

nd u

se

the s

ine a

nd c

osin

e

rule

s f

or

any t

riangle

use P

yth

agora

s’

Theore

m in 3

dim

ensio

ns

unders

tand a

nd u

se

the f

orm

ula

½ a

bsin

C for

the a

rea

of

a t

riangle

apply

tri

gonom

etr

ical

meth

ods t

o s

olv

e

pro

ble

ms in 3

dim

ensio

ns,

inclu

din

g

findin

g t

he a

ngle

betw

een a

lin

e a

nd a

pla

ne

The a

ngle

betw

een

two p

lanes w

ill not

be

requir

ed

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

26

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 N

um

ber a

nd

alg

eb

ra

Alg

eb

ra a

nd

fu

ncti

on

s

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4.9

M

en

su

rati

on

find p

eri

mete

rs a

nd

are

as o

f secto

rs o

f

cir

cle

s

Radia

n m

easure

is

exclu

ded

4.1

0 3

-D s

hap

es

an

d v

olu

me

find t

he s

urf

ace a

rea

and v

olu

me o

f a

sphere

and a

rig

ht

cir

cula

r cone u

sin

g

rele

vant

form

ula

e

convert

betw

een

volu

me m

easure

s

m3 →

cm

3 a

nd v

ice

vers

a

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

27

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 N

um

ber a

nd

alg

eb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

4.1

1 S

imil

arit

y

unders

tand t

hat

are

as

of

sim

ilar

figure

s a

re

in t

he r

atio o

f th

e

square

of

corr

espondin

g s

ides

unders

tand t

hat

volu

mes o

f sim

ilar

figure

s a

re in t

he r

atio

of

the c

ube o

f

corr

espondin

g s

ides

use a

reas a

nd

volu

mes o

f sim

ilar

figure

s in s

olv

ing

pro

ble

ms

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

28

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 N

um

ber a

nd

alg

eb

ra

5 V

ecto

rs a

nd

tran

sfo

rm

ati

on

geo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

5.1

V

ecto

rs

unders

tand t

hat

a

vecto

r has b

oth

magnitude a

nd

dir

ection

unders

tand a

nd u

se

vecto

r nota

tion

multip

ly v

ecto

rs b

y

scala

r quantities

add a

nd s

ubtr

act

vecto

rs

calc

ula

te t

he m

odulu

s

(magnitude)

of

a

vecto

r

find t

he r

esultant

of

two o

r m

ore

vecto

rs

apply

vecto

r m

eth

ods

for

sim

ple

geom

etr

ical

pro

ofs

The n

ota

tions ΟΑ

and

a w

ill be u

sed

ΟΑ

= 3

a,

AB

= 2

b,

BC

= c

so:

OC

= 3

a +

2b

+ c

CA

=

c –

2b

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

29

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

2 S

hap

e,

sp

ace a

nd

measu

res

5 V

ecto

rs a

nd

tran

sfo

rm

ati

on

geo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

5.2

Tran

sfo

rm

ati

on

geo

metr

y

See F

oundation T

ier.

Colu

mn v

ecto

rs m

ay

be u

sed t

o d

efine

transla

tions

6 S

tati

sti

cs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

6.1

G

rap

hic

al

rep

resen

tati

on

of

data

constr

uct

and

inte

rpre

t his

togra

ms

constr

uct

cum

ula

tive

frequency d

iagra

ms

from

tabula

ted d

ata

use c

um

ula

tive

frequency d

iagra

ms

For

continuous

vari

able

s w

ith u

nequal

cla

ss inte

rvals

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

30

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

3 H

an

dli

ng

data

6.2

S

tati

sti

cal

measu

res

estim

ate

the m

edia

n

from

a c

um

ula

tive

frequency d

iagra

m

unders

tand t

he

concept

of

a m

easure

of

spre

ad

find t

he inte

rquart

ile

range f

rom

a d

iscre

te

data

set

The t

erm

s ‘upper

quart

ile’ and ‘lo

wer

quart

ile’ m

ay b

e u

sed

estim

ate

the

inte

rquart

ile r

ange

from

a c

um

ula

tive

frequency d

iagra

m

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

31

UG

029447 –

Schem

e o

f W

ork

: Teachin

g I

GC

SE M

ath

em

atics a

longsid

e

Core

Math

em

atics 1

– I

ssue 1

– J

uly

2011 ©

Pears

on E

ducation L

imited 2

011

32

IG

CS

E M

ath

em

ati

cs (

Hig

her t

ier)

