Download - The mathematics overview and learning objectives A3 · increasing the reducing the increasing the degree of introduce and use a 5 Learning objectives 1 Mathematical processes and

3 The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010

The

Fram

ewor

k fo

r sec

onda

ry m

athe

mat

ics:

ove

rvie

w a

nd le

arni

ng o

bjec

tives

Ove

rvie

w o

f str

ands

Stra

nds

Sub-

stra

nds

Stra

nds

Sub-

stra

nds

1 M

athe

mat

ical

pro

cess

es a

nd a

pplic

atio

ns

3 A

lgeb

ra

1.1

Repr

esen

ting

3.1

Equa

tions

, for

mul

ae, e

xpre

ssio

ns a

nd id

entit

ies

1.2

Anal

ysin

g –

use

mat

hem

atic

al re

ason

ing

3.2

Sequ

ence

s, fu

nctio

ns a

nd g

raph

s

1.3

Anal

ysin

g –

use

appr

opria

te m

athe

mat

ical

pro

cedu

res

4 G

eom

etry

and

mea

sure

s

1.4

Inte

rpre

ting

and

eval

uatin

g 4.

1 G

eom

etric

al re

ason

ing

1.5

Com

mun

icat

ing

and

refle

ctin

g 4.

2 Tr

ansf

orm

atio

ns a

nd c

oord

inat

es

2 N

umbe

r 4.

3 Co

nstr

uctio

n an

d lo

ci

2.1

Plac

e va

lue,

ord

erin

g an

d ro

undi

ng

4.4

Mea

sure

s and

men

sura

tion

2.2

Inte

gers

, pow

ers a

nd ro

ots

5 St

atis

tics

2.3

Frac

tions

, dec

imal

s, pe

rcen

tage

s, ra

tio a

nd p

ropo

rtio

n 5.

1 Sp

ecify

ing

a pr

oble

m, p

lann

ing

and

colle

ctin

g da

ta

2.4

Num

ber o

pera

tions

5.

2 Pr

oces

sing

and

repr

esen

ting

data

2.5

Men

tal c

alcu

latio

n m

etho

ds

5.3

Inte

rpre

ting

and

disc

ussin

g re

sults

2.6

Writ

ten

calc

ulat

ion

met

hods

5.

4 Pr

obab

ility

2.7

Calc

ulat

or m

etho

ds

2.8

Chec

king

resu

lts

© Crown copyright 2009 01061-2009DOM-EN

5

Lear

ning

obj

ectiv

es

1 M

athe

mat

ical

pro

cess

es a

nd a

pplic

atio

nsSo

lve

prob

lem

s, e

xplo

re a

nd in

vest

igat

e in

a ra

nge

of co

ntex

ts

Incr

ease

the

chal

leng

e an

d bu

ild p

rogr

essi

on a

cros

s the

key

stag

e, a

nd fo

r gro

ups o

f pup

ils b

y:

incr

easin

g th

e co

mpl

exit

y of

the

appl

icat

ion,

e.g

. non

-rout

ine,

mul

ti-st

ep p

robl

ems,

exte

nded

enq

uirie

s

redu

cing

the

fam

iliar

ity

of th

e co

ntex

t, e.

g. n

ew c

onte

xts i

n m

athe

mat

ics,

cont

exts

dra

wn

from

oth

er su

bjec

ts, o

ther

asp

ects

of p

upils

’ liv

es

The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010


incr

easin

g th

e te

chni

cal d

eman

d of

the

mat

hem

atic

s req

uire

d, e

.g. m

ore

adva

nced

con

cept

s, m

ore

diff

icul

t pro

cedu

res

incr

easin

g th

e de

gree

of i

ndep

ende

nce

and

auto

nom

y in

pro

blem

-sol

ving

and

inve

stig

atio

n

Rep

rese

ntin

g 1.

1 iden

tify

the

nece

ssar

y in

form

atio

n to

un

ders

tand

or s

impl

ifya

cont

ext o

r pro

blem

; re

pres

ent p

robl

ems,

mak

ing

corr

ect u

se

of sy

mbo

ls, w

ords

, di

agra

ms,

tabl

es a

nd

grap

hs; u

se a

ppro

pria

te

proc

edur

es a

nd to

ols,

incl

udin

g IC

T

Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

iden

tify

the

mat

hem

atic

al fe

atur

esof

a c

onte

xt o

r pr

oble

m; t

ry o

ut a

nd

com

pare

mat

hem

atic

al

repr

esen

tatio

ns; s

elec

tap

prop

riate

pro

cedu

res

and

tool

s, in

clud

ing

ICT

brea

k do

wn

subs

tant

ial

task

s to

mak

e th

em

mor

e m

anag

eabl

e;

repr

esen

t pro

blem

s an

d sy

nthe

sise

info

rmat

ion

in

alge

brai

c, g

eom

etric

al

or g

raph

ical

form

; m

ove

from

one

form

to

ano

ther

to g

ain

a di

ffere

nt p

ersp

ectiv

eon

the

prob

lem

com

pare

and

eva

luat

ere

pres

enta

tions

;ex

plai

n th

e fe

atur

es

sele

cted

and

just

ify

the

choi

ce o

f re

pres

enta

tion

in

rela

tion

to th

e co

ntex

t

choo

se a

nd c

ombi

ne

repr

esen

tatio

ns fr

om a

ra

nge

of p

ersp

ectiv

es;

intr

oduc

e an

d us

e a

rang

e of

mat

hem

atic

alte

chni

ques

, the

mos

tef

ficie

nt fo

r ana

lysis

and

mos

t effe

ctiv

e fo

r co

mm

unic

atio

n

syst

emat

ical

ly m

odel

co

ntex

ts o

r pro

blem

s th

roug

h pr

ecise

an

d co

nsist

ent u

se

of sy

mbo

ls an

d re

pres

enta

tions

, and

su

stai

n th

is th

roug

hout

th

e w

ork

6

Ana

lysi

ng –

use

mat

hem

atic

al re

ason

ing

1.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion


01061-2009DOM-EN © Crown copyright 2009

clas

sify

and

visu

alise

pr

oper

ties a

nd

patt

erns

; gen

eral

ise

in si

mpl

e ca

ses b

y w

orki

ng lo

gica

lly; d

raw

simpl

e co

nclu

sions

an

d ex

plai

n re

ason

ing;

unde

rsta

nd th

e sig

nific

ance

of a

co

unte

r-exa

mpl

e;ta

ke a

ccou

nt o

f fe

edba

ck a

nd le

arn

from

mist

akes

visu

alise

and

m

anip

ulat

e dy

nam

ic

imag

es; c

onje

ctur

e an

d ge

nera

lise;

mov

ebe

twee

n th

e ge

nera

l an

d th

e pa

rtic

ular

tote

st th

e lo

gic

of a

n ar

gum

ent;

iden

tify

exce

ptio

nal c

ases

or

coun

ter-e

xam

ples

;m

ake

conn

ectio

ns w

ith

rela

ted

cont

exts

use

conn

ectio

ns w

ith

rela

ted

cont

exts

toim

prov

e th

e an

alys

is of

a

situa

tion

or p

robl

em;

pose

que

stio

ns a

nd

mak

e co

nvin

cing

ar

gum

ents

to ju

stify

ge

nera

lisat

ions

or

solu

tions

; rec

ogni

se th

e im

pact

of c

onst

rain

ts o

r as

sum

ptio

ns

iden

tify

a ra

nge

of st

rate

gies

and

ap

prec

iate

that

mor

eth

an o

ne a

ppro

ach

may

be

nece

ssar

y;

expl

ore

the

effe

cts

of v

aryi

ng v

alue

s and

lo

ok fo

r inv

aria

nce

and

cova

rianc

e in

mod

els

and

repr

esen

tatio

ns;

exam

ine

and

refin

ear

gum

ents

, con

clus

ions

an

d ge

nera

lisat

ions

;pr

oduc

e sim

ple

proo

fs

mak

e pr

ogre

ss b

yex

plor

ing

mat

hem

atic

al

task

s, de

velo

ping

an

d fo

llow

ing

alte

rnat

ive

appr

oach

es;

exam

ine

and

exte

ndge

nera

lisat

ions

; sup

port

as

sum

ptio

ns b

y cl

ear

argu

men

t and

follo

wth

roug

h a

sust

aine

dch

ain

of re

ason

ing,

incl

udin

g pr

oof

pres

ent r

igor

ous a

nd

sust

aine

d ar

gum

ents

;re

ason

indu

ctiv

ely,

dedu

ce a

nd p

rove

; ex

plai

n an

d ju

stify

as

sum

ptio

ns a

nd

cons

trai

nts

Ana

lysi

ng –

use

app

ropr

iate

mat

hem

atic

al p

roce

dure

s 1.

3W

ithin

the

appr

opria

te ra

nge

and

cont

ent:

mak

e ac

cura

te m

athe

mat

ical

dia

gram

s, gr

aphs

and

con

stru

ctio

ns o

n pa

per a

nd o

n sc

reen

; cal

cula

te a

ccur

atel

y, se

lect

ing

men

tal m

etho

ds o

r cal

cula

ting

devi

ces a

s app

ropr

iate

; man

ipul

ate

num

bers

, alg

ebra

ic e

xpre

ssio

ns a

nd e

quat

ions

, and

app

ly ro

utin

e al

gorit

hms;

use

accu

rate

not

atio

n, in

clud

ing

corr

ect

synt

ax w

hen

usin

g IC

T; re

cord

met

hods

, sol

utio

ns a

nd c

oncl

usio

ns; e

stim

ate,

app

roxi

mat

e an

d ch

eck

wor

king

7

Inte

rpre

ting

and

eval

uatin

g 1.

4 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

inte

rpre

t inf

orm

atio

n fro

m a

mat

hem

atic

al

repr

esen

tatio

n or

co

ntex

t; re

late

find

ings

to

the

orig

inal

con

text

; ch

eck

the

accu

racy

of

the

solu

tion;

exp

lain

an

d ju

stify

met

hods

an

d co

nclu

sions

;co

mpa

re a

nd e

valu

ate

appr

oach

es

use

logi

cal a

rgum

ent

to in

terp

ret t

he

mat

hem

atic

s in

a gi

ven

cont

ext o

r to

esta

blish

the

trut

h of

a st

atem

ent;

give

accu

rate

solu

tions

ap

prop

riate

to th

e co

ntex

t or p

robl

em;

eval

uate

the

effic

ienc

yof

alte

rnat

ive

stra

tegi

es

and

appr

oach

es

Com

mun

icat

ing

and

refle

ctin

g 1.

5

just

ify th

e m

athe

mat

ical

feat

ures

draw

n fro

m a

con

text

an

d th

e ch

oice

of

appr

oach

; gen

erat

efu

ller s

olut

ions

by

pres

entin

g a

conc

ise,

reas

oned

arg

umen

t us

ing

sym

bols,

di

agra

ms,

grap

hs a

nd

rela

ted

expl

anat

ions

revi

ew a

nd re

fine

own

findi

ngs a

nd

appr

oach

es o

n th

eba

sis o

f disc

ussio

ns

with

oth

ers;

look

for

and

refle

ct o

n ot

her

appr

oach

es a

nd b

uild

on

pre

viou

s exp

erie

nce

of si

mila

r situ

atio

ns

and

outc

omes

refin

e ow

n fin

ding

s an

d ap

proa

ches

on

the

basis

of d

iscus

sions

w

ith o

ther

s; re

cogn

ise

effic

ienc

y in

an

appr

oach

; rel

ate

the

curr

ent p

robl

em a

nd

stru

ctur

e to

pre

viou

ssit

uatio

ns



com

mun

icat

e ow

n fin

ding

s effe

ctiv

ely,

oral

ly a

nd in

writ

ing,

and

disc

uss a

nd

com

pare

app

roac

hes

and

resu

lts w

ith o

ther

s; re

cogn

ise e

quiv

alen

t ap

proa

ches

mak

e se

nse

of, a

nd

judg

e th

e va

lue

of, o

wn

findi

ngs a

nd th

ose

pres

ente

d by

oth

ers;

judg

e th

e st

reng

th o

f em

piric

al e

vide

nce

and

dist

ingu

ish b

etw

een

evid

ence

and

pro

of;

just

ify g

ener

alisa

tions

, ar

gum

ents

or s

olut

ions

use

a ra

nge

of fo

rms

to c

omm

unic

ate

findi

ngs e

ffect

ivel

y to

diffe

rent

aud

ienc

es;

revi

ew fi

ndin

gs a

nd

look

for e

quiv

alen

ce to

diffe

rent

pro

blem

s with

simila

r str

uctu

re

just

ify a

nd e

xpla

in

solu

tions

to p

robl

ems

invo

lvin

g an

unf

amili

ar

cont

ext o

r a n

umbe

r of

feat

ures

or

varia

bles

; com

men

t co

nstr

uctiv

ely

on

reas

onin

g, lo

gic,

proc

ess,

resu

lts a

nd

conc

lusio

ns

use

mat

hem

atic

alla

ngua

ge a

ndsy

mbo

ls ef

fect

ivel

y in

pr

esen

ting

conv

inci

ng

conc

lusio

ns o

r fin

ding

s; cr

itica

lly re

flect

on

own

lines

of e

nqui

ry

whe

n ex

plor

ing;

