The mathematics overview and learning objectives...

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

tive

s

Ove

rvie

w o

f str

ands

Stra

nds

Sub

-str

ands

St

rand

s Su

b-s

tran

ds

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

Ana

lysi

ng –

use

mat

hem

atic

al re

ason

ing

3.2

Sequ

ence

s, fu

nctio

ns a

nd g

raph

s

1.3

Ana

lysi

ng –

use

app

ropr

iate

mat

hem

atic

al p

roce

dure

s 4

Geo

met

ry a

nd m

easu

res

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 an

d m

ensu

ratio

n

2.2

Inte

gers

, pow

ers

and

root

s 5

Stat

isti

cs

2.3

Frac

tions

, dec

imal

s, p

erce

ntag

es, r

atio

and

pro

port

ion

5.1

Spec

ifyin

g a

prob

lem

, pla

nnin

g an

d co

llect

ing

data

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

ussi

ng 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

5

Lear

ning

obj

ecti

ves

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 c

onte

xts

Incr

ease

the

chal

leng

e an

d bu

ild p

rogr

essi

on a

cros

s th

e ke

y st

age,

and

for g

roup

s of

pup

ils b

y:

t�

incr

easi

ng th

e co

mpl

exit

y of

the

appl

icat

ion,

e.g

. non

-rou

tine,

mul

ti-st

ep p

robl

ems,

exte

nded

enq

uirie

s

t�

redu

cing

the

fam

iliar

ity

of th

e co

ntex

t, e.

g. n

ew c

onte

xts

in m

athe

mat

ics,

con

text

s dr

awn

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

t�

incr

easi

ng th

e te

chni

cal d

eman

d of

the

mat

hem

atic

s re

quire

d, e

.g. m

ore

adva

nced

con

cept

s, m

ore

diff

icul

t pro

cedu

res

t�

incr

easi

ng 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

g1.

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,

m

akin

g co

rrec

t use

of

sym

bols

, wor

ds,

diag

ram

s, ta

bles

and

gr

aphs

; use

app

ropr

iate

pr

oced

ures

and

tool

s,

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

in

form

atio

n in

al

gebr

aic,

geo

met

rical

or

gra

phic

al fo

rm;

mov

e fr

om o

ne fo

rm

to a

noth

er to

gai

n a

diff

eren

t per

spec

tive

on th

e pr

oble

m

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

lysi

san

d m

ost e

ffec

tive

for

com

mun

icat

ion

syst

emat

ical

ly m

odel

co

ntex

ts o

r pro

blem

s th

roug

h pr

ecis

e an

d co

nsis

tent

use

of

sym

bols

and

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

clas

sify

and

vis

ualis

e pr

oper

ties

and

patt

erns

; gen

eral

ise

in s

impl

e ca

ses

by

wor

king

logi

cally

; dra

wsi

mpl

e co

nclu

sion

s an

d ex

plai

n re

ason

ing;

unde

rsta

nd th

e si

gnifi

canc

e of

a

coun

ter-

exam

ple;

take

acc

ount

of

feed

back

and

lear

n fr

om m

ista

kes

visu

alis

e an

d 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-

exam

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 o

f a

situ

atio

n or

pro

blem

;po

se q

uest

ions

and

m

ake

conv

inci

ng

argu

men

ts to

just

ify

gene

ralis

atio

ns o

r so

lutio

ns; r

ecog

nise

the

impa

ct o

f con

stra

ints

or

assu

mpt

ions

iden

tify

a ra

nge

of s

trat

egie

s an

d 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 an

d 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 si

mpl

e pr

oofs

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

and

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

s1.