GC

E C

ore M

ath

em

ati

cs 1

un

it

AO

3 H

an

dli

ng

data

6 S

tati

sti

cs

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

6.3

P

ro

bab

ilit

y

dra

w a

nd u

se t

ree

dia

gra

ms

dete

rmin

e t

he

pro

bability t

hat

two o

r

more

independent

events

will both

occur

use s

imple

conditio

nal

pro

bability w

hen

com

bin

ing e

vents

Pic

kin

g t

wo b

alls o

ut

of

a b

ag,

one a

fter

the

oth

er,

without

repla

cem

ent

apply

pro

bability t

o

sim

ple

pro

ble

ms

Higher tier IGCSE Mathematics/GCE Core Mathematics 1 unit content summary

The table below is a summary of the IGCSE Mathematics Higher tier content and

GCE Core Mathematics 1 unit content that could be delivered/taught alongside each

other, in the first year of the course. This scheme of work should be used together

with the Higher tier course planner in the IGCSE Mathematics Teacher’s Guide.

The module numbers in the table below refer to the module numbers in the course

planner within the IGCSE Mathematics Teacher’s Guide. References to topic areas in

GCE Core Mathematics 1 unit are in bold.

Year 1 content summary

Module number Module Title

*Estimated

teaching

hours

Number 2 Powers and roots

C1: Use and manipulation of surds 5

1

Algebraic manipulation

C1:Algebraic manipulation of polynomials, including

expanding brackets and collecting like terms,

factorisation

5

3

Linear equations and simultaneous linear equations

C1: Simultaneous equations: analytical solution by

substitution

7

5

Linear graphs

C1: Equation of a straight line, including the forms

)( 11 xxmyy and 0 cbyax . Conditions for

two straight lines to be parallel or perpendicular to

each other

7

6

Integer sequences

C1: Sequences, including those given by a formula for

the nth term and those generated by a simple relation

of the form 1nx f( ) nx

This could be done in the first year as an extension of

integer sequences

5

8 Inequalities

C1: Solution of linear and quadratic inequalities 6

Algebra

9 Indices

C1: Laws of indices for all rational exponents. 5

*Teachers should be aware that the estimated teaching hours are approximate and

should only be used as a guideline.

UG029447 – Scheme of Work: Teaching IGCSE Mathematics alongside

Core Mathematics 1 – Issue 1 – July 2011 © Pearson Education Limited 2011

33

The table below is a summary of the IGCSE Mathematics Higher tier content and

GCE Core Mathematics 1 unit content that could be delivered/taught alongside each

other, in the second year of the course. This scheme of work should be used

together with the Higher tier course planner in the IGCSE Mathematics Teacher’s

Guide. The module numbers in the table below refer to the module numbers in the

course planner within the IGCSE Mathematics Teacher’s Guide. References to topic

areas in GCE Core Mathematics 1 unit are in bold.

Year 2 content summary

Module number Module Title

*Estimated

teaching

hours

7

Quadratic equations

C1: Completing the square. Solution of quadratic

graphs

10

11

Function notation

C1: Quadratic functions and their graphs. The

discriminant of the quadratic function.