sear

chfo

r and

app

reci

ate

mor

e el

egan

t for

ms

of c

omm

unic

atin

g ap

proa

ches

and

so

lutio

ns; c

onsid

er th

e ef

ficie

ncy

of a

ltern

ativ

e lin

es o

f enq

uiry

or

proc

edur

es

show

insig

ht in

to

the

mat

hem

atic

al

conn

ectio

ns in

the

cont

ext o

r pro

blem

; cr

itica

lly e

xam

ine

stra

tegi

es a

dopt

ed

and

argu

men

ts

pres

ente

d; c

onsid

er

the

assu

mpt

ions

in th

e m

odel

and

reco

gnise

lim

itatio

ns in

the

accu

racy

of r

esul

ts

and

conc

lusio

ns

Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

rout

inel

y re

view

and

re

fine

findi

ngs a

nd

appr

oach

es; i

dent

ifyho

w o

ther

con

text

s w

ere

diffe

rent

from

, or

simila

r to,

the

curr

ent

situa

tion

and

expl

ain

how

and

why

the

sam

e or

diff

eren

t str

ateg

ies

wer

e us

ed

9

2 N

umbe

r

Pla

ce v

alue

, ord

erin

g an

d ro

undi

ng

2.1



Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

unde

rsta

nd a

nd u

se

deci

mal

not

atio

n an

d pl

ace

valu

e; m

ultip

ly

and

divi

de in

tege

rsan

d de

cim

als b

y 10

, 10

0, 1

000,

and

exp

lain

th

e ef

fect

com

pare

and

ord

er

deci

mal

s in

diffe

rent

co

ntex

ts; k

now

that

whe

n co

mpa

ring

mea

sure

men

ts th

e un

its m

ust b

e th

e sa

me

roun

d po

sitiv

e w

hole

nu

mbe

rs to

the

near

est

10, 1

00 o

r 100

0, a

nd

deci

mal

s to

the

near

est

who

le n

umbe

r or o

ne

deci

mal

pla

ce

read

and

writ

e po

sitiv

e in

tege

r pow

ers o

f 10;

m

ultip

ly a

nd d

ivid

e in

tege

rs a

nd d

ecim

als

by 0

.1 a

nd 0

.01

orde

r dec

imal

s

roun

d po

sitiv

e nu

mbe

rsto

any

giv

en p

ower

of

10;

roun

d de

cim

als

to th

e ne

ares

t who

le

num

ber o

r to

one

or

two

deci

mal

pla

ces

exte

nd k

now

ledg

e of

inte

ger p

ower

s of

10;

reco

gnise

the

equi

vale

nce

of 0

.1, 101

and

10–1

; mul

tiply

and

di

vide

by

any

inte

ger

pow

er o

f 10

use

roun

ding

to m

ake

estim

ates

and

to g

ive

solu

tions

to p

robl

ems

to a

n ap

prop

riate

de

gree

of a

ccur

acy

conv

ert b

etw

een

ordi

nary

and

st

anda

rd in

dex

form

re

pres

enta

tions

, usin

g sig

nific

ant f

igur

es a

s ap

prop

riate

; jus

tify

the

repr

esen

tatio

n us

edan

d ch

oice

of a

ccur

acy

in re

latio

n to

the

prob

lem

and

aud

ienc

e fo

r the

solu

tion

enga

ge in

mat

hem

atic

al ta

sks

whe

re u

sing

num

bers

in

stan

dard

form

is es

sent

ial t

o th

e ca

lcul

atio

ns in

volv

ed;

criti

cally

exa

min

e th

e ef

fect

of n

umer

ical

re

pres

enta

tions

on

the

accu

racy

of

the

solu

tion,

e.g

.un

ders

tand

how

err

ors

can

be c

ompo

unde

d

in c

alcu

latio

ns

com

mun

icat

e th

e so

lutio

n to

a p

robl

em,

expl

aini

ng th

e lim

itatio

ns o

f acc

urac

y, us

ing

uppe

r and

lo

wer

bou

nds

10

Inte

gers

, pow

ers a

nd ro

ots

2.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1

unde

rsta

nd n

egat

ive

num

bers

as p

ositi

ons

on a

num

ber l

ine;

or

der,

add

and

subt

ract

inte

gers

in c

onte

xt

reco

gnise

and

use

mul

tiple

s, fa

ctor

s,pr

imes

(les

s tha

n 10

0),

com

mon

fact

ors,

high

est c

omm

onfa

ctor

s and

low

est

com

mon

mul

tiple

s in

simpl

e ca

ses;

use

simpl

ete

sts o

f div

isibi

lity

reco

gnise

the

first

few

tr

iang

ular

num

bers

;re

cogn

ise th

e sq

uare

s of

num

bers

to a

tle

ast 1

2!"!

12 a

nd th

e co

rres

pond

ing

root

s

add,

subt

ract

, mul

tiply

an

d di

vide

inte

gers

use

mul

tiple

s, fa

ctor

s,co

mm

on fa

ctor

s, hi

ghes

t com

mon

fa

ctor

s, lo

wes

t co

mm

on m

ultip

les a

nd

prim

es; f

ind

the

prim

e fa

ctor

dec

ompo

sitio

n of

a n

umbe

r, e.

g.

8000!=!2

6 !"!5

3

use

squa

res,

posit

ive

and

nega

tive

squa

re

root

s, cu

bes a

nd

cube

root

s, an

d in

dex

nota

tion

for s

mal

l po

sitiv

e in

tege

r pow

ers

use

the

prim

e fa

ctor

de

com

posit

ion

of

a nu

mbe

r

use

ICT

to e

stim

ate

squa

re ro

ots a

nd

cube

root

s

use

inde

x no

tatio

n fo

r in

tege

r pow

ers;

know

an

d us

e th

e in

dex

law

s fo

r mul

tiplic

atio

n an

d di

visio

n of

pos

itive

in

tege

r pow

ers

exam

ine

and

exte

nd

the

inde

x la

ws t

o es

tabl

ish th

e m

eani

ng

of n

egat

ive,

frac

tiona

l an

d ze

ro p

ower

s, in

clud

ing

use

of

surd

not

atio

n

exam

ine

and

exte

nd

the

inde

x la

ws t

o es

tabl

ish th

e m

eani

ng

of in

vers

e op

erat

ions

in

rela

tion

to in

dice

s, i.e

. th

e in

vers

e op

erat

ion

of ra

ising

a p

ositi

ve

num

ber t

o po

wer

n is

raisi

ng th

e re

sult

of th

is op

erat

ion

to p

ower

1 n

Exte

nsio

n

solv

e a

prob

lem

us

ing

ratio

nal a

nd

irrat

iona

l num

bers

, in

clud

ing

surd

s



11

Fra

ctio

ns, d

ecim

als,

per

cent

ages

, rat

io a

nd p

ropo

rtio

n 2.

3 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n



expr

ess a

smal

ler w

hole

nu

mbe

r as a

frac

tion

of

a la

rger

one

; sim

plify

fra

ctio

ns b

y ca

ncel

ling

all c

omm

on fa

ctor

s an

d id

entif

y eq

uiva

lent

fra

ctio

ns; c

onve

rt

term

inat

ing

deci

mal

s to

frac

tions

, e.g

. 23

=0.

2310

0 ; u

se

diag

ram

s to

com

pare

tw

o or

mor

e sim

ple

fract

ions

add

and

subt

ract

sim

ple

fract

ions

and

th

ose

with

com

mon

deno

min

ator

s;ca

lcul

ate

simpl

e fra

ctio

ns o

f qua

ntiti

es

and

mea

sure

men

ts(w

hole

-num

ber

answ

ers);

mul

tiply

a

fract

ion

by a

n in

tege

r

reco

gnise

that

a

recu

rrin

g de

cim

al is

a

fract

ion;

use

div

ision

to

conv

ert a

frac

tion

to a

de

cim

al; o

rder

frac

tions

by

writ

ing

them

with

a

com

mon

den

omin

ator

or

by

conv

ertin

g th

em

to d

ecim

als

unde

rsta

nd th

e eq

uiva

lenc

e of

sim

ple

alge

brai

c fra

ctio

ns;

know

that

a re

curr

ing

deci

mal

is a

n ex

act

fract

ion

expl

ain

the

patt

erns

fo

und

in re

curr

ing

deci

mal

s; ju

stify

w

hy d

ecim

als r

ecur

or

term

inat

e by

co

nsid

erin

g fa

ctor

s of

the

deno

min

ator

expl

ore

the

hist

oric

al

and

cultu

ral r

oots

of

the

num

ber s

yste

m

and

use

alge

bra

toju

stify

and

pro

ve so

me

of it

s fea

ture

s, e.

g. th

at

all r

ecur

ring

deci

mal

s ca

n be

exp

ress

ed a

s a

fract

ion

show

insig

ht in

to th

e in

finite

den

sity

of th

e nu

mbe

r lin

e; m

ake

sens

e of

the

proo

f tha

t #2

is ir

ratio

nal

add

and

subt

ract

fra

ctio

ns b

y w

ritin

g th

em w

ith a

com

mon

de

nom

inat

or; c

alcu

late

fra

ctio

ns o

f qua

ntiti

es

(frac

tion

answ

ers);

m

ultip

ly a

nd d

ivid

e an

in

tege

r by

a fra

ctio

n

use

effic

ient

met

hods

toad

d, su

btra

ct, m

ultip

lyan

d di

vide

frac

tions

,in

terp

retin

g di

visio

n as

a m

ultip

licat

ive

inve

rse;

canc

el co

mm

on fa

ctor

sbe

fore

mul

tiply

ing

or d

ivid

ing

unde

rsta

nd a

nd a

pply

ef

ficie

nt m

etho

ds to

add,

subt

ract

, mul

tiply

an

d di

vide

frac

tions

, in

terp

retin

g re

cipr

ocal

s as

mul

tiplic

ativ

ein

vers

es

12

Fra

ctio

ns, d

ecim

als,

per

cent

ages

, rat

io a

nd p

ropo

rtio

n (c

ontin

ued)

2.

3 Year

7

Year

8

unde

rsta

nd p

erce

ntag

e as

the

‘num

ber o

f par

ts

per 1

00’; c

alcu

late

sim

ple

perc

enta

ges

and

use

perc

enta

ges

to c

ompa

re si

mpl

e pr

opor

tions

reco

gnise

the

equi

vale

nce

of

perc

enta

ges,

fract

ions

an

d de

cim

als

unde

rsta

nd th

e re

latio

nshi

p be

twee

n ra

tio a

nd p

ropo

rtio

n;

use

dire

ct p

ropo

rtio

n in

sim

ple

cont

exts

; use

ra

tio n

otat

ion,

sim

plify

ra

tios a

nd d

ivid

e a

quan

tity

into

two

part

s in

a gi

ven

ratio

; so

lve

simpl

e pr

oble

ms

invo

lvin

g ra

tio a

nd

prop

ortio

n us

ing

info

rmal

stra

tegi

es

inte

rpre

t per

cent

age

as th

e op

erat

or ‘s

o m

any

hund

redt

hsof

’ and

exp

ress

one

gi

ven

num

ber a

s a

perc

enta

ge o

f ano

ther

;ca

lcul

ate

perc

enta

ges

and

find

the

outc

ome

of a

giv

en p

erce

ntag

e in

crea

se o

r dec

reas

e

use

the

equi

vale

nce

of fr

actio

ns, d

ecim

als

and

perc

enta

ges t

oco

mpa

re p

ropo

rtio

ns

appl

y un

ders

tand

ing

of th

e re

latio

nshi

p be

twee

n ra

tio a

nd

prop

ortio

n; si

mpl

ifyra

tios,

incl

udin

g th

ose

expr

esse

d in

diff

eren

t un

its, r

ecog

nisin

g lin

ks

with

frac

tion

nota

tion;

di

vide

a q

uant

ity in

totw

o or

mor

e pa

rts i

n a

give

n ra

tio; u

se th

e un

itary

met

hod

to

solv

e sim

ple

prob

lem

sin

volv

ing

ratio

and

di

rect

pro

port

ion

Year

9

Year

10

Year

11

Exte

nsio

n

reco

gnise

whe

n fra

ctio

ns o

r pe

rcen

tage

s are

ne

eded

to c

ompa

repr

opor

tions

; sol

ve

prob

lem

s inv

olvi

ng

perc

enta

ge c

hang

es

use

prop

ortio

nal

reas

onin

g to

solv

e pr

oble

ms,

choo

sing

the

corr

ect n

umbe

rs

to ta

ke a

s 100

%, o

r as

a w

hole

; com

pare

two

ratio

s; in

terp

ret a

nd

use

ratio

in a

rang

e

of c

onte

xts

iden

tify

whe

n a

prob

lem

in n

umbe

r, al

gebr

a, g

eom

etry

or st

atist

ics i

nvol

ves

prop

ortio

nalit

y; u

se

mul

tiplic

ativ

e m

etho

ds

fluen

tly in

the

solu

tion,

in

clud

ing

inve

rse

calc

ulat

ions

, e.g

. w

ith p

erce

ntag

es

mod

el re

al c

onte

xts

whe

re q

uant

ities

var

yin

dire

ct p

ropo

rtio

n,

incl

udin

g re

peat

edpr

opor

tiona

l cha

nge,

e.

g. g

row

th/d

ecay

; use

al

gebr

aic

met

hods

w

here

app

ropr

iate

and

co

nsid

er li

mita

tions

of

the

mod

el

(as i

n 2.

