Edexcel Level 1/Level 2 Certificate in Mathematics (KMA0) · Edexcel Level 1/Level 2 Certificate in...

Scheme of work: Bridging to GCE

Edexcel Level 1/Level 2 Certificate in Mathematics (KMA0)First examination 2012

Teaching the Level 1/Level 2 Certificate in Mathematics alongside Core Mathematics 1

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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. Edexcel does not endorse, approve or accept responsibility for the content of materials, which may be subject to change, or any opinions expressed therein. (Material may include textbooks, journals, magazines and other publications and websites.)

Authorised by Martin Stretton Prepared by Sharon Wood

Publications Code UG027243

All the material in this publication is copyright © Pearson Education Limited 2011

Introduction

The Edexcel Level 1/Level 2 Certificate in Mathematics 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 Level 1/Level 2 Certificate in Mathematics 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 Certificate 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 Certificate:

Sequences and series

Integration.

Contents

Mapping of Certificate in Mathematics Higher tier content to GCE Core Mathematics 1 unit content 1

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

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

UG

027243 –

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of

Work

: Tea

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ath

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ngsi

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Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

1

Map

pin

g of

Cer

tifi

cate

in M

athem

atic

s H

igher

tie

r co

nte

nt

to G

CE

Cor

e M

athem

atic

s 1 u

nit

con

tent

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

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nd

fu

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ion

s

1

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d t

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nu

mb

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syst

em

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tud

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: N

ote

s 1

A

lgeb

ra a

nd

fu

nct

ion

s W

hat

stu

den

ts n

eed

to

learn

N

ote

s

1.1

In

teg

ers

See

Foundat

ion T

ier

1.2

Fra

ctio

ns

See

Foundat

ion T

ier

1.3

Deci

mals

co

nve

rt r

ecurr

ing

dec

imal

s in

to f

ract

ions

3.0

= 31

,

0.23

33…

= 9021

1.4

Po

wers

an

d

roo

ts

under

stan

d t

he

mea

nin

g o

f su

rds

m

anip

ula

te s

urd

s,

incl

udin

g r

atio

nal

isin

g

the

den

om

inat

or

wher

e th

e den

om

inat

or

is a

pure

surd

Exp

ress

in t

he

form

a2:

82

, 18

+ 32

Exp

ress

in t

he

form

a

+ b2

: (

3 +

52

)2

Exte

nsi

on

to

pic

Use

an

d

man

ipu

lati

on

of

surd

s

Stu

den

ts s

ho

uld

be

ab

le t

o r

ati

on

alise

d

en

om

inato

rs

use

index

law

s to

si

mplif

y an

d e

valu

ate

num

eric

al e

xpre

ssio

ns

invo

lvin

g inte

ger

, fr

actional

and n

egat

ive

pow

ers

Eva

luat

e:

3 82 ,

21

625

,

23

251

Exte

nsi

on

to

pic

Law

s o

f in

dic

es

for

all r

ati

on

al

exp

on

en

ts

Th

e e

qu

ivale

nce

of

nm a a

nd

nm a

sh

ou

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Mat

hem

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Iss

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1 –

July

2011

© P

ears

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2011

2

Cert

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ate

in

Math

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Hig

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tier)

G

CE C

ore

Math

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AO

1 N

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an

d a

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Alg

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nd

fu

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1

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d t

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nu

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tud

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ote

s 1

A

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fu

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s W

hat

stu

den

ts n

eed

to

learn

N

ote

s

1.4

P

ow

ers

an

d

roo

ts c

on

tin

ued

ev

aluat

e H

ighes

t Com

mon F

acto

rs (

HCF)

an

d L

ow

est

Com

mon

Multip

les

(LCM

)

1.5

Set

lan

gu

ag

e

an

d n

ota

tio

n

under

stan

d s

ets

def

ined

in a

lgeb

raic

te

rms

under

stan

d a

nd u

se

subse

ts

under

stan

d a

nd u

se

the

com

ple

men

t of

a se

t

If A

is a

subse

t of

B,

then

A

B

Use

the

nota

tion A

UG

027243 –

Sch

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of

Work

: Tea

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1/L

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2 C

ertifica

te in M

ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

3

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

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an

d a

lgeb

ra

1

Nu

mb

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d t

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nu

mb

er

syst

em

S

tud

en

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

1.5

Set

lan

gu

ag

e a

nd

n

ota

tio

n

con

tin

ued

use

Ven

n d

iagra

ms

to

repre

sent

sets

and t

he

num

ber

of

elem

ents

in

sets

use

the

nota

tion n

(A)

for

the

num

ber

of

elem

ents

in t

he

set

A

use

set

s in

pra

ctic

al

situ

atio

ns

1.6

Perc

en

tag

es

use

rev

erse

per

centa

ges

repea

ted p

erce

nta

ge

chan

ge

In a

sal

e, p

rice

s w

ere

reduce

d b

y 30%

. The

sale

pri

ce o

f an

ite

m

was

£17.5

0.