10

12

Harder graphs

C1: Graphs of functions; sketching curves defined by

simple equations. Geometrical interpretation of

algebraic solution of equations. Use intersection points

of graphs of functions to solve equations. Knowledge

of the effect of simple transformations on the graphs

of y = f(x) as represented by y = af(x), y = f(x) + a,

y = f(x + a), y = f(ax)

10 Algebra

13

Calculus – Differentiation

C1: The derivative of f(x) as the gradient of the tangent

to the graph of f(x) at a point; the gradient of the

tangent as a limit; interpretation as a rate of change;

second order derivatives. Applications of

differentiation to gradients, tangents and normals

15

C1 topic 1

C1: Sequences and Series

Sequences could either be covered in Year 10 as an

extension of IGCSE/Certificate in Mathematics topic:

Integer sequence, or together with Arithmetic series in

the second year.

Arithmetic series, including the formula for the sum of

the first n natural numbers

10

C1 topic 2

C1: Calculus – Integration

Indefinite integration as the reverse of differentiation

integration of nx

15

*Teachers should be aware that the estimated teaching hours are approximate and should

only be used as a guideline.



34

Higher tier IGCSE Mathematics/GCE Core Mathematics 1 unit scheme of work

This scheme of work only contains the modules that could be extended through

teaching the Edexcel GCE Core Mathematics 1 unit alongside the IGCSE

Mathematics. It should be used alongside the IGCSE Mathematics course planner in

the Teacher’s Guide.

Number

Module 2 – Powers and roots [Year 1] Time: 4 – 6 hours

Target grades: A*/A/B/C

Content Area of specification

Squares and square roots 1.4

Cubes and cube roots 1.4

Using a calculator effectively to evaluate powers and roots 1.1

Powers of numbers – using index notation 1.4

Order of operations including powers (BIDMAS*) 1.1

Expressing a number as the product of powers of its prime factors 1.4

Using prime factors to evaluate Highest Common Factors (HCF) and

Lowest Common Multiples (LCM) 1.4

Understanding and using powers which are zero, negative or fractions 1.4

Recognising the relationship between fractional powers and roots 1.4

Using laws of indices to simplify and evaluate numerical expressions

involving integer, fractional and negative powers 1.4

Understanding the meaning of surds 1.4

Manipulating surds, including rationalising the denominator 1.4

*BIDMAS = Brackets, Indices, Division, Multiplication, Addition, Subtraction

A/A* notes/tips

In order for students to aspire to the top grades, it is essential that they

are able to use algebraic manipulation and index notation confidently

Remind students that when writing fractions, it is not usual to write surds in the

denominator, because without a calculator, it is not always easy to work out the

value of the fraction, eg 2

1 , but ‘rationalising’ the denominator will help clear

the surds from the denominator



35

Resources

Textbook References

Edexcel IGCSE

Mathematics A Student

Book 1

Unit 3: Number 3 page 117

Unit 3: Number 3 page 114 – 116

Edexcel IGCSE


Book 2


GCE Core Mathematics 1

Content Textbook reference

Write a number exactly as a surd 1.7

Rationalise the denominator of a fraction when it is a surd 1.8

Resources

Textbook References

Edexcel AS and A Level

Modular Mathematics Core

Mathematics 1

page 10 – 13



36

ALGEBRA

Module 1 – Algebraic manipulation [Year 1] Time: 4 – 6 hours

Target grades: A*/A/B/C/D


Multiplying a single term over a bracket 2.2

Factorising by taking out a single common factor 2.2

Finding and simplifying the product of two linear expressions,

eg (2x + 3)(3x – 1), (3x – 2y)(5x + 3y) 2.2

Factorising quadratic expressions, including the difference

of two squares 2.2

Adding and subtracting algebraic fractions, including simplifying

algebraic fractions by cancelling common factors 2.2

Numerator and/or the denominator may be numeric, linear or quadratic 2.2

Notes

Emphasise importance of using the correct symbolic notation, for example 3a rather

than 3 a or a3. Students should be aware that there may be a need to remove the

numerical HCF of a quadratic expression before factorising it in order to make

factorisation more obvious

A/A* notes/tips for Higher tier

Students need to be reminded that they should always factorise algebraic

expressions completely, setting their work out clearly

In order for students to work towards to the top grades, it is essential that they

are confidently able to manipulate algebraic expressions in a variety of

situations

When simplifying algebraic fractions, students should be encouraged to fully

factorise both the numerator and the denominator, where possible

A typical common error is for students to ‘cancel out’ the terms in x

Simplifying algebraic fractions is usually a challenging topic for many students.