4)

unde

rsta

nd a

nd u

se

dire

ct a

nd in

vers

e pr

opor

tion;

solv

e pr

oble

ms i

nvol

ving

in

vers

e pr

opor

tion

(incl

udin

g y ?

1/x

2 )

(as i

n 2.

4)



13

Num

ber o

pera

tions

2.

4 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n



unde

rsta

nd a

nd u

se th

e un

ders

tand

and

use

the

unde

rsta

nd th

e ef

fect

s re

cogn

ise a

nd u

se

mod

el re

al c

onte

xts

unde

rsta

nd a

nd u

se

rule

s of a

rithm

etic

and

ru

les o

f arit

hmet

ic a

nd

of m

ultip

lyin

g an

d re

cipr

ocal

s as a

w

here

qua

ntiti

es v

ary

dire

ct a

nd in

vers

e in

vers

e op

erat

ions

in

inve

rse

oper

atio

ns in

di

vidi

ng b

y nu

mbe

rs

mul

tiplic

ativ

e in

vers

e in

dire

ct p

ropo

rtio

n,

prop

ortio

n; so

lve

the

cont

ext o

f pos

itive

the

cont

ext o

f int

eger

s be

twee

n 0

and

1;

in c

onte

xts s

uch

asin

clud

ing

repe

ated

prob

lem

s inv

olvi

ng

inte

gers

and

dec

imal

s an

d fra

ctio

ns

cons

olid

ate

use

of th

e en

larg

emen

t; ex

plor

e pr

opor

tiona

l cha

nge,

in

vers

e pr

opor

tion

rule

s of a

rithm

etic

and

th

e be

havi

our o

f the

g

row

th/d

ecay

; use

e.

g.(in

clud

ing

y ?!1

/x 2 )

inve

rse

oper

atio

ns

reci

proc

al fu

nctio

n

alge

brai

c m

etho

ds

(as i

n 2.

3)

(y =!"

/x) f

or la

rge

and

whe

re a

ppro

pria

te a

nd

smal

l val

ues o

f x

cons

ider

lim

itatio

ns o

f th

e m

odel

(as i

n 2.

3)

use

the

orde

r of

use

the

orde

r of

unde

rsta

nd th

e or

der

oper

atio

ns, i

nclu

ding

op

erat

ions

, inc

ludi

ng

of p

rece

denc

e of

br

acke

ts

brac

kets

, with

mor

eop

erat

ions

, inc

ludi

ng

com

plex

cal

cula

tions

po

wer

s

14

Men

tal c

alcu

latio

n m

etho

ds

2.5 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion



reca

ll nu

mbe

rfa

cts,

incl

udin

g po

sitiv

e in

tege

r co

mpl

emen

ts to

100

an

d m

ultip

licat

ion

fact

s to

10 "

10, a

nd q

uick

ly

deriv

e as

soci

ated

divi

sion

fact

s

stre

ngth

en a

nd e

xten

d m

enta

l met

hods

of

calc

ulat

ion

to in

clud

e de

cim

als,

fract

ions

an

d pe

rcen

tage

s, ac

com

pani

ed w

here

ap

prop

riate

by

suita

ble

jott

ings

; sol

ve si

mpl

e pr

oble

ms m

enta

lly

mak

e an

d ju

stify

es

timat

es a

nd

appr

oxim

atio

ns o

f ca

lcul

atio

ns

reca

ll eq

uiva

lent

fra

ctio

ns, d

ecim

als

and

perc

enta

ges;

use

know

n fa

cts t

o de

rive

unkn

own

fact

s, in

clud

ing

prod

ucts

invo

lvin

g nu

mbe

rs

such

as 0

.7 a

nd 6

, and

an

d 8

0.03

stre

ngth

en a

nd e

xten

d m

enta

l met

hods

of

calc

ulat

ion,

wor

king

w

ith d

ecim

als,

fract

ions

, per

cent

ages

, sq

uare

s and

squa

re

root

s, cu

bes a

nd c

ube

root

s; so

lve

prob

lem

sm

enta

lly

use

know

n fa

cts t

o de

rive

unkn

own

fact

s; ex

tend

men

tal

met

hods

of c

alcu

latio

n,

wor

king

with

dec

imal

s, fra

ctio

ns, p

erce

ntag

es,

fact

ors,

pow

ers a

nd

root

s; so

lve

prob

lem

sm

enta

lly

sele

ct m

enta

l or

writ

ten

stra

tegi

es

and

calc

ulat

ing

devi

ces a

ppro

pria

te

to th

e st

age

of th

e pr

oble

m; c

alcu

late

accu

rate

ly w

ithre

cipr

ocal

s, po

wer

s, tr

igon

omet

rical

func

tions

and

num

bers

in

stan

dard

form

(as i

n 2.

6 an

d 2.

7)

sele

ct a

nd ju

stify

an

app

ropr

iate

and

ef

ficie

nt c

ombi

natio

n of

met

hods

of

calc

ulat

ion,

i.e.

men

tal,

writ

ten,

ICT

or c

alcu

lato

r to

solv

e pr

oble

ms

(as i

n 2.

6)

appr

ecia

te w

hen

resu

lts

of c

alcu

latio

ns c

an b

e m

ore

eleg

antly

and

ex

actly

com

mun

icat

edus

ing

surd

s and

!,

ratio

nalis

ing

a de

nom

inat

or w

here

ap

prop

riate

, e.g

. a

trig

onom

etric

also

lutio

n

(as i

n 2.

6)

mak

e an

d ju

stify

m

ake

and

just

ify

exam

ine

and

refin

ees

timat

es a

nd

estim

ates

and

es

timat

es a

nd

appr

oxim

atio

ns o

f ap

prox

imat

ions

of

appr

oxim

atio

ns o

f ca

lcul

atio

ns

calc

ulat

ions

ca

lcul

atio

ns in

volv

ing

roun

ding

15

Writ

ten

calc

ulat

ion

met

hods

2.

6 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

use

effic

ient

writ

ten

met

hods

to a

dd

and

subt

ract

who

le

num

bers

and

dec

imal

sw

ith u

p to

two

plac

es

mul

tiply

and

div

ide

thre

e-di

git b

y tw

o-di

git w

hole

num

bers

; ex

tend

to m

ultip

lyin

g an

d di

vidi

ng d

ecim

als

with

one

or t

wo

plac

es

by si

ngle

-dig

it w

hole

nu

mbe

rs

use

effic

ient

writ

ten

met

hods

to a

dd a

nd

subt

ract

inte

gers

and

de

cim

als o

f any

size

, in

clud

ing

num

bers

w

ith d

iffer

ing

num

bers

of

dec

imal

pla

ces



use

effic

ient

writ

ten

met

hods

for

mul

tiplic

atio

n an

d di

visio

n of

inte

gers

and

de

cim

als,

incl

udin

g by

de

cim

als s

uch

as 0

.6 o

r0.

06; u

nder

stan

d w

here

to

pos

ition

the

deci

mal

po

int b

y co

nsid

erin

g eq

uiva

lent

cal

cula

tions

use

effic

ient

writ

ten

met

hods

to a

dd a

nd

subt

ract

inte

gers

and

de

cim

als o

f any

size

; m

ultip

ly b

y de

cim

als;

divi

de b

y de

cim

als

by tr

ansf

orm

ing

to

divi

sion

by a

n in

tege

r

sele

ct m

enta

l or

writ

ten

stra

tegi

es

and

calc

ulat

ing

devi

ces a

ppro

pria

te

to th

e st

age

of th

e pr

oble

m; c

alcu

late

accu

rate

ly w

ithre

cipr

ocal

s, po

wer

s, tr

igon

omet

rical

func

tions

and

num

bers

in

stan

dard

form

(as i

n 2.

5 an

d 2.

7)

sele

ct a

nd ju

stify

an

app

ropr

iate

and

ef

ficie

nt c

ombi

natio

n of

met

hods

of

calc

ulat

ion,

i.e.

men

tal,

writ

ten,

ICT

or c

alcu

lato

r to

solv

e pr

oble

ms

(as i

n 2.

5)

appr

ecia

te w

hen

resu

lts

of c

alcu

latio

ns c

an b

e m

ore

eleg

antly

and

ex

actly

com

mun

icat

edus

ing

surd

s and

!,

ratio

nalis

ing

a de

nom

inat

or w

here

ap

prop

riate

, e.g

. a

trig

onom

etric

also

lutio

n

(as i

n 2.

5)

16

Cal

cula

tor m

etho

ds

2.7 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

carr

y ou

t cal

cula

tions

w

ith m

ore

than

one

step

usin

g br

acke

ts a

nd

the

mem

ory;

use

the

squa

re ro

ot a

nd si

gn

chan

ge k

eys

ente

r num

bers

and

in

terp

ret t

he d

ispla

y in

diff

eren

t con

text

s(d

ecim

als,

perc

enta

ges,

mon

ey, m

etric

m

easu

res)

carr

y ou

t mor

e di

ffic

ult

calc

ulat

ions

effe

ctiv

ely

and

effic

ient

ly u

sing

the

func

tion

keys

for

sign

chan

ge, p

ower

s, ro

ots a

nd fr

actio

ns;

use

brac

kets

and

th

e m

emor

y

use

a ca

lcul

ator

ef

ficie

ntly

and

ap

prop

riate

ly to

perf

orm

com

plex

ca

lcul

atio

ns w

ith

num

bers

of a

ny si

ze,

know

ing

not t

o ro

und

durin

g in

term

edia

te

step

s of a

cal

cula

tion;

us

e th

e co

nsta

nt, !

and

sign

chan

ge k

eys;

use

the

func

tion

keys

fo

r pow

ers,

root

s and

fra

ctio

ns; u

se b

rack

ets

and

the

mem

ory

sele

ct m

enta

l or

writ

ten

stra

tegi

es

and

calc

ulat

ing

devi

ces a

ppro

pria

te

to th

e st

age

of th

e pr

oble

m; c

alcu

late

accu

rate

ly w

ithre

cipr

ocal

s, po

wer

s, tr

igon

omet

rical

func

tions

and

num

bers

in

stan

dard

form

(as i

n 2.

5 an

d 2.

6)

criti

cally

exa

min

e al

tern

ativ

e m

etho

ds,

com

pare

stra

tegi

es fo

r:

calc

ulat

ing

(incl

udin

gca

lcul

atin

g de

vice

s)

chec

king

reco

gnise

the

limita

tions

of s

ome

appr

oach

es

(as i

n 2.

8)

refle

ct o

n a

solu

tion

to

a pr

oble

m c

omm

entin

g co

nstr

uctiv

ely

on th

e ch

oice

of c

alcu

latin

g st

rate

gies

(as i

n 2.

8)

ente

r num

bers

and

in

terp

ret t

he d

ispla

y in

diff

eren

t con

text

s(e

xten

d to

neg

ativ

enu

mbe

rs, f

ract

ions

, tim

e)



17

Che

ckin

g re

sults

2.

8 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

chec

k re

sults

by

cons

ider

ing

whe

ther

th

ey a

re o

f the

righ

t or

der o

f mag

nitu

dean

d by

wor

king

pr

oble

ms b

ackw

ards

sele

ct fr

om a

rang

e ch

eck

resu

lts u

sing

iden

tify

a ra

nge

of

of c

heck

ing

met

hods

, ap

prop

riate

met

hods

ch

ecki

ng st

rate

gies

and

in

clud

ing

estim

atin

g ap

prec

iate

that

mor

ein

con

text

and

usin

g th

an o

ne w

ay m

ay

inve

rse

oper

atio

ns

be n

eces

sary

in th

e co

ntex

t of t

he p

robl

em

criti

cally

exa

min

e al

tern

ativ

e m

etho

ds,

com

pare

stra

tegi

es fo

r:

calc

ulat

ing

(incl

udin

gca

lcul

atin

g de

vice

s)

chec

king

reco

gnise

the

limita

tions

of s

ome

appr

oach

es

(as i

n 2.

7)

refle

ct o

n a

solu

tion

to

a pr

oble

m c

omm

entin

g co

nstr

uctiv

ely

on th

e ch

oice

of c

heck

ing

stra

tegi

es

(as i

n 2.

7)



19

3 Al

gebr

a

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titie

s 3.