Cal

cula

te

the

ori

gin

al p

rice

of

the

item

Cal

cula

te t

he

tota

l per

centa

ge

incr

ease

w

hen

an incr

ease

of

30%

is

follo

wed

by

a dec

rease

of

20%

UG

027243 –

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eme

of

Work

: Tea

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g L

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1/L

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2 C

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te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

4

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

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an

d a

lgeb

ra

1

Nu

mb

ers

an

d t

he

nu

mb

er

syst

em

S

tud

en

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

1.6

Perc

en

tag

es

con

tin

ued

so

lve

com

pound

inte

rest

pro

ble

ms

To incl

ude

dep

reci

atio

n

1.7

Rati

o a

nd

p

rop

ort

ion

See

Foundat

ion T

ier.

1.8

D

eg

ree o

f acc

ura

cy

solv

e pro

ble

ms

usi

ng

upper

and low

er

bounds

wher

e va

lues

ar

e giv

en t

o a

deg

ree

of

accu

racy

The

dim

ensi

ons

of

a re

ctan

gle

are

12cm

an

d 8

cm t

o t

he

nea

rest

cm

. Cal

cula

te,

to 3

sig

nific

ant

figure

s,

the

smal

lest

poss

ible

ar

ea a

s a

per

centa

ge

of

the

larg

est

poss

ible

ar

ea

1.9

Sta

nd

ard

fo

rm

expre

ss n

um

ber

s in

the

form

a

10n w

her

e n

is

an inte

ger

and

1 ≤

a <

10

solv

e pro

ble

ms

invo

lvin

g s

tandar

d

form

150

000

000

=

1.5

1

08

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

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1/L

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2 C

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te in M

ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

5

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

1 N

um

bers

an

d t

he

nu

mb

er

syst

em

S

tud

en

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

1.1

0 A

pp

lyin

g

nu

mb

er

See

Foundat

ion T

ier

1.1

1 Ele

ctro

nic

ca

lcu

lato

rs

See

Foundat

ion T

ier

2 E

qu

ati

on

s,

form

ula

e a

nd

id

en

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

:

No

tes

1

Alg

eb

ra a

nd

fu

nct

ion

s W

hat

stu

den

ts

need

to

learn

N

ote

s

2.1

Use

of

sym

bo

ls

use

index

nota

tion

invo

lvin

g f

ract

ional

, neg

ativ

e an

d z

ero

pow

ers

Sim

plif

y:

32

364

t,

31

4321

aa

a

Exte

nsi

on

to

pic

law

s o

f in

dic

es

for

all r

ati

on

al

exp

on

en

ts

Th

e e

qu

ivale

nce

of

nm a a

nd

nm a

sh

ou

ld

be k

no

wn

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

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2 C

ertifica

te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

6

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

2 E

qu

ati

on

s,

form

ula

e a

nd

id

en

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 1

A

lgeb

ra a

nd

fu

nct

ion

s W

hat

stu

den

ts

need

to

learn

N

ote

s

2.2

Alg

eb

raic

m

an

ipu

lati

on

ex

pan

d t

he

pro

duct

of

two lin

ear

expre

ssio

ns

under

stan

d t

he

conce

pt

of

a quad

ratic

expre

ssio

n a

nd b

e ab

le

to f

acto

rise

such

ex

pre

ssio

ns

man

ipula

te a

lgeb

raic

fr

actions

wher

e th

e num

erat

or

and/o

r th

e den

om

inat

or

can b

e num

eric

, lin

ear

or

quad

ratic

(2x

+ 3

)(3x

– 1

)

(2x

– y)

(3x

+ y

)

Fact

ori

se:

x2 + 1

2x –

45

6 x2 –

5x

– 4

Exp

ress

as

a si

ngle

fr

action: 4

3

3

1

xx

3

)35(2

2

)14(3

x

x

xx

34

23

, x

x

1

2

1

3

12

21

xx

xx

Exte

nsi

on

to

pic

Alg

eb

raic

m

an

ipu

lati

on

of

po

lyn

om

ials

, in

clu

din

g

exp

an

din

g

bra

ckets

an

d

coll

ect

ing

lik

e

term

s,

fact

ori

sati

on

Stu

den

ts s

ho

uld

be

ab

le t

o u

se b

rack

ets

.

Fact

ori

sati

on

of

po

lyn

om

ials

of

deg

ree n

, n

≤ 3

, eg

xx

x3

42

3

.