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, s

elec

ting

men

tal m

etho

ds o

r cal

cula

ting

devi

ces

as a

ppro

pria

te; m

anip

ulat

e nu

mbe

rs, a

lgeb

raic

exp

ress

ions

and

equ

atio

ns, a

nd a

pply

rout

ine

algo

rithm

s; us

e ac

cura

te n

otat

ion,

incl

udin

g co

rrec

tsy

ntax

whe

n us

ing

ICT;

reco

rd m

etho

ds, s

olut

ions

and

con

clus

ions

; est

imat

e, a

ppro

xim

ate

and

chec

k w

orki

ng

7

Inte

rpre

ting

and

eva

luat

ing

1.4 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

inte

rpre

t inf

orm

atio

n fr

om 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

sion

s;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

give

n co

ntex

t or t

o es

tabl

ish

the

trut

h of

a s

tate

men

t; gi

veac

cura

te s

olut

ions

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

refl

ecti

ng

1.5

just

ify th

e m

athe

mat

ical

feat

ures

draw

n fr

om 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,

gra

phs

and

rela

ted

expl

anat

ions

revi

ew a

nd re

fine

own

findi

ngs

and

appr

oach

es o

n th

eba

sis

of d

iscu

ssio

ns

with

oth

ers;

look

for

and

refle

ct o

n ot

her

appr

oach

es a

nd b

uild

on

pre

viou

s ex

perie

nce

of s

imila

r situ

atio

ns

and

outc

omes

refin

e ow

n fin

ding

s an

d ap

proa

ches

on

the

basi

s of

dis

cuss

ions

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

ssi

tuat

ions

com

mun

icat

e ow

n fin

ding

s ef

fect

ivel

y,

oral

ly a

nd in

writ

ing,

and

disc

uss

and

com

pare

app

roac

hes

and

resu

lts w

ith o

ther

s; re

cogn

ise

equi

vale

nt

appr

oach

es

mak

e se

nse

of, a

nd

judg

e th

e va

lue

of, o

wn

findi

ngs

and

thos

e pr

esen

ted

by o

ther

s; ju

dge

the

stre

ngth

of

empi

rical

evi

denc

e an

d di

stin

guis

h be

twee

n ev

iden

ce a

nd p

roof

;ju

stify

gen

eral

isat

ions

, ar

gum

ents

or s

olut

ions

use

a ra

nge

of fo

rms

to c

omm

unic

ate

findi

ngs

effe

ctiv

ely

todi

ffer

ent a

udie

nces

; re

view

find

ings

and

lo

ok fo

r equ

ival

ence

todi

ffer

ent p

robl

ems

with

sim

ilar s

truc

ture

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

lusi

ons

use

mat

hem

atic

alla

ngua

ge a

ndsy

mbo

ls e

ffec

tivel

y in

pr

esen

ting

conv

inci

ng

conc

lusi

ons

or fi

ndin

gs;

criti

cally

refle

ct o

n ow

n lin

es o

f enq

uiry

w

hen

expl

orin

g; s

earc

hfo

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

onsi

der t

he

effic

ienc

y of

alte

rnat

ive

lines

of e

nqui

ry o

r pr

oced

ures

show

insi

ght i

nto

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

onsi

der

the

assu

mpt

ions

in th

e m

odel

and

reco

gnis

e lim

itatio

ns in

the

accu

racy

of r

esul

ts

and

conc

lusi

ons

Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

rout

inel

y re

view

and

re

fine

findi

ngs

and

appr

oach

es; i

dent

ifyho

w o

ther

con

text

s w

ere

diff

eren

t fro

m, o

r si

mila

r to,

the

curr

ent

situ

atio

n an

d ex

plai

n ho

w a

nd w

hy th

e sa

me

or d

iffer

ent s

trat

egie

s w

ere

used

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

by 1

0,

100,

100

0, a

nd e

xpla

in

the

effe

ct

com

pare

and

ord

er

deci

mal

s in

diff

eren

t 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

of 1

0;

mul

tiply

and

div

ide

inte

gers

and

dec

imal

s by

0.1

and

0.0

1

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

gnis

e th

e eq

uiva

lenc

e 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

, usi

ng

sign

ifica

nt fi

gure

s as

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

sol

utio

n

enga

ge in

mat

hem

atic

al ta

sks

whe

re u

sing

num

bers

in

sta

ndar

d fo

rmis

ess

entia

l to

the

calc

ulat

ions

invo

lved

; cr

itica

lly e

xam

ine

the

effe

ct o

f num

eric

al

repr

esen

tatio

ns

on th

e ac

cura

cy o

f th

e so

lutio

n, e

.g.

unde

rsta

nd h

ow e

rror

s ca

n be

com

poun

ded

in

cal

cula

tions

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,

usin

g up

per a

nd

low

er b

ound

s

10

Inte

gers

, pow

ers

and

root

s2.