1



Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

use

lett

er sy

mbo

ls to

reco

gnise

that

lett

er

dist

ingu

ish th

e pr

esen

t con

vinc

ing

exam

ine

and

refin

eus

e sy

mbo

ls an

d re

pres

ent u

nkno

wn

sym

bols

play

diff

eren

t di

ffere

nt ro

les p

laye

d al

gebr

aic

argu

men

ts to

alge

brai

c ar

gum

ents

re

pres

enta

tions

nu

mbe

rs o

r var

iabl

es;

role

s in

equa

tions

,by

lett

er sy

mbo

ls in

ju

stify

gen

eral

isatio

ns

pres

ente

d to

exp

lain

cons

isten

tly to

pre

sent

know

the

mea

ning

s fo

rmul

ae a

nd fu

nctio

ns;

equa

tions

, ide

ntiti

es,

or so

lutio

ns; r

elat

e ge

omet

rical

and

a

form

al p

roof

, e.g

. of

the

wor

ds te

rm,

know

the

mea

ning

s fo

rmul

ae a

nd fu

nctio

ns

argu

men

ts to

the

num

eric

al p

rope

rtie

s; de

rivin

g th

e fo

rmul

a ex

pres

sion

and

equa

tion

of th

e w

ords

form

ula

stru

ctur

e of

the

cont

ext

choo

se a

nd c

ombi

ne

for s

olvi

ng q

uadr

atic

an

d fu

nctio

n or

pro

blem

; pro

duce

re

pres

enta

tions

toeq

uatio

ns

simpl

e pr

oofs

pr

esen

t a c

onvi

ncin

g pr

oof

unde

rsta

nd th

atun

ders

tand

that

use

inde

x no

tatio

n fo

r us

e al

gebr

aic

appr

ecia

te th

e al

gebr

aic

oper

atio

nsal

gebr

aic

oper

atio

ns,

inte

ger p

ower

s and

re

pres

enta

tion

toge

nera

lity

of th

e fo

rms

follo

w th

e ru

les o

f in

clud

ing

the

use

of

simpl

e in

stan

ces o

f sy

nthe

sise

know

n ru

les

a +

b =

c an

d ab

= c

, ar

ithm

etic

br

acke

ts, f

ollo

w th

e th

e in

dex

law

s of

arit

hmet

ic, i

nclu

ding

w

here

eac

h te

rm c

an

rule

s of a

rithm

etic

; use

th

e co

mm

utat

ive

and

itsel

f be

an e

xpre

ssio

n;

inde

x no

tatio

n fo

r sm

all

dist

ribut

ive

law

s; ju

stify

us

e th

is in

sight

into

posit

ive

inte

ger p

ower

s th

ese

gene

ralis

atio

ns,

stru

ctur

e to

dev

elop

us

ing

spat

ial

e.g.

flu

ency

in tr

ansf

orm

ing

repr

esen

tatio

ns; u

se

mor

e co

mpl

ex

alge

brai

c ar

gum

ent t

oeq

uatio

ns

gene

ralis

e th

e in

dex

law

s for

mul

tiplic

atio

n an

d di

visio

n to

incl

ude

zero

, neg

ativ

e an

d fra

ctio

nal p

ower

s

20

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titie

s (co

ntin

ued)

3.

1 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

simpl

ify li

near

alg

ebra

ic

expr

essio

ns b

y co

llect

ing

like

term

s;m

ultip

ly a

sing

le te

rm

over

a b

rack

et (i

nteg

er

coef

ficie

nts)

simpl

ify o

r tra

nsfo

rm

linea

r exp

ress

ions

by

colle

ctin

g lik

e te

rms;

mul

tiply

a si

ngle

term

ov

er a

bra

cket

simpl

ify o

r tra

nsfo

rm

alge

brai

c ex

pres

sions

by

taki

ng o

ut si

ngle

-te

rm c

omm

on fa

ctor

s;ad

d sim

ple

alge

brai

c fra

ctio

ns

deve

lop

fluen

cy in

tr

ansf

orm

ing

linea

r ex

pres

sions

; exp

and

the

prod

uct o

f tw

o lin

ear e

xpre

ssio

ns

of th

e fo

rm x

± n

and

fact

orise

sim

ple

quad

ratic

exp

ress

ions

;es

tabl

ish id

entit

ies s

uch

as th

e di

ffere

nce

of

two

squa

res;

com

pare

an

d ev

alua

te d

iffer

ent

repr

esen

tatio

ns o

f the

sa

me

cont

ext;

iden

tify

equi

vale

nt e

xpre

ssio

ns

and

conf

irm b

y tr

ansf

orm

atio

n

expa

nd a

nd fa

ctor

ise

quad

ratic

exp

ress

ions

;sim

plify

or t

rans

form

al

gebr

aic

fract

ions

, b

y fa

ctor

ising

and

e.

g.ca

ncel

ling

com

mon

fa

ctor

s; co

mpa

re a

nd

eval

uate

diff

eren

t re

pres

enta

tions

of t

he

sam

e co

ntex

t; id

entif

yeq

uiva

lent

exp

ress

ions

an

d co

nfirm

this

bytr

ansf

orm

atio

n



21

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titie

s (co

ntin

ued)

3.

1 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

cons

truc

t and

so

lve

simpl

e lin

ear

equa

tions

with

inte

ger

coef

ficie

nts (

unkn

own

on o

ne si

de o

nly)

usin

g an

app

ropr

iate

met

hod

inve

rse

oper

atio

ns)

(e.g

.

cons

truc

t and

solv

e lin

ear e

quat

ions

with

inte

ger c

oeff

icie

nts

(unk

now

n on

eith

er o

r bo

th si

des,

with

out a

nd

with

bra

cket

s) u

sing

appr

opria

te m

etho

ds

inve

rse

oper

atio

ns,

(e.g

.tr

ansf

orm

ing

both

sid

es in

sam

e w

ay)

use

grap

hs a

nd se

t up

equ

atio

ns to

solv

e sim

ple

prob

lem

s in

volv

ing

dire

ctpr

opor

tion

cons

truc

t and

solv

e lin

ear e

quat

ions

with

inte

ger c

oeff

icie

nts

(with

and

with

out

brac

kets

, neg

ativ

e sig

ns a

nyw

here

in th

e eq

uatio

n, p

ositi

ve o

r ne

gativ

e so

lutio

n)

use

alge

brai

c m

etho

ds

to so

lve

prob

lem

s in

volv

ing

dire

ctpr

opor

tion;

rela

teal

gebr

aic

solu

tions

to

gra

phs o

f the

eq

uatio

ns; u

se IC

T

as a

ppro

pria

te

cons

truc

t lin

ear

equa

tions

and

sim

ple

linea

r ine

qual

ities

(one

va

riabl

e) to

repr

esen

t re

al-li

fe si

tuat

ions

or

mat

hem

atic

alpr

oble

ms;

solv

e lin

ear e

quat

ions

an

d in

equa

litie

s, re

pres

entin

g th

e so

lutio

n in

the

cont

ext

of th

e pr

oble

m

cons

truc

t sim

ple

quad

ratic

equ

atio

ns

to re

pres

ent r

eal-

life

situa

tions

or

mat

hem

atic

al p

robl

ems

and

solv

e th

em

usin

g fa

ctor

isatio

n,

grap

hica

l or t

rial a

ndim

prov

emen

t met

hods

;ju

stify

the

num

ber

of so

lutio

ns u

sing

alge

brai

c or

gra

phic

al

argu

men

ts a

nd se

lect

appr

opria

te so

lutio

ns,

inte

rpre

ting

thei

r ac

cura

cy

repr

esen

t rea

l-lif

e sit

uatio

ns o

r m

athe

mat

ical

pro

blem

s in

volv

ing:

mor

e co

mpl

ex

quad

ratic

eq

uatio

ns,

choo

sing

an

appr

opria

te

met

hod

ofso

lutio

n in

clud

ing

com

plet

ing

the

squa

re a

nd u

se o

f th

e fo

rmul

a

dire

ct o

r inv

erse

pr

opor

tion,

in

clud

ing

y ?

x 2 , y

?

1/x 2

rela

te a

lgeb

raic

so

lutio

ns to

gra

phic

al

repr

esen

tatio

n of

the

func

tions



22

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titie

s (co

ntin

ued)

3.

1 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

use

syst

emat

ic tr

ial a

nd

impr

ovem

ent m

etho

ds

and

ICT

tool

s to

find

appr

oxim

ate

solu

tions

to

equ

atio

ns su

ch a

s x 2 +

x =

20

expl

ore

way

s of

cons

truct

ing

mod

els

of re

al-li

fe si

tuat

ions

by d

raw

ing

grap

hsan

d co

nstru

ctin

g al

gebr

aic e

quat

ions

and

in

equa

litie

s

(See

obj

ectiv

e ab

ove

(S

ee o

bjec

tive

abov

e

(See

obj

ectiv

e ab

ove

fo

r pro

gres

sion)

fo

r pro

gres

sion)

fo

r pro

gres

sion)

cons

truc

t a p

air o

f sim

ulta

neou

s lin

ear

equa

tions

to re

pres

ent

real

-life

situ

atio

ns

or m

athe

mat

ical

pr

oble

ms;

exam

ine

and

com

pare

al

gebr

aic

met

hods

of

solu

tion;

use

gra

phic

al

repr

esen

tatio

n to

expl

ain

why

the

inte

rsec

tion

of tw

o lin

es g

ives

the

com

mon

so

lutio

n an

d w

hy so

me

case

s hav

e no

com

mon

so

lutio

n an

d ot

hers

have

an

infin

ite n

umbe

r

sele

ct a

nd ju

stify

op

timum

met

hods

fo

r sol

ving

a p

air

of si

mul

tane

ous

linea

r equ

atio

ns in

a

varie

ty o

f con

text

s; co

nstr

uct s

ever

al

linea

r ine

qual

ities

in

one

and

two

varia

bles

to re

pres

ent

real

-life

situ

atio

ns

or m

athe

mat

ical

prob

lem

s; so

lve

the

ineq

ualit

ies

grap

hica

lly, i

dent

ifyin

g an

d in

terp

retin

g th

e so

lutio

n se

t in

the

cont

ext o

f the

pro

blem

solv

e m

ore

com

plex

pa

irs o

f sim

ulta

neou

s eq

uatio

ns g

ener

ated

fro

m re

al-li

fe c

onte

xts

or g

eom

etric

al

inve

stig

atio

ns,

incl

udin

g pa

irs w

here

on

e is

linea

r and

the

othe

r is q

uadr

atic

or o

f th

e fo

rm x

2 + y

2 = r2



23

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titie

s (co

ntin

ued)

3.

1 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

use

simpl

e fo

rmul

ae

from

mat

hem

atic

s an

d ot

her s

ubje

cts;

subs

titut

e po

sitiv

ein

tege

rs in

to li

near

ex

pres

sions

and

fo

rmul

ae a

nd, i

n sim

ple

case

s, de

rive

a fo

rmul

a

use

form

ulae

from

m

athe

mat

ics a

nd o

ther

su

bjec

ts; s

ubst

itute

inte

gers

into

sim

ple

form

ulae

, inc

ludi

ng

exam

ples

that

lead

to

an

equa

tion

toso

lve;

subs

titut

epo

sitiv

e in

tege

rs in

toex

pres

sions

invo

lvin

g sm

all p

ower

s, e.

g.

3x 2 +

4 o

r 2x3 ; d

eriv

e sim

ple

form

ulae

use

form

ulae

from

m

athe

mat

ics a

nd

othe

r sub

ject

s;su

bstit

ute

num

bers

in

to e

xpre

ssio

ns a

nd

form

ulae

; der

ive

a fo

rmul

a an

d, in

sim

ple

case

s, ch

ange

its

subj

ect

deriv

e fo

rmul

ae, e

.g.

in th

e co

ntex

t of

men

sura

tion;

inte

rpre

t a

rang

e of

form

ulae

draw

n fro

m re

al-li

fe

cont

exts

and

oth

er

subj

ects

, rel

atin

gth

e va

riabl

es to

the

cont

ext a

nd d

escr

ibin

g th

eir b

ehav

iour

; so

lve

prob

lem

s by

man

ipul

atin

g fo

rmul

ae

deriv

e an

d us

e fo

rmul

ae th

at in

volv

e m

ore

varia

bles

or

mor

e co

mpl

ex

alge

brai

c ex

pres

sions

;m

anip

ulat

e fo

rmul

ae

in o

rder

to re

ach

a so

lutio

n, sh

ow in

sight

in

to th

e m

athe

mat

ical

co

nnec

tions

, e.g

. usin

g th

e co

ntex

t and

the

form

ulae

to e

xpla

in th

e pr

opor

tiona

l effe

ct o

f va

ryin

g va

lues



24

Seq

uenc

es, f

unct

ions

and

gra

phs

3.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

desc

ribe

inte

ger

sequ

ence

s; ge

nera

tete

rms o

f a si

mpl

e se

quen

ce, g

iven

a

rule

(e.g

. fin

ding

a

term

from

the

prev

ious

te

rm, f

indi

ng a

term

gi

ven

its p

ositi

on in

th

e se

quen

ce)

gene

rate

term

s of

a lin

ear s

eque

nce

usin

g te

rm-to

-term

and

posit

ion-

to-te

rm

rule

s, on

pap

er a

nd

usin

g a

spre

adsh

eet o

r gr

aphi

cs c

alcu

lato

r

gene

rate

term

s of a

se

quen

ce u

sing

term

-to

-term

and

pos

ition

-to

-term

rule

s, on

pap

er

and

usin

g IC

T

gene

rate

sequ

ence

s fro

m p

atte

rns o

r pr

actic

al c

onte

xts a

nd

desc

ribe

the

gene

ral

term

in si

mpl

e ca

ses

deve

lop,

com

pare

and

ev

alua

te a

lgeb

raic

and

sp

atia

l rep

rese

ntat

ions

of

situ

atio

ns th

at

gene

rate

sequ

ence

s; in

terp

ret,

dedu

ce a

nd

just

ify g

ener

alisa

tions

fo

r the

nth

term

of

linea

r and

qua

drat

ic

sequ

ence

s, in

clud

ing

the

prop

ertie

s of

squa

re a

nd tr

iang

ular

num

bers



use

linea

r exp

ress

ions

to d

escr

ibe

the

nth

term

of a

sim

ple

arith

met

icse

quen

ce, ju

stify

ing

its fo

rm b

y re

ferr

ing

toth

e ac

tivity

or p

ract

ical

cont

ext f

rom

whi

ch it

was

gen

erat

ed

gene

rate

sequ

ence

s fro

m p

ract

ical

con

text

san

d w

rite

and

just

ify a

n ex

pres

sion

to d

escr

ibe

the

nth

term

of a

n ar

ithm

etic

sequ

ence

25

Seq

uenc

es, f

unct

ions

and

gra

phs (

cont

inue

d)