Th

e

no

tati

on

f(x

). (

Use

of

the f

act

or

theo

rem

is

no

t re

qu

ired

)

Fact

ori

se a

nd s

implif

y:

12

42

2

x

xx

x

UG

027243 –

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of

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: Tea

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1/L

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2 C

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te in M

ath

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alo

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Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

7

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

2 E

qu

ati

on

s,

form

ula

e a

nd

id

en

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

2.3

Exp

ress

ion

s an

d

form

ula

e

under

stan

d t

he

pro

cess

of

man

ipula

ting

form

ula

e to

chan

ge

the

subje

ct,

to incl

ude

case

s w

her

e th

e su

bje

ct m

ay a

ppea

r tw

ice,

or

a pow

er o

f th

e su

bje

ct o

ccurs

v2 = u

2 + 2

gs;

m

ake

s th

e su

bje

ct

m =

atat

11

;

mak

e t t

he

subje

ct

V =

34πr

3 ;

mak

e r

the

subje

ct

glT

2

;

mak

e l t

he

subje

ct

2.4

Lin

ear

eq

uati

on

s See

Foundat

ion T

ier.

4

17x

=

2 –

x,

25

3

)2(

6

)32(

xx

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

8

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

2 E

qu

ati

on

s,

form

ula

e a

nd

id

en

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

2.5

Pro

po

rtio

n

set

up p

roble

ms

invo

lvin

g d

irec

t or

inve

rse

pro

port

ion a

nd

rela

te a

lgeb

raic

so

lution

s to

gra

phic

al

repre

senta

tion o

f th

e eq

uat

ions

To incl

ude

only

the

follo

win

g:

y

x,

y

x1,

y

x2 ,

y

21 x

,

y

x3 ,

y

x

2.6

Sim

ult

an

eo

us

lin

ear

eq

uati

on

s ca

lcula

te t

he

exac

t so

lution

of

two

sim

ultan

eous

equat

ions

in t

wo

unkn

owns

inte

rpre

t th

e eq

uat

ions

as lin

es a

nd t

he

com

mon s

olu

tion a

s th

e poin

t of

inte

rsec

tion

3x –

4y

= 7

2x

– y

= 8

2x +

3y

= 1

7 3x

– 5

y =

35

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

9

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

2 E

qu

ati

on

s,

form

ula

e a

nd

id

en

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 1

A

lgeb

ra a

nd

fu

nct

ion

s W

hat

stu

den

ts n

eed

to

learn

N

ote

s

2.7

Qu

ad

rati

c eq

uati

on

s so

lve

quad

ratic

equat

ions

by

fact

ori

sation

solv

e quad

ratic

equat

ions

by

usi

ng t

he

quad

ratic

form

ula

form

and s

olv

e quad

ratic

equat

ions

from

dat

a giv

en in a

co

nte

xt

2x2 –

3x

+ 1

= 0

,

x(3x

– 2

) =

5

New

to

pic

Co

mp

leti

ng

th

e

squ

are

. S

olv

e

qu

ad

rati

c eq

uati

on

s

So

lve q

uad

rati

c eq

uati

on

s b

y

com

ple

tin

g t

he

squ

are

so

lve

sim

ultan

eous

equat

ions

in t

wo

unkn

ow

ns,

one

equat

ion b

eing lin

ear

and t

he

oth

er b

eing

quad

ratic

y =

2x

– 11

and

x2 +

y2 =

25

y =

11x

– 2

and

y =

5x2

New

to

pic

Sim

ult

an

eo

us

eq

uati

on

s: a

naly

tica

l so

luti

on

by

sub

stit

uti

on

Fo

r exam

ple

, w

here

o

ne e

qu

ati

on

is

lin

ear

an

d o

ne

eq

uati

on

is

qu

ad

rati

c

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

10

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

2 E

qu

ati

on

s,

form

ula

e a

nd

id

en

titi

es

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 1

A

lgeb

ra a

nd

fu

nct

ion

s W

hat

stu

den

ts n

eed

to

learn

N

ote

s

2.8

In

eq

ualiti

es

solv

e quad

ratic

ineq

ual

itie

s in

one

unkn

ow

n a

nd r

epre

sent

the

solu

tion s

et o

n a

num

ber

lin

e

iden

tify

har

der

ex

ample

s of re

gio

ns

def

ined

by

linea

r in

equal

itie

s

For

exam

ple

, b

ax

>d

cx

,

x2 ≤ 2

5, 4

x2 > 2

5

Shad

e th

e re

gio

n

def

ined

by

the

ineq

ual

itie

s x

≤ 4

,

y ≤

2x

+ 1

,

5x +

2y

≤ 2

0

New

to

pic

So

luti

on

of

lin

ear

an

d q

uad

rati

c in

eq

ualiti

es

Fo

r exam

ple

,

ax +

b, c

x +

d,

02

r

qxpx

,

bax

rqx

px

2

UG

027243 –

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eme

of

Work

: Tea

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evel

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evel

2 C

ertifica

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ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