2 Year

7

Year

8

Year

9

Year

10

Year

11

unde

rsta

nd n

egat

ive

num

bers

as

posi

tions

on

a n

umbe

r lin

e;

orde

r, ad

d an

d su

btra

ctin

tege

rs in

con

text

reco

gnis

e an

d us

em

ultip

les,

fact

ors,

prim

es (l

ess t

han

100)

,co

mm

on fa

ctor

s,hi

ghes

t com

mon

fact

ors a

nd lo

wes

tco

mm

on m

ultip

les i

nsi

mpl

e ca

ses;

use

sim

ple

test

s of d

ivisi

bilit

y

reco

gnis

e th

e fir

st fe

w

tria

ngul

ar n

umbe

rs;

reco

gnis

e 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,

sub

trac

t, m

ultip

ly

and

divi

de in

tege

rs

use

mul

tiple

s, fa

ctor

s,co

mm

on fa

ctor

s,

high

est c

omm

on

fact

ors,

low

est

com

mon

mul

tiple

s an

d pr

imes

; fin

d th

e pr

ime

fact

or d

ecom

posi

tion

of a

num

ber,

e.g.

80

00 =

26 ×

53

use

squa

res,

pos

itive

and

nega

tive

squa

re

root

s, c

ubes

and

cu

be ro

ots,

and

inde

x no

tatio

n fo

r sm

all

posi

tive

inte

ger p

ower

s

use

the

prim

e fa

ctor

de

com

posi

tion

of

a nu

mbe

r

use

ICT

to e

stim

ate

squa

re ro

ots

and

cu

be ro

ots

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

visi

on o

f pos

itive

in

tege

r pow

ers

exam

ine

and

exte

nd

the

inde

x la

ws

to

esta

blis

h th

e m

eani

ng

of n

egat

ive,

frac

tiona

l an

d ze

ro p

ower

s,

incl

udin

g us

e of

su

rd n

otat

ion

exam

ine

and

exte

nd

the

inde

x la

ws

to

esta

blis

h th

e m

eani

ng

of in

vers

e op

erat

ions

in

rela

tion

to in

dice

s, i.

e.

the

inve

rse

oper

atio

n of

rais

ing

a po

sitiv

e nu

mbe

r to

pow

er n

isra

isin

g th

e re

sult

of th

is

oper

atio

n to

pow

er 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

n2.

3 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

expr

ess

a sm

alle

r who

le

num

ber a

s a

frac

tion

of

a la

rger

one

; sim

plify

fr

actio

ns b

y ca

ncel

ling

all c

omm

on fa

ctor

s an

d id

entif

y eq

uiva

lent

fr

actio

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 si

mpl

e fr

actio

ns

add

and

subt

ract

si

mpl

e fr

actio

ns a

nd

thos

e w

ith c

omm

onde

nom

inat

ors;

calc

ulat

e si

mpl

e fr

actio

ns o

f qua

ntiti

es

and

mea

sure

men

ts(w

hole

-num

ber

answ

ers)