3.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

expr

ess s

impl

e ex

pres

s sim

ple

find

the

inve

rse

of a

co

mpa

re g

raph

ical

, fu

nctio

ns in

wor

ds,

func

tions

alg

ebra

ical

ly

linea

r fun

ctio

n al

gebr

aic

and

then

usin

g sy

mbo

ls;an

d re

pres

ent t

hem

ge

omet

rical

re

pres

ent t

hem

in

in m

appi

ngs o

r on

a re

pres

enta

tions

,m

appi

ngs

spre

adsh

eet

incl

udin

g m

appi

ng

diag

ram

s, to

exp

lain

th

e ef

fect

of:

rota

ting

the

line

y =

mx

+ c

thro

ugh

90° a

bout

any

po

int

refle

ctin

g th

e lin

ey

= m

x +

c in

the

line

y =

x

deriv

e pr

oper

ties o

f pe

rpen

dicu

lar l

ines

and

of

the

inve

rse

func

tion

gene

rate

coo

rdin

ate

gene

rate

poi

nts i

n al

l ge

nera

te p

oint

s and

expl

ore

grap

hs o

f ex

plor

e co

nnec

tions

ex

plor

e gr

aphs

of

pairs

that

satis

fy a

fo

ur q

uadr

ants

and

plo

t pl

ot g

raph

s of l

inea

rfu

nctio

ns o

f the

form

be

twee

n th

e fo

rm o

f ex

pone

ntia

l and

sim

ple

linea

r rul

e; p

lot

the

grap

hs o

f lin

ear

func

tions

, whe

re y

isy

= xn (n

an

inte

ger)

the

equa

tion

and

the

trig

onom

etric

alan

d re

cogn

ise th

eir

resu

lting

gra

phs o

f fu

nctio

ns a

nd re

cogn

iseth

e gr

aphs

of s

impl

e fu

nctio

ns, w

here

y is

give

n im

plic

itly

in te

rms

linea

r fun

ctio

ns, w

here

give

n ex

plic

itly

in te

rms

of x

(e.g

. ay

+ bx

= 0

, ch

arac

teris

tic sh

apes

; qu

adra

tic a

nd c

ubic

th

eir c

hara

cter

istic

y is

give

n ex

plic

itly

of x

, on

pape

r and

y

+ bx

+ c

= 0)

, on

pape

rva

ry th

e va

lues

of a

, b

func

tions

such

as:

shap

es; a

pply

to th

ein

term

s of x

, on

usin

g IC

T; re

cogn

ise

and

usin

g IC

T; fin

d th

ean

d c

in fu

nctio

ns su

ch

grap

h y =

f(x)

the

y =

(x +

2)(x

– 5

) pa

per a

nd u

sing

ICT;

that

equ

atio

ns o

f gr

adie

nt o

f lin

es g

iven

as y

= ax

2 + c,

tr

ansf

orm

atio

ns

y =

(x –

2)(x

2 + 7

x + 12

)re

cogn

ise st

raig

ht-li

neth

e fo

rm y

= m

x +

cby

equ

atio

ns o

f the

y =

ax 3 +

c,

y =

f(x) +

a, y

= a

f(x),

grap

hs p

aral

lel t

o th

e

corr

espo

nd to

stra

ight

-fo

rm y

= m

x +

c, gi

ven

y =

(x +

b)2 u

sing

a y

= f(x

+ a

), y

= f(a

x) fo

r

y =

x 2 – 2

x +

1 x-

axis

or y

-axi

s lin

e gr

aphs

va

lues

for m

and

c gr

aph

plot

ter t

o ex

plai

n lin

ear,

quad

ratic

, sin

e

y =

x 3 + 3

how

this

tran

sfor

ms

and

cosin

e fu

nctio

ns;

the

grap

h us

e a

grap

h pl

otte

r to

incl

ude

feat

ures

such

ex

plai

n th

e ef

fect

of

as ro

ots o

f the

equ

atio

n,

tran

sfor

mat

ions

on

the

inte

rcep

ts a

nd tu

rnin

g gr

aph

and

gene

ralis

epo

ints

to

oth

er fu

nctio

ns



26

Seq

uenc

es, f

unct

ions

and

gra

phs (

cont

inue

d)

3.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

plot

and

inte

rpre

t the

co

nstr

uct l

inea

r co

nstr

uct f

unct

ions

sk

etch

and

inte

rpre

t ap

ply

know

ledg

e of

se

t up

a m

athe

mat

ical

gr

aphs

of s

impl

e lin

ear

func

tions

aris

ing

from

ar

ising

from

real

-life

gr

aphs

that

mod

el re

al-

mat

hem

atic

al fu

nctio

ns

mod

el o

f a re

al-li

fe

func

tions

aris

ing

from

re

al-li

fe p

robl

ems

prob

lem

s and

plo

t the

ir lif

e sit

uatio

ns, i

nclu

ding

to

pro

blem

s inv

olvi

ng:

cont

ext o

r pro

blem

, re

al-li

fe si

tuat

ions

, e.g

. an

d pl

ot th

eir

corr

espo

ndin

g gr

aphs

; th

ose

gene

rate

d id

entif

ying

the

optim

isat

ion,

co

nver

sion

grap

hs

corr

espo

ndin

g gr

aphs

; in

terp

ret g

raph

s aris

ing

from

oth

er su

bjec

tsva

riabl

es a

nd th

eir

usin

g nu

mer

ical

, di

scus

s and

inte

rpre

t fro

m re

al si

tuat

ions

, e.g

. su

ch a

s sci

ence

;fu

nctio

nal r

elat

ions

hip;

al

gebr

aic

and

grap

hs a

risin

g fro

m

time

serie

s gra

phs

use

mat

hem

atic

alus

e gr

aphs

and

gr

aphi

cal,

real

situ

atio

ns, e

.g.

argu

men

t to

just

ify

sket

ches

to e

xpla

in

tech

niqu

es,

dist

ance

–tim

e gr

aphs

fe

atur

es o

f the

ir sh

apes

th

e be

havi

our o

f the

in

clud

ing

max

ima

varia

bles

and

to e

xpla

in

and

min

ima

or ju

stify

the

effe

ct o

f

usin

g IC

T to

fit

assu

mpt

ions

in

a cu

rve

to d

ata

the

mod

el

from

a re

al c

onte

xt

such

as a

scie

nce

expe

rimen

t re

peat

edpr

opor

tiona

l ch

ange

, e.g

. co

mpo

und

inte

rest

use

ICT

to e

xplo

re th

e gr

aphi

cal r

epre

sent

atio

n of

alg

ebra

ic e

quat

ions

an

d to

inte

rpre

t how

pr

oper

ties o

f the

gra

ph

are r

elat

ed to

feat

ures

of

the

equa

tion,

p

aral

lel a

nd

e.g.

perp

endi

cula

r lin

es



27

Seq

uenc

es, f

unct

ions

and

gra

phs (

cont

inue

d)

3.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion



inte

rpre

t the

mea

ning

of

vario

us p

oint

s and

se

ctio

ns o

f stra

ight

-lin

e gr

aphs

, incl

udin

g in

terc

epts

and

inte

rsec

tions

, e.g

. sol

ving

sim

ulta

neou

s lin

ear

equa

tions

29

4 G

eom

etry

and

mea

sure

s

Geo

met

rical

reas

onin

g4.

1 Year

7

use

corr

ectly

the

voca

bula

ry, n

otat

ion

and

labe

lling

co

nven

tions

for l

ines

, an

gles

and

shap

es

iden

tify

para

llel a

nd

perp

endi

cula

r lin

es;

know

the

sum

of a

ngle

s at

a p

oint

, on

a st

raig

ht

line

and

in a

tria

ngle

; re

cogn

ise v

ertic

ally

op

posit

e an

gles

iden

tify

alte

rnat

e an

gles

and

corr

espo

ndin

g an

gles

; un

ders

tand

a p

roof

th

at: th

e an

gle

sum

of a

tr

iang

le is

180

° and

of

a q

uadr

ilate

ral

is 36

0°

the

exte

rior a

ngle

of

a tr

iang

le is

eq

ual t

o th

e su

m

of th

e tw

o in

terio

r op

posit

e an

gles



Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

dist

ingu

ish b

etw

een

conv

entio

ns,

defin

ition

s and

der

ived

prop

ertie

s

expl

ain

how

to fi

nd,

calc

ulat

e an

d us

e:

the

sum

s of t

he

inte

rior a

nd

exte

rior a

ngle

s of

qua

drila

tera

ls,

pent

agon

s and

he

xago

ns

the

inte

rior a

nd

exte

rior a

ngle

s of

regu

lar p

olyg

ons

know

the

defin

ition

of

a ci

rcle

and

the

nam

es

of it

s par

ts; e

xpla

in

why

insc

ribed

regu

lar

poly

gons

can

be

cons

truc

ted

by e

qual

di

visio

ns o

f a c

ircle

exam

ine

and

exam

ine

and

crea

te

pres

ent r

igor

ous a

nd

refin

e ar

gum

ents

chai

ns o

f ded

uctiv

e su

stai

ned

argu

men

ts

in so

lutio

ns to

reas

onin

g in

solu

tions

in

the

solu

tion

of

geom

etric

al p

robl

ems,

to m

ore

com

plex

ge

omet

rical

pro

blem

s;di

stin

guish

ing

betw

een

geom

etric

al p

robl

ems

cons

truc

t for

mal

pr

actic

al d

emon

stra

tion

geom

etric

al p

roof

s an

d pr

oof;

prod

uce

simpl

e pr

oofs

use

dyna

mic

imag

es to

ex

amin

e th

e po

ints

dem

onst

rate

inva

riant

an

d lin

es u

sed

to c

reat

e re

latio

nshi

ps b

etw

een

exam

ine

and

crea

te

stan

dard

con

stru

ctio

ns

radi

i, ch

ords

and

pr

oofs

of t

he c

ircle

an

d us

e th

e co

nditi

ons

tang

ents

in c

ircle

s;th

eore

ms;

use

circ

le

of c

ongr

uenc

e to

deve

lop

argu

men

ts

theo

rem

s to

solv

e pr

esen

t a p

roof

that

the

to e

xpla

in a

nd ju

stify

pr

oble

ms

stan

dard

con

stru

ctio

ns

simpl

e ci

rcle

pro

pert

ies

are

exac

t an

d th

eore

ms

30

Geo

met

rical

reas

onin

g (c

ontin

ued)

4.