11

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Seq

uen

ces

an

d s

eri

es

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 3

S

eq

uen

ces

an

d s

eri

es

Wh

at

stu

den

ts n

eed

to

le

arn

N

ote

s

3.1

Seq

uen

ces

use

lin

ear

expre

ssio

ns

to d

escr

ibe

the

nth

term

of

an a

rith

met

ic

sequen

ce

1,

3,

5,

7,

9,

…

nth t

erm

= 2

n –

1

New

to

pic

New

to

pic

Seq

uen

ces,

in

clu

din

g

tho

se g

iven

by a

fo

rmu

la

for

the n

th t

erm

an

d t

ho

se

gen

era

ted

by a

sim

ple

re

lati

on

of

the f

orm

1nx =

f(

nx)

Ari

thm

eti

c se

ries,

in

clu

din

g t

he f

orm

ula

fo

r th

e s

um

of

the f

irst

n

natu

ral n

um

bers

Th

e g

en

era

l te

rm

an

d t

he s

um

of

n te

rms

of

the s

eri

es

are

req

uir

ed

. Th

e

pro

of

of

the s

um

fo

rmu

la s

ho

uld

be

kn

ow

n

Un

ders

tan

din

g o

f

no

tati

on

wil

l b

e

exp

ect

ed

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

12

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 1

A

lgeb

ra a

nd

fu

nct

ion

s W

hat

stu

den

ts n

eed

to

le

arn

N

ote

s

3.2

Fu

nct

ion

n

ota

tio

n

under

stan

d t

he

conce

pt

that

a f

unct

ion is

a m

appin

g b

etw

een

elem

ents

of

two s

ets

use

funct

ion n

ota

tions

of

the

form

f(

x) =

… a

nd

f :

x

…

under

stan

d t

he

term

s dom

ain a

nd r

ange

and

whic

h v

alues

may

nee

d

to b

e ex

cluded

fro

m t

he

dom

ain

under

stan

d a

nd f

ind t

he

com

posi

te f

unct

ion f

g an

d t

he

inve

rse

funct

ion

f 1

f (x)

=

x1,

excl

ude

x = 0

f(x)

=

3x

,

excl

ude

x <

–3

‘ fg’ w

ill m

ean ‘do

g

firs

t, t

hen

f’

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

13

C

ert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

3.3

Gra

ph

s plo

t an

d d

raw

gra

phs

with e

quat

ion:

y =

Ax3 +

Bx2 +

Cx

+ D

in

whic

h:

(i)

the

const

ants

are

in

teger

s an

d s

om

e co

uld

be

zero

(ii)

the

lett

ers

x an

d y

can b

e re

pla

ced

with a

ny

oth

er t

wo

lett

ers

y =

x3 ,

y =

3x3 –

2x2 +

5x

– 4,

y =

2x3 –

6x

+ 2

,

V =

60w

(60

– w

)

New

to

pic

Qu

ad

rati

c fu

nct

ion

s an

d

their

gra

ph

s

Th

e d

iscr

imin

an

t o

f a

qu

ad

rati

c fu

nct

ion

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

14

C

ert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

3.3

G

rap

hs

con

tin

ued

or:

y =

Ax3 +

Bx2 +

Cx

+ D

+

E/x

+ F

/x2

in w

hic

h:

(i)

the

const

ants

are

num

eric

al a

nd a

t le

ast

thre

e of

them

are

zer

o

(ii)

the

lett

ers

x an

d y

ca

n b

e re

pla

ced

with a

ny

oth

er t

wo

lett

ers

find t

he

gra

die

nts

of

non-l

inea

r gra

phs

y =

x1

, x

0,

y =

2x2 +

3x

+ 1

/x,

x

0,

y =

x1

(3x2 –

5),

x

0,

W =

25 d

, d

0

By

dra

win

g a

tan

gen

t

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

15

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 1

A

lgeb

ra a

nd

fu

nct

ion

s W

hat

stu

den

ts n

eed

to

le

arn

N

ote

s

3.3

Gra

ph

s co

nti

nu

ed

find t

he

inte

rsec

tion

poin

ts o

f tw

o gra

phs,

one

linea

r ( y

1) a

nd o

ne

non-l

inea

r (y

2),

and

reco

gnis

e th

at t

he

solu

tion

s co

rres

pond

to t

he

solu

tion

s of

y 2 –

y1

= 0

The

x-va

lues

of

the

inte

rsec

tion o

f th

e tw

o

gra

phs:

y =

2x

+ 1

y =

x2 +

3x

– 2

are

the

solu

tions

of:

x2 + x

– 3

= 0

Exte

nsi

on

to

pic

Gra

ph

s o

f fu

nct

ion

s;

sketc

hin

g c

urv

es

defi

ned

b

y s

imp

le e

qu

ati

on

s.