A key point is that algebraic fractions are actually generalised arithmetic, and

that the same rules apply



37

Resources

Textbook References

Edexcel IGCSE


Book 1

Unit 1: Algebra 1 page 11 – 12



Edexcel IGCSE

Mathematics A Student Book 2

Unit 5: Algebra 5 (Revision)

page 346 – 347



Simplify expressions by collecting like terms 1.1

Expand an expression by multiplying each term inside the

bracket by the term outside 1.3

Resources

Textbook References



Mathematics 1

page 2, 4 – 6



38

Module 3 – Linear equations and simultaneous linear equations [Year 1]

Time: 6 – 8 hours

Target grades: B/C/D


Inverse operations 2.4

Understanding and use of ‘balancing’ methods 2.4

Solving simple linear equations 2.4

Solving linear equations:

with two or more operations 2.4

with the unknown on both sides 2.4

with brackets 2.4

with negative or fractional coefficients 2.4

with combinations of these 2.4

Setting up and solving simple linear equations to solve problems, including

finding the value of a variable which is not the subject of the formula 2.4

Solving simple simultaneous linear equations, including cases

where one or both of the equations must be multiplied 2.6

Interpreting the equations as lines and their common solution

as the point of intersection 2.6

Prior knowledge

Algebra: Modules 1 and 2

The idea that some operations are ‘opposite’ to each other

Notes

Students need to realise that not all linear equations can be solved easily by either

observation or trial and improvement; a formal method is often needed

Students should leave their answers in fractional form where appropriate

Resources

Textbook References

Edexcel IGCSE


Book 1


Unit 2: Graphs 2 page 79 – 80




39



Solve simultaneous linear equations by elimination 3.1

Solve simultaneous linear equations by substitution 3.2

Use the substitution method to solve simultaneous equations

whereone equation is linear and the other is quadratic 3.3

Resources

Textbook References


Modular Mathematics Core Mathematics 1

page 28 – 31



40

Module 5 – Linear graphs [Year 1] Time: 6 – 8 hours

Target grades: A/B/C/D


Recognising that equations of the form x = a and y = b correspond to

straight line graphs parallel to the y-axis and to the x-axis respectively 3.3

Completing tables of values and drawing graphs with equations of the

form y = mx + c where the values of m and c are given and m may be an

integer or a fraction 3.3

Drawing straight line graphs with equations in which y is given implicitly in

terms of x, for example x + y = 7 3.3

Calculating the gradient of a straight line given its equation of the

coordinates of two points on the line 3.3

Recognising that graphs with equations of the form y = mx + c

are straight line graphs with gradient m and intercept (0, c) on the y-axis 3.3

Finding the equation of a straight line given the coordinates of two points

on the line 3.3

Finding the equation of a straight line parallel to a given line 3.3

Prior knowledge

Algebra: Modules 1, 2, 3 and 4

Notes

Axes should be labelled on graphs and a ruler should be used to draw linear graphs

Science experiments/work could provide results which give linear graphs

Resources

Textbook References

Edexcel IGCSE





41



Write the equation of a straight line in the form of y = mx + c or ax + by + c = 0 5.1