1

Year

8

Year

9

Year

10

Year

7

iden

tify

and

use

angl

e,

side

and

sym

met

ry

prop

ertie

s of t

riang

les

and

quad

rilat

eral

s; ex

plor

e ge

omet

rical

pr

oble

ms i

nvol

ving

th

ese

prop

ertie

s, ex

plai

ning

reas

onin

g or

ally

, usin

g st

ep-

by-s

tep

dedu

ctio

n su

ppor

ted

by d

iagr

ams

solv

e ge

omet

rical

pr

oble

ms u

sing

side

and

angl

e pr

oper

ties

of e

quila

tera

l, iso

scel

es a

nd ri

ght-

angl

ed tr

iang

les a

nd

spec

ial q

uadr

ilate

rals,

ex

plai

ning

reas

onin

g w

ith d

iagr

ams a

nd te

xt;

clas

sify

quad

rilat

eral

s by

thei

r geo

met

rical

pr

oper

ties

know

that

if tw

o 2-

D

shap

es a

re c

ongr

uent

, co

rres

pond

ing

sides

an

d an

gles

are

equ

al

solv

e pr

oble

ms

usin

g pr

oper

ties o

f an

gles

, of p

aral

lel a

nd

inte

rsec

ting

lines

and

of

tria

ngle

s and

oth

erpo

lygo

ns, j

ustif

ying

in

fere

nces

and

ex

plai

ning

reas

onin

g w

ith d

iagr

ams a

nd te

xt

unde

rsta

ndco

ngru

ence

and

ex

plor

e sim

ilarit

y

inve

stig

ate

Pyth

agor

as’

theo

rem

, usin

g a

varie

ty

of m

edia

, thr

ough

its

hist

oric

and

cultu

ral

root

s, in

clud

ing

‘pic

ture

’ pr

oofs

solv

e ge

omet

rical

pr

oble

ms u

sing

prop

ertie

s of l

ines

, an

gles

, pol

ygon

san

d ci

rcle

s; ju

stify

ar

gum

ents

and

so

lutio

ns u

sing

dedu

ctiv

e re

ason

ing

draw

infe

renc

es a

bout

pr

oper

ties o

f sim

ilar

2-D

shap

es a

nd u

se

prop

ortio

nal r

easo

ning

to

solv

e ge

omet

rical

an

d tr

igon

omet

rical

prob

lem

s

visu

alise

and

man

ipul

ate

dyna

mic

imag

es a

ndus

e sc

ale

draw

ing

toin

vest

igat

e ar

eas o

fsq

uare

s on

sides

of

right

-ang

led

and

non

right

-ang

led

tria

ngle

s,re

latin

g fin

ding

s to

Pyth

agor

as’ t

heor

em;

use

Pyth

agor

as’ t

heor

emto

solv

e pr

oble

ms i

n 2-

Dan

d sim

ple

3-D

cas

es

Year

11

form

alise

exi

stin

gkn

owle

dge

of li

nes,

angl

es a

nd p

olyg

ons b

y:

usin

g th

eco

ngru

ence

cond

ition

s (SS

S, S

AS,

RHS,

ASA

) to

dedu

cefa

mili

ar p

rope

rtie

sof

tria

ngle

s and

quad

rilat

eral

s, e.

g.an

isos

cele

s tria

ngle

has t

wo

equa

lan

gles

expl

aini

ngw

hy st

anda

rdco

nstr

uctio

ns w

ork,

e.g.

obs

ervi

ng th

atlin

es jo

inin

g po

ints

whe

re co

mpa

ss a

rcs

mee

t are

side

s of a

rhom

bus

Exte

nsio

n

(see

obj

ectiv

e ab

ove

for p

rogr

essio

n)

enga

ge w

ith a

nd

expl

ain

the

stag

es o

f a

varie

ty o

f pro

ofs o

f Py

thag

oras

’ the

orem

;us

e Py

thag

oras

’ th

eore

m to

solv

e m

ore

com

plex

3-D

pro

blem

s

pres

ent a

nd ju

stify

a

form

al p

roof

of

Pyth

agor

as’ t

heor

em



31

Geo

met

rical

reas

onin

g (c

ontin

ued)

4.

1 Year

7

Year

8

Year

9

Year

10

Year

11

use

2-D

repr

esen

tatio

ns

visu

alise

3-D

shap

es

to v

isual

ise 3

-D sh

apes

fro

m th

eir n

ets;

use

and

dedu

ce so

me

of

geom

etric

al p

rope

rtie

s th

eir p

rope

rtie

s of

cub

oids

and

shap

es

mad

e fro

m c

uboi

ds;

use

simpl

e pl

ans a

nd

elev

atio

ns

visu

alise

and

use

2-D

re

pres

enta

tions

of 3

-D

obje

cts;

anal

yse

3-D

sh

apes

thro

ugh

2-D

pr

ojec

tions

, inc

ludi

ng

plan

s and

ele

vatio

ns

visu

alise

and

des

crib

e pr

oper

ties o

f poi

nts,

lines

and

pla

nes i

n 3-

D

spac

e, in

clud

ing

cros

s se

ctio

ns c

reat

ed b

y sli

cing

a 3

-D sh

ape

visu

alise

and

m

anip

ulat

e im

ages

to

est

ablis

h tr

igon

omet

rical

rela

tions

hips

by:

gene

ratin

g tr

iang

les u

sing

a ro

tatin

g un

it ra

dius

(circ

le, c

entr

e th

eor

igin

)

iden

tifyi

ng th

e pr

oper

ties o

f sim

ilar t

riang

les

form

ed b

yen

larg

emen

ts o

f th

e ci

rcle

use

trig

onom

etric

al

rela

tions

hips

to so

lve

simpl

e pr

oble

ms i

n 2-

D,

incl

udin

g be

arin

gs

deriv

e th

e fo

rmul

a $

ab si

nC fo

r the

area

of a

tria

ngle

; us

e tr

igon

omet

rical

re

latio

nshi

ps to

solv

e m

ore

com

plex

2-

D p

robl

ems a

nd

prob

lem

s in

3-D

, suc

has

the

angl

e be

twee

n a

line

and

a pl

ane

Exte

nsio

n

draw

, ske

tch

and

com

pare

the

grap

hsof

trig

onom

etric

alfu

nctio

ns a

nd

tran

sfor

mat

ions

of

thes

e gr

aphs

; pro

ve th

e sin

e an

d co

sine

rule

s an

d us

e th

em to

solv

e 2-

D a

nd 3

-D p

robl

ems

in a

rang

e of

con

text

s



32

Tra

nsfo

rmat

ions

and

coor

dina

tes

4.2

Year

8

Year

9

Year

10

Year

11

Year

7

unde

rsta

nd a

nd u

se th

e la

ngua

ge a

nd n

otat

ion

asso

ciat

ed w

ithre

flect

ions

, tra

nsla

tions

and

rota

tions

reco

gnise

and

visu

alise

th

e sy

mm

etrie

s of a

2-D

sh

ape

tran

sfor

m 2

-D sh

apes

by

: refle

ctin

g in

giv

en

mirr

or li

nes

rota

ting

abou

t a

give

n po

int

tran

slatin

g

expl

ore

thes

e tr

ansf

orm

atio

ns a

nd

sym

met

ries u

sing

ICT

iden

tify

all t

he

sym

met

ries o

f 2-D

sh

apes

tran

sfor

m 2

-D sh

apes

by

rota

tion,

refle

ctio

nan

d tr

ansla

tion,

on

pape

r and

usin

g IC

T

try

out m

athe

mat

ical

re

pres

enta

tions

of

simpl

e co

mbi

natio

ns o

f th

ese

tran

sfor

mat

ions

iden

tify

refle

ctio

n sy

mm

etry

in 3

-D

shap

es

reco

gnise

that

tran

slatio

ns, r

otat

ions

an

d re

flect

ions

pr

eser

ve le

ngth

and

an

gle,

and

map

obj

ects

on

to c

ongr

uent

im

ages

expl

ore

and

com

pare

m

athe

mat

ical

repr

esen

tatio

ns o

f co

mbi

natio

ns o

f tr

ansla

tions

, rot

atio

ns

and

refle

ctio

ns o

f 2-D

sh

apes

, on

pape

r and

us

ing

ICT

devi

se in

stru

ctio

ns fo

r a

com

pute

r to

gene

rate

and

tran

sfor

m sh

apes

use

prec

ise la

ngua

ge

and

nota

tion

to d

escr

ibe

and

gene

ralis

e th

e re

sults

of c

ombi

ning

tr

ansf

orm

atio

ns o

f 2-D

sh

apes

on

pape

r and

us

ing

ICT,

incl

udin

g:

rota

tions

abo

ut

any

poin

t

refle

ctio

ns in

any

lin

e

tran

slatio

ns u

sing

vect

or n

otat

ion

a tr

ansf

orm

atio

n an

d its

inve

rse

gene

rate

and

ana

lyse

pa

tter

ns, e

.g. I

slam

ic

desig

ns

expl

ain

and

dem

onst

rate

grap

hica

lly th

e ef

fect

s of c

ombi

ning

tr

ansla

tions

, usin

g ve

ctor

not

atio

n,

incl

udin

g: th

e ru

le fo

rad

ditio

n of

vec

tors

scal

ar

mul

tiplic

atio

n of

a

vect

or (r

epea

ted

addi

tion)

Exte

nsio

n

expl

ain

and

dem

onst

rate

grap

hica

lly th

e ef

fect

s of c

ombi

ning

tr

ansla

tions

, usin

g ve

ctor

not

atio

n,

incl

udin

g:

the

diffe

renc

e of

tw

o ve

ctor

s

the

resu

ltant

of

two

vect

ors

the

com

mut

ativ

e an

d as

soci

ativ

epr

oper

ties o

f ve

ctor

add

ition

solv

e sim

ple

geom

etric

al p

robl

ems

in 2

-D u

sing

vect

ors



33

Tra

nsfo

rmat

ions

and

coor

dina

tes (

cont

inue

d)

4.2

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n



Year

7

use

conv

entio

ns

and

nota

tion

for

2-D

coo

rdin

ates

in

all f

our q

uadr

ants

;fin

d co

ordi

nate

s of

poin

ts d

eter

min

edby

geo

met

rical

in

form

atio

n

unde

rsta

nd a

nd u

se th

e la

ngua

ge a

nd n

otat

ion

asso

ciat

ed w

ithen

larg

emen

t; en

larg

e 2-

D sh

apes

, giv

en a

ce

ntre

of e

nlar

gem

ent

and

a po

sitiv

e in

tege

r sc

ale

fact

or; e

xplo

re

enla

rgem

ent u

sing

ICT

mak

e sc

ale

draw

ings

find

the

mid

poin

t of

the

line

segm

ent A

B,

give

n th

e co

ordi

nate

s of

poi

nts A

and

B

enla

rge

2-D

shap

es,

give

n a

cent

re o

f en

larg

emen

t and

a

posit

ive

inte

ger s

cale

fa

ctor

, on

pape

r and

us

ing

ICT;

iden

tify

the

scal

e fa

ctor

of

an e

nlar

gem

ent

as th

e ra

tio o

f the

le

ngth

s of a

ny tw

o co

rres

pond

ing

line

segm

ents

; rec

ogni

se

that

enl

arge

men

tspr

eser

ve a

ngle

but

not

le

ngth

, and

und

erst

and

the

impl

icat

ions

of

enla

rgem

ent f

or

perim

eter

enla

rge

2-D

shap

es

usin

g po

sitiv

e,

fract

iona

l and

neg

ativ

e sc

ale

fact

ors,

on

pape

r and

usin

g IC

T; u

se re

cipr

ocal

s as

a m

ultip

licat

ive

inve

rse

in th

e co

ntex

t of

enl

arge

men

t; re

cogn

ise th

e sim

ilarit

yof

resu

lting

shap

es

and

expl

ain

the

effe

ct

of e

nlar

gem

ent o

n pe

rimet

er

use

and

inte

rpre

t map

s an

d sc

ale

draw

ings

in th

e co

ntex

t of

mat

hem

atic

s and

oth

er

subj

ects

use

the

coor

dina

te

grid

to so

lve

prob

lem

s in

volv

ing

tran

slatio

ns,

rota

tions

, ref

lect

ions

an

d en

larg

emen

ts

appl

y th

e pr

oper

ties

of si

mila

r tria

ngle

s and

Pyth

agor

as’ t

heor

em

to so

lvin

g pr

oble

ms

pres

ente

d on

a 2

-D

coor

dina

te g

rid;

use

a 3-

D c

oord

inat

e gr

id to

repr

esen

t sim

ple

shap

es

enla

rge

3-D

shap

es;

iden

tify

and

expl

ain

the

effe

cts o

f enl

arge

men

t on

are

as a

nd v

olum

es

of si

mila

r sha

pes

and

solid

s; re

late

this

unde

rsta

ndin

g to

pr

actic

al c

onte

xts,

e.g.

in

bio

logy

34

Con

stru

ctio

n an

d lo

ci

4.3 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion



use

a ru

ler a

nd

prot

ract

or to

:

mea

sure

and

dra

w

lines

to th

e ne

ares

tm

illim

etre

and

angl

es, i

nclu

ding

re

flex

angl

es, t

oth

e ne

ares

t deg

ree

cons

truc

t a

tria

ngle

, giv

en

two

sides

and

the

incl

uded

ang

le

(SAS

) or t

wo

angl

es

and

the

incl

uded

sid

e (A

SA)

use

stra

ight

edg

e an

d co

mpa

sses

to co

nstr

uct:

the

mid

poin

t and

pe

rpen

dicu

lar

bise

ctor

of a

line

se

gmen

t

the

bise

ctor

of

an a

ngle

the

perp

endi

cula

r fro

m a

poi

nt to

a

line

the

perp

endi

cula

r fro

m a

poi

nt o

n

a lin

e

a tr

iang

le, g

iven

th

ree

sides

(SSS

)

use

stra

ight

edg

e an

d co

mpa

sses

to c

onst

ruct

tr

iang

les,

give

n rig

ht

angl

e, h

ypot

enus

e an

d sid

e (R

HS)

use

ICT

to e

xplo

re

cons

truc

tions

of

tria

ngle

s and

oth

er2-

D sh

apes

find

the

locu

s of a

poi

nt

that

mov

es a

ccor

ding

to

a si

mpl

e ru

le, b

oth

by re

ason

ing

and

by

usin

g IC

T

use

prop

ertie

s of

2-D

and

3-D

shap

es

to m

ake

accu

rate

co

nstr

uctio

ns o

n pa

per

and

usin

g IC

T; in

clud

ing

cons

truc

ting

tria

ngle

s fro

m c

ombi

natio

nsof

side

and

ang

lefa

cts,

revi

ewin

g an

d ge

nera

lisin

g fin

ding

sto

iden

tify

whi

ch o

f th

ese

cond

ition

s def

ine

uniq

ue c

onst

ruct

ions

use

ICT

to e

xplo

re

use

ICT

to e

xplo

re

cons

truc

tions

th

ese

cons

truc

tions

use

rule

r and

pro

trac

tor

to c

onst

ruct

sim

ple

nets

of 3

-D sh

apes

, c

uboi

d, re

gula

r e.