Geo

metr

ical

inte

rpre

tati

on

of

alg

eb

raic

so

luti

on

of

eq

uati

on

s. U

se

inte

rsect

ion

po

ints

of

gra

ph

s o

f fu

nct

ion

s to

so

lve e

qu

ati

on

s

Fu

nct

ion

s to

in

clu

de

sim

ple

cu

bic

fu

nct

ion

s an

d t

he

reci

pro

cal fu

nct

ion

xky

, w

ith

0

x

Kn

ow

led

ge o

f th

e

term

asy

mp

tote

is

exp

ect

ed

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

16

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

3.3

G

rap

hs

con

tin

ued

calc

ula

te t

he

gra

die

nt

of

a st

raig

ht

line

giv

en

the

Car

tesi

an

coord

inat

es o

f tw

o

poin

ts

Sim

ilarly,

the

x-va

lues

of

the

inte

rsec

tion o

f th

e tw

o g

raphs:

y =

5

y =

x3 –

3x2 +

7

are

the

solu

tions

of:

x3 – 3

x2 + 2

= 0

re

cognis

e th

at

equat

ions

of th

e fo

rm

y =

mx

+ c

are

str

aight

line

gra

phs

with

gra

die

nt

m a

nd

inte

rcep

t on t

he

y axi

s at

the

poin

t ( 0

, c)

find t

he

equat

ion o

f a

stra

ight

line

par

alle

l to

a

giv

en lin

e

Find t

he

equat

ion o

f th

e st

raig

ht

line

thro

ugh

(1,

7)

and (

2,

9)

Exte

nsi

on

to

pic

Eq

uati

on

s o

f a s

traig

ht

lin

e,

incl

ud

ing

th

e

form

s)

(2

12

1x

xm

yy

an

d

0

cby

ax

To

in

clu

de:

1)

the e

qu

ati

on

of

a

lin

e t

hro

ug

h t

wo

g

iven

po

ints

2

) th

e e

qu

ati

on

of

a

lin

e p

ara

llel (o

r p

erp

en

dic

ula

r) t

o a

g

iven

lin

e t

hro

ug

h

a g

iven

po

int.

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

17

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Co

ord

inate

geo

metr

y in

th

e (

x, y

)

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 2

C

oo

rdin

ate

g

eo

metr

y in

th

e (

x, y

)

Wh

at

stu

den

ts n

eed

to

le

arn

N

ote

s

Exte

nsi

on

to

pic

Co

nd

itio

ns

for

two

st

raig

ht

lin

es

to b

e

para

llel o

r p

erp

en

dic

ula

r to

each

oth

er

Fo

r exam

ple

, th

e

lin

e p

erp

en

dic

ula

r to

th

e lin

e

184

3

y

x,

thro

ug

h t

he p

oin

t ,2(

)3,

has

the

eq

uati

on

)2(

343

x

y

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alongsi

de

Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

18

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Co

ord

inate

geo

metr

y in

th

e (

x, y

)

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 2

C

oo

rdin

ate

g

eo

metr

y in

th

e (

x, y

)

Wh

at

stu

den

ts n

eed

to

le

arn

N

ote

s

3.3

Gra

ph

s co

nti

nu

ed

Exte

nsi

on

to

pic

Kn

ow

led

ge o

f th

e e

ffect

o

f si

mp

le t

ran

sfo

rmati

on

s o

n t

he g

rap

h o

f y

= f

(x)

as

rep

rese

nte

d b

y y

= a

f(x)

,

y =

f (

x) +

a,

y =

f (

x +

a),

y

= f

(ax

)

Stu

den

ts s

ho

uld

b

e a

ble

to

ap

ply

o

ne o

f th

ese

tr

an

sfo

rmati

on

s

to a

ny f

un

ctio

n

(qu

ad

rati

cs,

cub

ics,

re

cip

roca

l) a

nd

sk

etc

h t

he r

esu

ltin

g

gra

ph

.

Giv

en

th

e g

rap

h o

f

an

y f

un

ctio

n y

= f

(x)

stu

den

ts s

ho

uld

be

ab

le t

o s

ketc

h t

he

gra

ph

resu

ltin

g

fro

m o

ne o

f th

ese

tran

sfo

rmati

on

s.

UG

027243 –

Sch

eme

of

Work

: Tea

chin

g L

evel

1/L

evel

2 C

ertifica

te in M

ath

ematics

alo

ngsi

de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

19

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Dif

fere

nti

ati

on

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 4

D

iffe

ren

tiati

on

W

hat

stu

den

ts n

eed

to

le

arn

No

tes

3.4

Calc

ulu

s D

iffe

ren

tiati

on

under

stan

d t

he

conce

pt

of

a va

riab

le

rate

of

chan

ge

diffe

rentiat

e in

teger

pow

ers

of x

det

erm

ine

gra

die

nts

, ra

tes

of

chan

ge,

tu

rnin

g p

oin

ts

(max

ima

and m

inim

a)

by

diffe

rentiat

ion a

nd

rela

te t

hes

e to

gra

phs

dis

tinguis

h b

etw

een

max

ima

and m

inim

a by

consi

der

ing t

he

gen

eral

shap

e of

the

gra

ph

y =

x +

x9

Find t

he

Car

tesi

an

coord

inat

es o

f th

e m

axim

um

and

min

imum

poi

nts

Exte

nsi

on

to

pic

Th

e d

eri

vati

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

gra

die

nt

of

the t

an

gen

t as

a lim

it;

inte

rpre

tati

on

as

a

rate

of

chan

ge;

seco

nd

o

rder

deri

vati

ves

Fo

r exam

ple

,

kn

ow

led

ge t

hat

dxdy

is t

he r

ate

of

chan

ge

of

y wit

h r

esp

ect

to

x

UG

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Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