Work out the gradient m of the line joining the point with

coordinates (x , y 1 ) to the point with the coordinates (x , y )

by using the formula

m =

1 2 2

12

12

xx

yy

5.2

Find the equation of a line with gradient m that passes through

the point with coordinates (x 1 , y 1 ) by using the formula

)( 11 xxmyy 5.3

Find the equation of the line that passes through the points with

the coordinates (x 1 , y 1 ) and (x 2 , y 2 ) by using the formula

12

1

12

1

xx

xx

yy

yy

5.4

Work out the gradient of a line that is perpendicular to the line

y = mx + c 5.5

Resources

Textbook References



page 74 – 90



42

Module 6 – Integer sequences [Year 1] Time: 4 – 6 hours

Target grades: B/C/D


Using term-to-term and position-to-term definitions to generate

the terms of a sequence 3.1

Finding and using linear expressions to describe the nth term

of an arithmetic sequence 3.1

Resources

Textbook References

Edexcel IGCSE


Unit 5: Sequences 5

page 254 – 264



A series of numbers following a set rule is called a sequence 6.1

Know a formula for the nth term of a sequence (eg 13 nU n )

to find any term in the sequence 6.2

Know the rule to get from one term to the next, and use this

information to produce a recurrence relationship

(or recurrence formula) 6.3

A sequence that increases by a constant amount each time is

called an arithmetic sequence 6.4

Resources

Textbook References



Mathematics 1

page 92 – 100



43

Module 7 – Quadratic equations [Year 2] Time: 9 – 11 hours

Target grade: A*/A/B/C


Solving quadratic equations by factorisation 2.7

Solving quadratic equations by using the quadratic formula 2.7

Setting up and solving quadratic equations from data given in a context 2.7

Solving exactly, by elimination of an unknown, two simultaneous equations

in two unknowns, one of which is linear in each unknown and the other is

linear in one unknown and quadratic in the other 2.7

Solving exactly, by elimination of an unknown, two simultaneous equations

in two unknowns, one of which is linear in each unknown and the other is

linear in one unknown and the other is of the form x2 + y2 = r2 2.7

Prior knowledge


Notes

Remind students that they should factorise a quadratic before using the formula

A/A* notes/tips

Remind students that it is important to always factorise completely before

resorting to using the quadratic formula

When applying the quadratic formula, students must substitute the correct

values into the formula. They should be reminded that rounding or truncating

during the process leads to inaccurate solutions

Often solving equations with algebraic fractions is a challenge for most students,