g.te

trah

edro

n, sq

uare

-ba

sed

pyra

mid

, tr

iang

ular

pris

m

find

simpl

e lo

ci, b

oth

by re

ason

ing

and

by

usin

g IC

T, to

pro

duce

sh

apes

and

pat

hs, e

.g.

an e

quila

tera

l tria

ngle

visu

alise

and

des

crib

e th

e lo

cus o

f a p

oint

th

at m

oves

acc

ordi

ng

to a

mor

e co

mpl

ex

rule

; exp

lain

the

path

usin

g ac

cura

te

geom

etric

al v

ocab

ular

yan

d no

tatio

n an

d us

e a

varie

ty o

f med

ia,

incl

udin

g dy

nam

ic

geom

etry

soft

war

e,

sket

ches

and

gra

phs

crea

te a

cha

in o

f re

ason

ing

to d

educ

eth

e eq

uatio

n of

a

circ

le b

y ap

plyi

ng

Pyth

agor

as’ t

heor

em to

th

e lo

cus o

f a p

oint

35

Mea

sure

s and

men

sura

tion

4.4 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

choo

se a

nd u

se u

nits

of

mea

sure

men

t to

mea

sure

, est

imat

e,

calc

ulat

e an

d so

lve

prob

lem

s in

ever

yday

cont

exts

; con

vert

one

m

etric

uni

t to

anot

her,

gra

ms t

o ki

logr

ams;

e.g.

read

and

inte

rpre

t sca

les

on a

rang

e of

mea

surin

g in

stru

men

ts

dist

ingu

ish b

etw

een

and

estim

ate

the

size

of a

cute

,ob

tuse

and

refle

x an

gles

choo

se a

nd u

se u

nits

of

mea

sure

men

t to

mea

sure

, est

imat

e,

calc

ulat

e an

d so

lve

prob

lem

s in

a ra

nge

of

cont

exts

; kno

w ro

ugh

met

ric e

quiv

alen

ts o

f im

peria

l mea

sure

s in

com

mon

use

, suc

h

as m

iles,

poun

ds (l

b)

and

pint

s

use

bear

ings

to sp

ecify

di

rect

ion

solv

e pr

oble

ms i

nvol

ving

m

easu

rem

ents

in a

va

riety

of c

onte

xts;

conv

ert b

etw

een

area

m

easu

res (

e.g.

mm

2 to

cm2 , c

m2 to

m2 , a

nd v

ice

vers

a) a

nd b

etw

een

volu

me

mea

sure

s (e.

g.

mm

3 to c

m3 , c

m3 to

m3 ,

and

vice

ver

sa)

Inte

rpre

t and

exp

lore

co

mbi

ning

mea

sure

s in

to ra

tes o

f cha

nge

in

ever

yday

cont

exts

(e.g

. km

pe

r hou

r, pe

nce

per m

etre

); us

e co

mpo

und

mea

sure

s to

com

pare

in re

al-li

fe

cont

exts

(e.g

. tra

vel g

raph

s an

d va

lue

for m

oney

), us

ing

ICT

as a

ppro

pria

te.

inte

rpre

t and

use

co

mpo

und

mea

sure

s, in

clud

ing

from

oth

er

subj

ects

and

real

life

; so

lve

prob

lem

s inv

olvi

ng

rate

s; co

nver

t bet

wee

n co

mpo

und

mea

sure

s, ch

oosin

g un

its m

ost

suite

d to

the

solu

tion

mak

e co

nnec

tions

be

twee

n th

e co

ntin

uity

of th

e nu

mbe

r lin

e an

d co

ntin

uous

mea

sure

s; cr

itica

lly e

xam

ine

the

mea

sure

men

ts u

sed

in a

pr

oble

m a

nd th

eir e

ffect

on th

e ac

cura

cy o

f the

so

lutio

n, e

.g. u

nder

stan

d ho

w e

rror

s can

be

com

poun

ded

com

mun

icat

e th

e so

lutio

nto

a p

robl

em

invo

lvin

g m

easu

rem

ent,

expl

aini

ng th

e lim

itatio

ns o

f ac

cura

cy u

sing

uppe

r and

lo

wer

bou

nds



36

Mea

sure

s and

men

sura

tion

(con

tinue

d)

4.4 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

know

and

use

the

deriv

e an

d us

e fo

rmul

ae

form

ula

for t

he a

rea

of

for t

he a

rea

of a

a re

ctan

gle;

cal

cula

tetr

iang

le, p

aral

lelo

gram

th

e pe

rimet

er a

nd a

rea

and

trap

eziu

m;

of sh

apes

mad

e fro

m

calc

ulat

e ar

eas o

fre

ctan

gles

co

mpo

und

shap

es

solv

e pr

oble

ms

invo

lvin

g m

ore

com

plex

shap

esan

d so

lids,

incl

udin

gse

gmen

ts o

f circ

les a

ndfru

stum

s of c

ones

calc

ulat

e th

e su

rfac

e ar

ea o

f cub

es a

nd

cubo

ids

know

and

use

the

form

ula

for t

he v

olum

e of

a c

uboi

d; c

alcu

late

volu

mes

and

surf

ace

area

s of c

uboi

ds a

nd

shap

es m

ade

from

cu

boid

s

know

and

use

the

form

ulae

for t

heci

rcum

fere

nce

and

area

of

a c

ircle

calc

ulat

e th

e su

rfac

e ar

ea a

nd v

olum

e of

rig

ht p

rism

s

pres

ent a

con

cise

re

ason

ed a

rgum

ent t

ode

rive

form

ulae

for:

leng

ths o

f circ

ular

ar

cs

area

s of s

ecto

rs o

f a

circ

le

surf

ace

area

of

a cy

linde

r

volu

me

of a

cy

linde

r

solv

e pr

oble

ms

invo

lvin

g th

e us

e of

th

ese

form

ulae

pres

ent a

con

cise

re

ason

ed a

rgum

ent

whe

n de

rivin

gfo

rmul

ae fo

r the

surf

ace

area

s of

pyra

mid

s and

con

es;

expl

ore

conn

ectio

ns

betw

een:

form

ulae

for

the

volu

me

of a

py

ram

id a

nd th

e re

late

d cu

boid

form

ulae

for

the

surf

ace

area

an

d vo

lum

e of

a

sphe

re a

nd th

eci

rcum

scrib

ed a

nd

insc

ribed

cub

es

solv

e pr

oble

ms

invo

lvin

g th

e us

e of

th

ese

form

ulae



37

5 St

atis

tics

5.1S

peci

fyin

g a

prob

lem

, pla

nnin

g an

d co

llect

ing



Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

sugg

est p

ossib

le

answ

ers,

give

n a

ques

tion

that

can

be

addr

esse

d by

stat

istic

al

met

hods

disc

uss a

pro

blem

that

ca

n be

add

ress

ed b

y st

atis

tical

met

hods

an

d id

entif

y re

late

d qu

estio

ns to

exp

lore

sugg

est a

pro

blem

toex

plor

e us

ing

stat

istic

al

met

hods

, fra

me

ques

tions

and

raise

co

njec

ture

s

inde

pend

ently

dev

ise a

su

itabl

e pl

an fo

r a m

ore

com

plex

stat

istic

al

proj

ect,

sele

ctin

g su

itabl

e hy

poth

eses

to

addr

ess t

he p

robl

em

eval

uate

pos

sible

di

ffic

ultie

s with

pl

anne

d ap

proa

ches

; ad

just

the

proj

ect

plan

acc

ordi

ngly

,in

clud

ing

reco

nsid

erin

g hy

poth

eses

deci

de w

hich

dat

a w

ould

be

rele

vant

to a

n en

quiry

and

pos

sible

so

urce

s

deci

de w

hich

dat

a to

col

lect

to a

nsw

er

a qu

estio

n an

d th

e de

gree

of a

ccur

acy

need

ed; i

dent

ifypo

ssib

le so

urce

s;co

nsid

er a

ppro

pria

te

sam

ple

size

disc

uss h

ow d

iffer

ent

sets

of d

ata

rela

te to

th

e pr

oble

m; i

dent

ifypo

ssib

le p

rimar

y or

se

cond

ary

sour

ces;

dete

rmin

e th

e sa

mpl

e siz

e an

d m

ost

appr

opria

te d

egre

e of

ac

cura

cy

just

ify th

e sa

mpl

ing

met

hod

sele

cted

, id

entif

y po

ssib

le

sour

ces o

f bia

s and

pl

an h

ow to

min

imise

it

iden

tify

prac

tical

prob

lem

s suc

h as

non-

resp

onse

or

miss

ing

data

and

refin

e ap

proa

ches

to m

inim

ise

thei

r im

pact

on

the

valid

ity o

f the

resu

lts

38

5.1S

peci

fyin

g a

prob

lem

, pla

nnin

g an

d co

llect

ing

(con

tinue

d)

Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

plan

how

to c

olle

ct a

nd

orga

nise

smal

l set

s of

data

from

surv

eys a

nd

expe

rimen

ts:

desig

n da

ta

colle

ctio

n sh

eets

or q

uest

ionn

aire

s to

use

in a

sim

ple

surv

ey

cons

truc

tfre

quen

cy ta

bles

fo

r gat

herin

g di

scre

te d

ata,

gr

oupe

d w

here

appr

opria

te

in e

qual

cla

ss

inte

rval

s

plan

how

to c

olle

ct

the

data

; con

stru

ct

frequ

ency

tabl

es w

itheq

ual c

lass

inte

rval

s for

ga

ther

ing

cont

inuo

us

data

and

two-

way

ta

bles

for r

ecor

ding

di

scre

te d

ata

desig

n a

surv

ey o

r ex

perim

ent t

o ca

ptur

e th

e ne

cess

ary

data

from

one

or m

ore

sour

ces;

desig

n, tr

ial

and

if ne

cess

ary

refin

eda

ta c

olle

ctio

n sh

eets

;co

nstr

uct t

able

s for

ga

ther

ing

larg

e di

scre

tean

d co

ntin

uous

sets

of

raw

dat

a, c

hoos

ing

suita

ble

clas

s int

erva

ls;de

sign

and

use

two-

way

tabl

es

gath

er d

ata

from

sp

ecifi

ed se

cond

ary

sour

ces,

incl

udin

g pr

inte

d ta

bles

and

list

s, an

d IC

T-ba

sed

sour

ces,

incl

udin

g th

e in

tern

et



deci

de o

n th

e be

st

met

hods

for t

estin

g th

e hy

poth

eses

; se

lect

, jus

tify

and

use

the

data

-gat

herin

g te

chni

que

mos

tap

prop

riate

to th

e co

ntex

t, de

cidi

ng

betw

een

a ra

nge

of so

urce

s: p

rimar

y (o

bser

vatio

n, co

ntro

lled

expe

rimen

t, da

talo

ggin

g) a

nd se

cond

ary

(spr

eads

heet

dat

a,

prin

ted

tabl

es, l

ists)

sele

ct, j

ustif

y an

d us

e th

e da

ta-g

athe

ring

tech

niqu

e ap

prop

riate

to

com

plex

and

un

fam

iliar

pro

blem

s, id

entif

ying

pot

entia

l ba

rrie

rs a

nd li

mita

tions

;id

entif

y w

hat e

xtra

in

form

atio

n m

ay b

e re

quire

d to

pur

sue

a fu

rthe

r lin

e of

enq

uiry

sele

ct a

nd c

ritic

ally

ev

alua

te a

sam

plin

g sc

hem

e an

d a

met

hod

to in

vest

igat

e a

popu

latio

n, in

clud

ing

rand

om a

nd st

ratif

ied

sam

plin

g; e

xpla

in th

e ef

fect

on

relia

bilit

y

and

valid

ity

colle

ct sm

all s

ets o

f co

llect

dat

a us

ing

a da

ta fr

om su

rvey

s su

itabl

e m

etho

d (e

.g.

and

expe

rimen

ts, a

sob

serv

atio

n, c

ontr

olle

d pl

anne

d ex

perim

ent,

data

logg

ing

usin

g IC

T)


Pro

cess

ing

and

repr

esen

ting

data

5.