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ears

on E

duca

tion L

imited

2011

20

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Dif

fere

nti

ati

on

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 4

D

iffe

ren

tiati

on

co

nti

nu

ed

W

hat

stu

den

ts n

eed

to

le

arn

No

tes

3.4

Calc

ulu

s D

iffe

ren

tiati

on

co

nti

nu

ed

apply

cal

culu

s to

lin

ear

kinem

atic

s an

d

to o

ther

sim

ple

pra

ctic

al p

roble

ms

The

dis

pla

cem

ent,

s

met

res,

of

a par

ticl

e fr

om

a f

ixed

poin

t 0

afte

r t s

econds

is g

iven

by:

s = 2

4t2 –

t3 ,

0 ≤

t ≤

20

Find e

xpre

ssio

ns

for

the

velo

city

and t

he

acce

lera

tion

Exte

nsi

on

to

pic

K

no

wle

dg

e o

f th

e

chain

ru

le i

s n

ot

req

uir

ed

Th

e n

ota

tio

n f

/ (x)

may b

e u

sed

Fo

r exam

ple

, fo

r 1

n

, th

e a

bilit

y t

o

dif

fere

nti

ate

exp

ress

ion

s su

ch a

s )1

)(52(

xx

an

d

21

2

3

35 xx

x

is

exp

ect

ed

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Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

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tion L

imited

2011

21

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Dif

fere

nti

ati

on

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 4

D

iffe

ren

tiati

on

co

nti

nu

ed

W

hat

stu

den

ts n

eed

to

le

arn

No

tes

New

co

nce

pts

Ap

plica

tio

ns

of

dif

fere

nti

ati

on

to

g

rad

ien

ts,

tan

gen

ts a

nd

n

orm

als

Use

of

dif

fere

nti

ati

on

to

fi

nd

eq

uati

on

s o

f ta

ng

en

ts a

nd

n

orm

als

at

speci

fic

po

ints

on

a c

urv

e

Inte

gra

tio

n

5

Inte

gra

tio

n

Wh

at

stu

den

ts n

eed

to

le

arn

N

ote

s

3.4

Calc

ulu

s D

iffe

ren

tiati

on

co

nti

nu

ed

New

to

pic

Ind

efi

nit

e in

teg

rati

on

as

the r

evers

e o

f d

iffe

ren

tiati

on

inte

gra

tio

n o

f n x

Stu

den

ts s

ho

uld

kn

ow

th

at

a

con

stan

t o

f in

teg

rati

on

is

req

uir

ed

.

UG

027243 –

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Mat

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atic

s 1 –

Iss

ue

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July

2011

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imited

2011

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Cert

ific

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Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

1 N

um

ber

an

d a

lgeb

ra

Dif

fere

nti

ati

on

3 S

eq

uen

ces,

fu

nct

ion

s an

d

gra

ph

s

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s 5

In

teg

rati

on

co

nti

nu

ed

W

hat

stu

den

ts n

eed

to

le

arn

No

tes

3.4

Calc

ulu

s D

iffe

ren

tiati

on

co

nti

nu

ed

N

ew

to

pic

Fo

r exam

ple

, th

e

ab

ilit

y t

o in

teg

rate

exp

ress

ion

s su

ch

as:

21

23

21

x

x a

nd

21

2 )2(

x

x

is

exp

ect

ed

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Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

Pea

rson E

duca

tion L

imited

2011

23

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 N

um

ber

an

d a

lgeb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

4.1

Lin

es

an

d

tria

ng

les

See

Foundat

ion T

ier.

4.2

Po

lyg

on

s See

Foundat

ion T

ier.

4.3

Sym

metr

y

See

Foundat

ion T

ier.

4.4

Measu

res

See

Foundat

ion T

ier.

4.5

Co

nst

ruct

ion

See

Foundat

ion T

ier.