however they should be encouraged to show their working out through using a

few lines of correct algebra. Remind students of the value of retaining the

structure of the equation throughout their working, rather than merely treating

the algebra as an expression to be simplified

Resources

Textbook References

Edexcel IGCSE



Edexcel IGCSE


Book 2





44



Plot graphs of quadratic equations 2.1

Solve quadratic equations using factorising 2.2

Write quadratic expressions in another form by

completing the square 2.3

Solve quadratic equations by completing the square 2.4

Solve quadratic equations by using the formula 2.5

Resources

Textbook References



page 17 – 23



45

Module 8 – Inequalities [Year 1] Time: 5 – 7 hours

Target grades: A/B/C


Understanding and using the symbols >, <, ≥ and ≤ 2.8

Understanding and using the convention for open and closed

intervals on a number line 2.8

Solving simple linear inequalities in one variable, including

‘double-ended’ inequalities 2.8

Representing on a number line the solution set of simple linear

inequalities 2.8

Finding the integer solutions of simple linear inequalities 2.8

Using regions to represent simple linear inequalities in one variable 2.8

Using regions to represent the solution set to several linear

inequalities in one or two variables 2.8

Solving quadratic inequalities in one unknown and representing the

solution set on a number line 2.8

Prior knowledge

Algebra: Modules 3, 5 and 7

Resources

Textbook References

Edexcel IGCSE


Book 1

Unit 2: Algebra 2 page 74 – 78,

81 – 86

Edexcel IGCSE



Unit 5: Algebra 5 page 356



46



Solve linear inequalities using similar methods to those for

solvinglinear equations 3.4

Solve quadratic inequalities 3.5

Resources

Textbook References



Mathematics 1

page 31 – 39



47

Module 9 – Indices [Year 1] Time: 4 – 6 hours

Target grades: A/B/C/D


Using index notation for positive integer powers 2.1

Substituting positive and negative numbers into expressions and

formulae with quadratic and/or cubic terms 2.1

Completing tables of values and drawing graphs of quadratic functions 3.3

Using index notation with positive, negative and fractional powers to

simplify expressions 2.1

Prior knowledge


Resources

Textbook References

Edexcel IGCSE


Book 1

Unit 2: Number 2 page 60,

73 – 74


Edexcel IGCSE





Simplify expressions and functions by using rules of indices (powers) 1.2

Extend rules of indices to all rational exponents 1.6

Resources

Textbook References



Mathematics 1

page 3,4 and 8,9



48

Module 11 – Function notation [Year 2] Time: 9 – 11 hours

Target grades: A*/A/B


Understanding the concept that a function is a mapping between

elements of two sets 3.2

Using function notation of the form f(x) = … and f x: 3.2

Understanding the terms domain and range 3.2

Understanding which parts of the domain may need to be excluded 3.2

Understanding and using composite function fg and inverse function f –1 3.2

Prior knowledge

Algebra: Modules 1, 2 and 3

A/A* notes/tips

This tends to be demanding topic for students and in order to deepen their

understanding of how to apply their knowledge of functions in different types of

questions, they should be given plenty of practice

Students may need to be reminded that f(x) = y

When solving f(x) = g(x), given the graphs of both functions, remind students

that they should give their answers as solutions of x

Remind students that when one function is followed by another, the result is a

composite function, eg fg(x) means do f first followed by g, where the domain of

f is the range of g

Students need to understand, and be able to, use the concepts of domain and

range, as this will enable them to develop an appropriate working knowledge of

functions. In particular, students must be familiar with the concept that division

by zero is undefined,

eg for g(x) = 2

1

x, 02 , which means

x = 2 must be excluded from the domain of g

x

For inverse functions, remind students that the inverse of f(x) is the function

that ‘undoes’ whatever f(x) has done, and that the notation f1(x) is used

It is helpful to remind students that if the inverse function is not obvious then:

– Step 1: write the function as y =…

– Step 2: change any x to y, and any y to x

– Step 3: make y the subject, giving the inverse function



49

Resources

Textbook References

Edexcel IGCSE


Book 2




Transform the curve of a function f(x) by simple translations 4.5

Transform the curve of a function f(x) by simple stretches 4.6

Perform simple transformations on a given sketch of a function 4.7

Resources

Textbook References

Edexcel AS and A Level Modular Mathematics Core

Mathematics 1

page 55 – 65



50

Module 12 – Harder graphs [Year 2] Time: 9 – 11 hours



Plotting and drawing graphs with equation y = Ax3 + Bx2 + Cx + D in which

(i) the constants are integers and some could be zero

(ii) the letters x and y can be replaced with any other two letters 3.3

Plotting and drawing graphs with equation

y = Ax3 + Bx2 + Cx + D + x

E +

2x

F

in which

(i) the constants are integers and at least three of them are zero

(ii) the letters x and y can be replaced with any other two letters 3.3

Finding the gradients of non-linear graphs by drawing a tangent 3.3

Finding the intersection points of two graphs, one linear (y1) and one

non-linear (y2) and recognising that the solutions correspond to y2 – y1 = 0 3.3

Prior knowledge

Algebra: Modules 1, 2, 3, 5 and 9

Notes

Students should be made aware that they should not use rulers to join plotted

points on non-linear graphs

When plotting points or reading off values from a graph, the scales on the axes

should be checked carefully

A/A* notes/tips

Remind students that when finding an estimate for the gradient of a graph

y = f(x) at given point, a tangent drawn at this point is helpful, although a

related, correct division, to find the gradient, is required to gain top marks in a