2 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

calc

ulat

e st

atist

ics f

or

smal

l set

s of d

iscre

teda

ta: fin

d th

e m

ode,

m

edia

n an

d ra

nge,

an

d th

e m

odal

cl

ass f

or g

roup

ed

data

calc

ulat

e th

e m

ean,

incl

udin

g fro

m a

sim

ple

frequ

ency

tabl

e,us

ing

a ca

lcul

ator

fo

r a la

rger

num

ber

of it

ems

calc

ulat

e st

atist

ics

for s

ets o

f disc

rete

an

d co

ntin

uous

da

ta, i

nclu

ding

with

a

calc

ulat

or a

nd

spre

adsh

eet;

reco

gnise

w

hen

it is

appr

opria

te

to u

se th

e ra

nge,

mea

n,

med

ian

and

mod

e an

d,

for g

roup

ed d

ata,

the

mod

al c

lass

calc

ulat

e st

atist

ics

and

sele

ct th

ose

mos

tap

prop

riate

to th

e pr

oble

m o

r whi

ch

addr

ess t

he q

uest

ions

po

sed

use

an a

ppro

pria

te

rang

e of

stat

istic

al

met

hods

to e

xplo

re

and

sum

mar

ise la

rge

data

sets

, jus

tifyi

ngth

e ch

oice

s mad

e;

incl

ude

grou

ping

da

ta, e

stim

atin

g an

d fin

ding

the

mea

n,

med

ian,

qua

rtile

s and

in

terq

uart

ile ra

nge

proc

ess d

ata

draw

n fro

m p

robl

ems

invo

lvin

g se

ason

ality

an

d tr

ends

in a

tim

e se

ries;

choo

se a

nd

com

bine

stat

istic

al

met

hods

to a

naly

se

the

prob

lem

, inc

ludi

ngm

ovin

g av

erag

es


Pro

cess

ing

and

repr

esen

ting

data

(con

tinue

d)

5.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0

cons

truc

t, on

pap

er

and

usin

g IC

T, g

raph

san

d di

agra

ms t

ore

pres

ent d

ata,

in

clud

ing:

bar-l

ine

grap

hs

freq

uenc

y di

agra

ms f

or

grou

ped

disc

rete

data

sim

ple

pie

char

ts

cons

truc

t gra

phic

al

repr

esen

tatio

ns, o

n pa

per a

nd u

sing

ICT,

and

iden

tify

whi

ch

are

mos

t use

ful i

n th

e co

ntex

t of t

he p

robl

em,

incl

udin

g:

pie

char

ts fo

r ca

tego

rical

dat

a

bar c

hart

s and

fre

quen

cy

diag

ram

s for

di

scre

te a

nd

cont

inuo

us d

ata

sim

ple

line

grap

hs

for t

ime

serie

s

sim

ple

scat

ter

grap

hs

stem

-and

-leaf

di

agra

ms

sele

ct, c

onst

ruct

an

d m

odify

, on

pape

r and

usin

g IC

T,su

itabl

e gr

aphi

cal

repr

esen

tatio

ns to

pr

ogre

ss a

n en

quiry

an

d id

entif

y ke

y fe

atur

es p

rese

nt in

the

data

. Inc

lude

:

line

grap

hs fo

r tim

e se

ries

scat

ter g

raph

s to

deve

lop

furt

her

unde

rsta

ndin

g of

co

rrel

atio

n

cons

truc

t on

pape

r an

d us

ing

ICT

suita

ble

grap

hica

l re

pres

enta

tions

,in

clud

ing:

hist

ogra

ms

for g

roup

ed

cont

inuo

us d

ata

with

equ

al c

lass

in

terv

als

cum

ulat

ive

frequ

ency

tabl

es

and

diag

ram

s

box

plot

s

scat

ter g

raph

s and

lin

es o

f bes

t fit

(b

y ey

e)

just

ify th

eir s

uita

bilit

y w

ith re

fere

nce

to th

e co

ntex

t of t

he p

robl

em

and

the

audi

ence

wor

k th

roug

h th

e en

tire

hand

ling

data

cycl

e to

expl

ore

rela

tions

hips

w

ithin

bi-v

aria

te d

ata,

in

clud

ing

appl

icat

ions

to g

loba

l citi

zens

hip,

e.g

. ho

w fa

ir is

our s

ocie

ty?

Year

11

Exte

nsio

n

choo

se a

nd c

ombi

ne

suita

ble

grap

hica

l re

pres

enta

tions

topr

ogre

ss a

n un

fam

iliar

or

non

-rou

tine

enqu

iry,

incl

udin

g hi

stog

ram

s w

ith e

qual

or u

nequ

al

clas

s int

erva

ls

use

prec

ise a

nd

cons

isten

t gra

phic

alre

pres

enta

tion

topr

ogre

ss a

n un

fam

iliar

an

d no

n-ro

utin

e en

quiry



41

Inte

rpre

ting

and

disc

ussi

ng re

sults

5.

3

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n



Year

7

inte

rpre

t dia

gram

s and

gr

aphs

(inc

ludi

ng p

ie

char

ts) a

nd d

raw

sim

ple

conc

lusio

ns b

ased

on

the

shap

e of

gra

phs

and

simpl

e st

atist

ics f

or

a sin

gle

dist

ribut

ion

com

pare

two

simpl

e di

strib

utio

ns u

sing

the

rang

e an

d on

e of

the

mod

e, m

edia

n or

mea

n

writ

e a

shor

t rep

ort o

f a

stat

istic

al e

nqui

ry,

incl

udin

g ap

prop

riate

di

agra

ms,

grap

hs a

nd

char

ts, u

sing

ICT

as

appr

opria

te; j

ustif

y th

e ch

oice

of p

rese

ntat

ion

inte

rpre

t tab

les,

grap

hs a

nd d

iagr

ams

for d

iscre

te a

nd

cont

inuo

us d

ata,

re

latin

g su

mm

ary

stat

istic

s and

find

ings

to

the

ques

tions

bei

ng

expl

ored

com

pare

two

dist

ribut

ions

usin

g th

e ra

nge

and

one

or m

ore

of th

e m

ode,

med

ian

and

mea

n

writ

e ab

out a

nd

disc

uss t

he re

sults

of a

st

atis

tical

enq

uiry

usin

g IC

T as

app

ropr

iate

;ju

stify

the

met

hods

us

ed

inte

rpre

t gra

phs a

nd

diag

ram

s and

mak

ein

fere

nces

to su

ppor

t or

cas

t dou

bt o

n in

itial

co

njec

ture

s; ha

ve a

ba

sic u

nder

stan

ding

of

corr

elat

ion

com

pare

two

or

mor

e di

strib

utio

ns

and

mak

e in

fere

nces

, us

ing

the

shap

e of

th

e di

strib

utio

ns a

nd

appr

opria

te st

atist

ics

revi

ew in

terp

reta

tions

and

resu

lts o

f a

stat

istic

al e

nqui

ry o

n th

e ba

sis o

f disc

ussio

ns;

com

mun

icat

e th

ese

inte

rpre

tatio

ns a

nd

resu

lts u

sing

sele

cted

ta

bles

, gra

phs a

nd

diag

ram

s

find

patt

erns

and

in

terp

ret a

nd c

ompa

re

expl

ain

and

just

ify

exce

ptio

ns a

nd e

xpla

in

dist

ribut

ions

, inc

ludi

ng

assu

mpt

ions

and

an

omal

ies;

incl

udin

g cu

mul

ativ

e fre

quen

cy

cons

trai

nts;

incl

ude

inte

rpre

tatio

n of

di

agra

ms;

mak

e an

d in

terp

reta

tion

and

soci

al st

atist

ics a

nd

disc

uss i

nfer

ence

s, co

mpa

rison

of

eval

uatio

n of

the

usin

g th

e sh

ape

of

hist

ogra

ms w

ith

stre

ngth

of a

ssoc

iatio

nth

e di

strib

utio

ns a

nd

uneq

ual c

lass

inte

rval

sw

ithin

bi-v

aria

te d

ata

mea

sure

s of a

vera

ge

(cor

rela

tion,

line

s of

and

spre

ad, i

nclu

ding

be

st fi

t) m

edia

n an

d qu

artil

es

eval

uate

the

resu

lts

criti

cally

exa

min

e us

e st

atist

ical

of a

stat

istic

al

stra

tegi

es a

dopt

ed

anal

ysis

effe

ctiv

ely

in

enqu

iry; r

evie

w a

nd

and

argu

men

ts

pres

entin

g co

nvin

cing

ju

stify

or r

efin

e th

e pr

esen

ted,

rela

ting

conc

lusio

ns; c

ritic

ally

ch

oice

of s

tatis

tical

th

em to

the

orig

inal

re

flect

on

own

lines

re

pres

enta

tions

and

hy

poth

eses

; rec

ogni

se

of e

nqui

ry; s

earc

h re

late

sum

mar

ised

data

the

limita

tions

of a

ny

for a

nd a

ppre

ciat

e to

the

ques

tions

bei

ng

assu

mpt

ions

and

the

mor

e el

egan

t for

ms

expl

ored

ef

fect

s tha

t var

ying

of

com

mun

icat

ing

assu

mpt

ions

cou

ld

conc

lusio

ns

have

on

conc

lusio

ns

draw

n fro

m d

ata

anal

ysis


Pro

babi

lity

5.4 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

use

voca

bula

ry a

nd

idea

s of p

roba

bilit

y, dr

awin

g on

exp

erie

nce

inte

rpre

t the

resu

lts

of a

n ex

perim

ent

usin

g th

e la

ngua

ge o

f pr

obab

ility

; app

reci

ate

that

rand

om p

roce

sses

ar

e un

pred

icta

ble

inte

rpre

t res

ults

in

volv

ing

unce

rtai

nty

and

pred

ictio

n id

entif

y w

hen

the

even

ts in

a p

robl

em

are

mut

ually

exc

lusiv

e or

inde

pend

ent;

use

and

inte

rpre

t tre

edi

agra

ms t

o re

pres

ent

outc

omes

of c

ombi

ned

even

ts a

nd to

info

rm

the

calc

ulat

ion

of

thei

r pro

babi

litie

s;de

cide

whe

n to

add

an

d w

hen

to m

ultip

lypr

obab

ilitie

s

inte

rpre

t the

effe

cton

pro

babi

lity

of

cont

exts

invo

lvin

g se

lect

ion

with

and

w

ithou

t rep

lace

men

t; ch

oose

and

com

bine

re

pres

enta

tions

to

com

mun

icat

e pr

obab

ilitie

s as p

art o

f a

solu

tion

to a

pro

blem

reco

gnise

whe

n an

d ho

w to

wor

k w

ithpr

obab

ilitie

s ass

ocia

ted

with

inde

pend

ent

and

mut

ually

excl

usiv

e ev

ents

whe

n in

terp

retin

g da

ta

unde

rsta

nd a

nd u

se

the

prob

abili

ty sc

ale

from

0 to

1; f

ind

and

just

ify p

roba

bilit

ies

base

d on

equ

ally

like

ly

outc

omes

in si

mpl

e co

ntex

ts; i

dent

ify a

ll th

e po

ssib

le m

utua

lly

excl

usiv

e ou

tcom

es o

f a

singl

e ev

ent

know

that

if th

e pr

obab

ility

of a

n ev

ent

occu

rrin

g is

p th

en th

e pr

obab

ility

of i

t not

oc

curr

ing

is 1!%!

p; u

se

diag

ram

s and

tabl

es to

re

cord

in a

syst

emat

ic

way

all

poss

ible

m

utua

lly e

xclu

sive

outc

omes

for s

ingl

e ev

ents

and

for t

wo

succ

essiv

e ev

ents

iden

tify

all t

he m

utua

lly

excl

usiv

e ou

tcom

es

of a

n ex

perim

ent;

know

that

the

sum

of

pro

babi

litie

s of a

ll m

utua

lly e

xclu

sive

outc

omes

is 1

and

us

e th

is w

hen

solv

ing

prob

lem

s


43

Pro

babi

lity

(con

tinue

d)

5.4 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

estim

ate

prob

abili

ties

by c

olle

ctin

g da

ta fr

om

a sim

ple

expe

rimen

tan

d re

cord

ing

it in

a

frequ

ency

tabl

e;co

mpa

re e

xper

imen

tal

and

theo

retic

al

prob

abili

ties i

n sim

ple

cont

exts

com

pare

est

imat

ed

expe

rimen

tal

prob

abili

ties

with

theo

retic

al

prob

abili

ties,

reco

gnisi

ng th

at:

if an

exp

erim

ent

is re

peat

ed th

e ou

tcom

e m

ay, a

nd

usua

lly w

ill, b

e di

ffere

nt

incr

easin

g th

e nu

mbe

r of t

imes

an

exp

erim

ent i

sre

peat

ed g

ener

ally

le

ads t

o be

tter

es

timat

es o

f pr

obab

ility

com

pare

exp

erim

enta

l an

d th

eore

tical

pr

obab

ilitie

s in

a ra

nge

of c

onte

xts;

appr

ecia

teth

e di

ffere

nce

betw

een

mat

hem

atic

al

expl

anat

ion

and

expe

rimen

tal e

vide

nce

use

rela

tive

frequ

ency

as

an

estim

ate

of

prob

abili

ty, i

nclu

ding

sim

ulat

ion

usin

g IC

T to

gen

erat

e la

rger

sa

mpl

es; d

iscus

s its

relia

bilit

y ba

sed

on sa

mpl

e siz

e an

d us

e to

inte

rpre

t and

co

mpa

re o

utco

mes

of

expe

rimen

ts

expl

ore

a re

leva

nt a

nd

purp

osef

ul p

robl

em

invo

lvin

g un

cert

aint

y;

estim

ate

risk

bym

odel

ling

real

eve

nts

thro

ugh

simul

atio

n;

just

ify d

ecisi

ons b

ased

on

exp

erim

enta

lpr

obab

ility

and

co

mm

ent o

n th

e ef

fect

of a

ssum

ptio

ns

and

sam

ple

size

on th

e re

liabi

lity

of

conc

lusio

ns