UG

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eme

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: Tea

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ath

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Core

Mat

hem

atic

s 1 –

Iss

ue

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July

2011

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imited

2011

24

Cert

ific

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in

Math

em

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cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 N

um

ber

an

d a

lgeb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

4.6

Cir

cle

pro

pert

ies

under

stan

d a

nd u

se

the

inte

rnal

and

exte

rnal

inte

rsec

ting

chord

pro

per

ties

re

cognis

e th

e te

rm

cycl

ic q

uad

rila

tera

l under

stan

d a

nd u

se

angle

pro

per

ties

of

the

circ

le incl

udin

g:

an

gle

subte

nded

by

an a

rc a

t th

e ce

ntr

e of

a ci

rcle

is

twic

e th

e an

gle

su

bte

nded

at

any

poin

t on t

he

rem

ainin

g p

art

of

the

circ

um

fere

nce

an

gle

subte

nded

at

the

circ

um

fere

nce

by

a dia

met

er is

a ri

ght

angle

Form

al p

roof of

thes

e th

eore

ms

is n

ot

requir

ed

UG

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eme

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Work

: Tea

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evel

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ath

ematics

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de

Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

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tion L

imited

2011

25

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 N

um

ber

an

d a

lgeb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

4.6

Cir

cle

pro

pert

ies

con

tin

ued

an

gle

s in

the

sam

e se

gm

ent

are

equal

th

e su

m o

f th

e opposi

te a

ngle

s of

a cy

clic

quad

rila

tera

l is

180

th

e al

tern

ate

segm

ent

theo

rem

4.7

Geo

metr

ical

reaso

nin

g

pro

vide

reas

ons,

usi

ng

stan

dar

d g

eom

etri

cal

stat

emen

ts,

to s

upport

num

eric

al v

alues

for

angle

s obta

ined

in a

ny

geo

met

rica

l co

nte

xt

invo

lvin

g lin

es,

poly

gons

and c

ircl

es

UG

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Work

: Tea

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evel

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ath

ematics

alongsi

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Core

Mat

hem

atic

s 1 –

Iss

ue

1 –

July

2011

© P

ears

on E

duca

tion L

imited

2011

26

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 N

um

ber

an

d a

lgeb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

4.8

Tri

go

no

metr

y

an

d P

yth

ag

ora

s’

Th

eo

rem

under

stan

d a

nd u

se

sine,

cosi

ne

and

tangen

t of

obtu

se

angle

s

under

stan

d a

nd u

se

angle

s of

elev

atio

n

and d

epre

ssio

n

under

stan

d a

nd u

se

the

sine

and c

osi

ne

rule

s fo

r an

y tr

iangle

use

Pyt

hag

ora

s’

Theo

rem

in 3

dim

ensi

ons

under

stan

d a

nd u

se

the

form

ula

½

abs

in C

for

the

area

of

a tr

iangle

apply

tri

gonom

etri

cal

met

hods

to s

olv

e pro

ble

ms

in 3

dim

ensi

ons,

incl

udin

g

findin

g t

he

angle

bet

wee

n a

lin

e an

d a

pla

ne

The

angle

bet

wee

n

two p

lanes

will

not

be

requir

ed

UG

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eme

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Work

: Tea

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Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

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imited

2011

27

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 N

um

ber

an

d a

lgeb

ra

Alg

eb

ra a

nd

fu

nct

ion

s

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

4.9

M

en

sura

tio

n

find p

erim

eter

s an

d

area

s of

sect

ors

of

circ

les

Rad

ian m

easu

re is

excl

uded

4.1

0 3

-D s

hap

es

an

d v

olu

me

find t

he

surf

ace

area

an

d v

olu

me

of

a sp

her

e an

d a

rig

ht

circ

ula

r co

ne

usi

ng

rele

vant

form

ula

e

conve

rt b

etw

een

volu

me

mea

sure

s m

3 →

cm

3 a

nd v

ice

vers

a

UG

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Core

Mat

hem

atic

s 1 –

Iss

ue

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July

2011

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imited

2011

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Cert

ific

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Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 N

um

ber

an

d a

lgeb

ra

4 G

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

4.1

1 S

imil

ari

ty

under

stan

d t

hat

are

as

of

sim

ilar

figure

s ar

e in

the

ratio o

f th

e sq

uar

e of

corr

espondin

g s

ides

under

stan

d t

hat

vo

lum

es o

f si

mila

r figure

s ar

e in

the

ratio

of

the

cube

of

corr

espondin

g s

ides

use

are

as a

nd

volu

mes

of

sim

ilar

figure

s in

solv

ing

pro

ble

ms

UG

027243 –

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Work

: Tea

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evel

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Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

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duca

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imited

2011

29

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 N

um

ber

an

d a

lgeb

ra

5 V

ect

ors

an

d

tran

sfo

rmati

on

g

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

5.1

Vect

ors

under

stan

d t

hat

a

vect

or

has

both

m

agnitude

and

dir

ection

under

stan

d a

nd u

se

vect

or

nota

tion

multip

ly v

ecto

rs b

y sc

alar

quan

tities

add a

nd s

ubtr

act

vect

ors

calc

ula

te t

he

modulu

s (m

agnitude)

of

a ve

ctor

find t

he

resu

ltan

t of

two o

r m

ore

vec

tors

apply

vec

tor

met

hods

for

sim

ple

geo

met

rica

l pro

ofs

The

nota

tions ΟΑ

and

a w

ill b

e use

d

ΟΑ

= 3

a, A

B=

2b,

BC=

c

so:

OC

= 3

a +

2b

+ c

CA

=

c –

2b

UG

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Core

Mat

hem

atic

s 1 –

Iss

ue

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July

2011

© P

ears

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duca

tion L

imited

2011

30

Cert

ific

ate

in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

2 S

hap

e,

space

an

d m

easu

res

5 V

ect

ors

an

d

tran

sfo

rmati

on

g

eo

metr

y

Stu

den

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

5.2

Tra

nsf

orm

ati

on

g

eo

metr

y

See

Foundat

ion T

ier.