; mul

tiply

a

frac

tion

by a

n in

tege

r

reco

gnis

e th

at a

re

curr

ing

deci

mal

is a

fr

actio

n; u

se d

ivis

ion

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 fr

actio

ns;

know

that

a re

curr

ing

deci

mal

is a

n ex

act

frac

tion

expl

ain

the

patt

erns

fo

und

in re

curr

ing

deci

mal

s; ju

stify

w

hy d

ecim

als

recu

r or

term

inat

e by

co

nsid

erin

g fa

ctor

s of

th

e de

nom

inat

or

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 s

ome

of it

s fe

atur

es, e

.g. t

hat

all r

ecur

ring

deci

mal

s ca

n be

exp

ress

ed a

s

a fr

actio

n

show

insi

ght i

nto

the

infin

ite d

ensi

ty o

f the

nu

mbe

r lin

e; m

ake

sens

e of

the

proo

f tha

t √2

is ir

ratio

nal

add

and

subt

ract

fr

actio

ns b

y w

ritin

g th

em w

ith a

com

mon

de

nom

inat

or; c

alcu

late

fr

actio

ns o

f qua

ntiti

es

(frac

tion

answ

ers)

; m

ultip

ly a

nd d

ivid

e an

in

tege

r by

a fr

actio

n

use

effic

ient

met

hods

toad

d, su

btra

ct, m

ultip

lyan

d di

vide

frac

tions

,in

terp

retin

g di

visi

on a

sa

mul

tiplic

ativ

e in

vers

e;ca

ncel

com

mon

fact

ors

befo

re m

ultip

lyin

gor

div

idin

g

unde

rsta

nd a

pply

ef

ficie

nt m

etho

ds to

add,

sub

trac

t, m

ultip

ly

and

divi

de fr

actio

ns,

inte

rpre

ting

reci

proc

als

as m

ultip

licat

ive

inve

rses

12

Fra

ctio

ns, d

ecim

als,

per

cent

ages

, rat

io a

nd p

ropo

rtio

n (c

onti

nued

)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

si

mpl

e pe

rcen

tage

s an

d us

e pe

rcen

tage

s to

com

pare

sim

ple

prop

ortio

ns

reco

gnis

e th

e eq

uiva

lenc

e of

pe

rcen

tage

s, fr

actio

ns

and

deci

mal

s

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

and

divi

de a

qu

antit

y in

to tw

o pa

rts

in a

giv

en ra

tio;

solv

e si

mpl

e pr

oble

ms

invo

lvin

g ra

tio a

nd

prop

ortio

n us

ing

info

rmal

str

ateg

ies

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

toco

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; s

impl

ifyra

tios,

incl

udin

g th

ose

expr

esse

d in

diff

eren

t un

its, r

ecog

nisi

ng li

nks

with

frac

tion

nota

tion;

di

vide

a q

uant

ity in

totw

o or

mor

e pa

rts

in

a gi

ven

ratio

; use

the

unita

ry m

etho

d to

so

lve

sim

ple

prob

lem

sin

volv

ing

ratio

and

di

rect

pro

port

ion

Year

9

Year

10

Year

11

Exte

nsio

n

reco

gnis

e w

hen

frac

tions

or

perc

enta

ges

are

need

ed to

com

pare

prop

ortio

ns; s

olve

pr

oble

ms

invo

lvin

g pe

rcen

tage

cha

nges

use

prop

ortio

nal

reas

onin

g to

sol

ve

prob

lem

s, c

hoos

ing

the

corr

ect n

umbe

rs

to ta

ke a

s 10

0%, 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 s

tatis

tics

invo

lves

pr

opor

tiona

lity;

use

m

ultip

licat

ive

met

hods

flu

ently

in th

e so

lutio

n,

incl

udin

g in

vers

e ca

lcul

atio

ns, e

.g.

with

per

cent

ages

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;

sol

ve

prob

lem

s in

volv

ing

inve

rse

prop

ortio

n (in

clud

ing

y ?

1/x

2 )

(as i

n 2.

4)

13

Num

ber o

pera

tion

s2.