question

Students should recognise that cubic graphs have distinctive shapes that

depend on the coefficient of 3x

Students should recognise that reciprocal graphs have x as the denominator,

and that they produce a type of curve called a hyperbola. An awareness of the

concept of the smallest (minimum) value of y, and the value of x where this

happens on the graph, is helpful

Students should appreciate that an accurately drawn graph can be used to solve

equations that may prove difficult to solve by other methods. They should also

appreciate that most graphs of real-life situations are curves rather than

straight lines. Information on rates of change can still be found by drawing a

tangent to a curve, and using this to estimate the gradient of the curve at this

point



51

Students should recognise that the algebraic method is more accurate than the

graphical method of solving simultaneous equations, in particular when one

equation is linear and the other equation is nonlinear

Resources

Textbook References

Edexcel IGCSE


Book 2





Sketch graphs of quadratic equations and solve problems using

the discriminant 2.6

Sketch cubic curve of the form y = Ax3 + Bx2 + Cx + D 4.1

Sketch and interpret graphs of the cubic form y = x3 4.2

Sketch the reciprocal function x

ky , where k is a constant 4.3

Sketch curves of functions to show points of intersection and

solutions to equations 4.4

Resources

Textbook References


Modular Mathematics

Core Mathematics 1

page 23 – 25, 42 – 55



52

Module 13 – Calculus [Year 2] Time: 14 – 16 hours



Understanding the concept of a variable rate of change 3.4

Differentiating integer powers of x 3.4

Determining gradients, rates of change, maxima and minima by

differentiation and relating these to graphs 3.4

Applying calculus to linear kinematics and to other simple practical

problems 3.4

Prior knowledge

Algebra; Modules 1, 2, 5, 9 and 12

Notes

When applying calculus to linear kinematics, the reverse of differentiation will not

be required

A/A* notes/tips

Student should understand that the process of finding the gradient of a curve

is called differentiation, where the result is the derivative or the gradient

function, and that the gradient of a curve can also be represented by dx

dy

Students should be encouraged to set their work out appropriately, maintaining

the structure of their solution, as this will aid their understanding, and revision,

of the topic, particularly as it increases in complexity

Students need to understand the turning points are points on the curve where

the gradient is zero. They should also be able to distinguish between a minimum

turning point and a maximum turning point

Students need to be able to apply their knowledge of differentiation to the

motion of a particle in a straight line, including speed and acceleration

Resources

Textbook References

Edexcel IGCSE





53


Differentiation


Estimate the gradient of the tangent 7.1

Find the formula for the gradient of the function f(x) = 7.2 nx

Find the gradient formula for a function such as f(x) = 7.3 384 2 xx

Find the gradient formula for a function such as f(x) = 2

1

23 xxx 7.4

Expand or simplify polynomial functions so they are easier to

differentiate 7.5

Repeat the process of differentiation to give a second derivative 7.6

Find the rate of change of a function f at a particular points using

f /(x) and substituting in the value of x 7.7

Use differentiation to find the gradient of a tangent to a curve and

then find the equation of the tangent and the normal to the curve

at a specified point 7.8

Resources

Textbook References


Modular Mathematics

Core Mathematics 1

page 113 – 131



54

C1 topic

Module 1 – Sequences and Series [Year 2] Time: 9 – 11 hours



Arithmetic series are formed by adding together the terms of an

arithmetic sequence 6.5

Find the sum of an arithmetic series 6.6

Use to signify ‘the sum of’ 6.7

Resources

Textbook References



page 100 – 110



55

C1 topic

Module 2 – Integration [Year 2] Time: 14 – 16 hours

GCE unit Core Mathematics 1


Integrate functions of the form f(x) = where n ℝ and

a is a constant 8.1

nax

Apply the principle of integration separately to each term of dx

dy 8.2

Use the integral sign 8.3

Simplify an expression into separate terms of the form 8.4 nx

Find the constant of integration, c, when given any point (x, y)

that the curve of the function passes through 8.5

Resources

Textbook References


Modular Mathematics

Core Mathematics 1

page 134 – 142

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56

Publications Code UG029447 July 2011

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