Colu

mn v

ecto

rs m

ay

be

use

d t

o d

efin

e

tran

slat

ions

6 S

tati

stic

s S

tud

en

ts s

ho

uld

be

tau

gh

t to

: N

ote

s

6.1

Gra

ph

ical

rep

rese

nta

tio

n

of

data

const

ruct

and

inte

rpre

t his

togra

ms

const

ruct

cum

ula

tive

fr

equen

cy d

iagra

ms

from

tab

ula

ted d

ata

use

cum

ula

tive

fr

equen

cy d

iagra

ms

For

continuou

s va

riab

les

with u

neq

ual

cl

ass

inte

rval

s

UG

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Core

Math

ematics

1 –

Iss

ue

1 –

July

2011 ©

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imited

2011

31

Cert

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in

Math

em

ati

cs (

Hig

her

tier)

G

CE C

ore

Math

em

ati

cs 1

un

it

AO

3 H

an

dli

ng

data

6.2

Sta

tist

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UG027243 – Scheme of Work: Teaching Level 1/Level 2 Certificate in Mathematics alongside

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

33

Higher tier Mathematics Certificate/GCE Core Mathematics 1 unit content summary The table below is a summary of the Mathematics Certificate 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 Mathematics Certificate Teacher’s Guide. The module numbers in the table below refer to the module numbers in the course planner within the Mathematics Certificate 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.

UG027243 – Scheme of Work: Teaching Level 1/Level 2 Certificate in Mathematics

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

34

The table below is a summary of the Mathematics Certificate 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 Mathematics Certificate Teacher’s Guide. The module numbers in the table below refer to the module numbers in the course planner within the Mathematics Certificate 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.



35

Higher tier Mathematics Certificate/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 Mathematics Certificate. It should be used alongside the Mathematics Certificate 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



36

Resources

Textbook References

Edexcel IGCSE Mathematics A Student Book 1

Unit 3: Number 3 page 117 Unit 3: Number 3 page 114 – 116


Unit 2: Number 2 page 66 – 70

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



37

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



38

Resources

Textbook References


Unit 1: Algebra 1 page 11 – 12 Unit 2: Algebra 2 page 65 – 67 Unit 3: Algebra 3 page 121 – 123


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


page 2, 4 – 6



39

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


Unit 1: Algebra 1 page 12 – 17 Unit 2: Graphs 2 page 79 – 80 Unit 3: Algebra 3 page 126 – 130



40



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


page 28 – 31



41

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


Unit 1: Graphs 1 page 19 – 27



42



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 1 , y 1 ) to the point with the coordinates (x 2 , y 2 )

by using the formula

m = 12

12

xxyy

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

xxxx

yyyy

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



43

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


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


page 92 – 100



44

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


Unit 5: Algebra 5 page 248 – 251


Unit 2: Algebra 2 page 71 – 80 Unit 3: Algebra 3 page 176 – 182



45



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



46

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


Unit 2: Algebra 2 page 74 – 78, 81 – 86


Unit 2: Algebra 2 page 81 – 84 Unit 5: Algebra 5 page 356



47



Solve linear inequalities using similar methods to those for solvinglinear equations 3.4

Solve quadratic inequalities 3.5

Resources

Textbook References


page 31 – 39



48

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


Unit 2: Number 2 page 60, 73 – 74 Unit 4: Graphs 4 page 185 – 190


Unit 2: Number 2 page 66 – 70



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


page 3,4 and 8,9



49

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 x , which means

x = 2 must be excluded from the domain of g

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 f 1 (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



50

Resources

Textbook References


Unit 3: Algebra 3 page 183 – 197



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


page 55 – 65



51

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 + xE

+ 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



52

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


Unit 1: Graphs 1 page 19 – 27 Unit 3: Graphs 3 page 198 – 209



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 xky , 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


page 23 – 25, 42 – 55



53

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 dxdy

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


Unit 4: Graphs 4 page 268 – 287



54


Differentiation


Estimate the gradient of the tangent 7.1

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

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

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

123 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


page 113 – 131



55

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



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C1 topic

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

GCE unit Core Mathematics 1


Integrate functions of the form f(x) = nax where n ℝ and a is a constant 8.1

Apply the principle of integration separately to each term of dxdy

8.2

Use the integral sign 8.3

Simplify an expression into separate terms of the form nx 8.4

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


page 134 – 142

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