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

and

use

mod

el re

al c

onte

xts

unde

rsta

nd a

nd u

se

rule

s of

arit

hmet

ic a

nd

rule

s of

arit

hmet

ic a

nd

of m

ultip

lyin

g an

d re

cipr

ocal

s as

a

whe

re q

uant

ities

var

ydi

rect

and

inve

rse

inve

rse

oper

atio

ns in

in

vers

e op

erat

ions

in

divi

ding

by

num

bers

m

ultip

licat

ive

inve

rse

in d

irect

pro

port

ion,

pr

opor

tion;

sol

ve

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

such

as

incl

udin

g re

peat

edpr

oble

ms

invo

lvin

g in

tege

rs a

nd d

ecim

als

and

frac

tions

co

nsol

idat

e us

e of

the

enla

rgem

ent;

expl

ore

prop

ortio

nal c

hang

e,

inve

rse

prop

ortio

n ru

les

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 ) in

vers

e op

erat

ions

re

cipr

ocal

func

tion

al

gebr

aic

met

hods

(a

s in

2.3)

(y

=��

/x) f

or la

rge

and

whe

re a

ppro

pria

te a

nd

smal

l val

ues

of x

co

nsid

er li

mita

tions

of

the

mod

el

(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

lati

on 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

, and

qui

ckly

de

rive

asso

ciat

eddi

visi

on fa

cts

stre

ngth

en a

nd e

xten

d m

enta

l met

hods

of

calc

ulat

ion

to in

clud

e de

cim

als,

frac

tions

an

d pe

rcen

tage

s,

acco

mpa

nied

whe

re

appr

opria

te b

y su

itabl

e jo

ttin

gs; s

olve

sim

ple

prob

lem

s 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

fr

actio

ns, d

ecim

als

and

perc

enta

ges;

use

know

n fa

cts

to

deriv

e un

know

n fa

cts,

in

clud

ing

prod

ucts

invo

lvin

g nu

mbe

rs

such

as

0.7

and

6, a

nd

and

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,

fr

actio

ns, p

erce

ntag

es,

squa

res

and

squa

re

root

s, c

ubes

and

cub

ero

ots;

solv

e pr

oble

ms

men

tally

use

know

n fa

cts

to

deriv

e un

know

n fa

cts;

exte

nd m

enta

l m

etho

ds o

f cal

cula

tion,

w

orki

ng w

ith d

ecim

als,

fr

actio

ns, p

erce

ntag

es,

fact

ors,

pow

ers

and

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

appr

opria

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,

trig

onom

etric

alfu

nctio

ns a

nd n

umbe

rs

in s

tand

ard

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 an

d π,

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

Wri

tten

cal

cula

tion

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 s

ingl

e-di

git 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

of a

ny s

ize,

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

visi

on o

f int

eger

s an

d de

cim

als,

incl

udin

g by

de

cim

als

such

as

0.6

or0.

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

of a

ny s

ize;

m

ultip

ly b

y de

cim

als;

divi

de b

y de

cim

als

by tr

ansf

orm

ing

to

divi

sion

by

an in

tege

r

sele

ct m

enta

l or

writ

ten

stra

tegi

es

and

calc

ulat

ing

devi

ces

appr

opria

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,

trig

onom

etric

alfu

nctio

ns a

nd n

umbe

rs

in s

tand

ard

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 an

d π,

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

usi

ng b

rack

ets

and

the

mem

ory;

use

the

squa

re ro

ot a

nd s

ign

chan

ge k

eys

ente

r num

bers

and

in

terp

ret t

he d

ispl

ay

in d

iffer

ent c

onte

xts

(dec

imal

s, p

erce

ntag

es,

mon

ey, m

etric

m

easu

res)

carr

y ou

t mor

e di

ffic

ult

calc

ulat

ions

eff

ectiv

ely

and

effic

ient

ly u

sing

th

e fu

nctio

n ke

ys fo

r si

gn c

hang

e, p

ower

s,

root

s an

d 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 s

ize,

kn

owin

g no

t to

roun

d du

ring

inte

rmed

iate

st

eps

of a

cal

cula

tion;

us

e th

e co

nsta

nt, π

and

sign

cha

nge

keys

; us

e th

e fu

nctio

n ke

ys

for p

ower

s, ro

ots

and

frac

tions

; use

bra

cket

s an

d th

e m

emor

y

sele

ct m

enta

l or

writ

ten

stra

tegi

es

and

calc

ulat

ing

devi

ces

appr

opria

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,

trig

onom

etric

alfu

nctio

ns a

nd n

umbe

rs

in s

tand

ard

form

(as i

n 2.

5 an

d 2.

6)

criti

cally

exa

min

e al

tern

ativ

e m

etho

ds,

com

pare

str

ateg

ies

for:

t�

calc

ulat

ing

(incl

udin

gca

lcul

atin

g de

vice

s)

t�

chec

king

reco

gnis

e th

e lim

itatio

ns o

f som

e ap

proa

ches

(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

ispl

ay

in d

iffer

ent c

onte

xts

(ext

end

to n

egat

ive

num

bers

, fra

ctio

ns,

time)

17

Che

ckin

g re

sult

s2.

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

back

war

ds

sele

ct fr

om a

rang

e ch

eck

resu

lts u

sing

id

entif

y a

rang

e of

of

che

ckin

g m

etho

ds,

appr

opria

te m

etho

ds

chec

king

str

ateg

ies

and

incl

udin

g es

timat

ing

appr

ecia

te th

at m

ore

in c

onte

xt a

nd u

sing

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

str

ateg

ies

for:

t�

calc

ulat

ing

(incl

udin

gca

lcul

atin

g de

vice

s)

t�

chec

king

reco

gnis

e th

e lim

itatio

ns o

f som

e ap

proa

ches

(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 A

lgeb

ra

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titi

es

3.1

Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

use

lett

er s

ymbo

ls to

reco

gnis

e th

at le

tter

di

stin

guis

h th

e pr

esen

t con

vinc

ing

exam

ine

and

refin

eus

e sy

mbo

ls a

nd

repr

esen

t unk

now

n sy

mbo

ls p

lay

diff

eren

t di

ffer

ent r

oles

pla

yed

alge

brai

c ar

gum

ents

toal

gebr

aic

argu

men

ts

repr

esen

tatio

ns

num

bers

or v

aria

bles

;ro

les

in e

quat

ions

,by

lett

er s

ymbo

ls in

ju

stify

gen

eral

isat

ions

pr

esen

ted

to e

xpla

inco

nsis

tent

ly to

pre

sent

know

the

mea

ning

s fo

rmul

ae a

nd fu

nctio

ns;

equa

tions

, ide

ntiti

es,

or s

olut

ions

; rel

ate

geom

etric

al a

nd

a fo

rmal

pro

of, e

.g.

of th

e w

ords

term

, kn

ow th

e m

eani

ngs

form

ulae

and

func

tions

ar

gum

ents

to th

e nu

mer

ical

pro

pert

ies;

deriv

ing

the

form

ula

expr

essio

n an

d eq

uatio

n of

the

wor

ds fo

rmul

a st

ruct

ure

of th

e co

ntex

t ch

oose

and

com

bine

fo

r sol

ving

qua

drat

ic

and

func

tion

or p

robl

em; p

rodu

ce

repr

esen

tatio

ns to

equa

tions

si

mpl

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 an

d re

pres

enta

tion

toge

nera

lity

of th

e fo

rms

follo

w th

e ru

les

of

incl

udin

g th

e us

e of

si

mpl

e in

stan

ces

of

synt

hesi

se k

now

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

arit

hmet

ic; u

se

the

com

mut

ativ

e an

d its

elf b

e an

exp

ress

ion;

in

dex

nota

tion

for s

mal

l di

strib

utiv

e la

ws;

just

ify

use

this

insi

ght i

nto

posi

tive

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 fo

r mul

tiplic

atio

n an

d di

visi

on to

incl

ude

zero

, neg

ativ

e an

d fr

actio

nal p

ower

s

20

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titi

es (c

onti

nued

)3.

1 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

sim

plify

line

ar a

lgeb

raic

ex

pres

sion

s by

co

llect

ing

like

term

s;m

ultip

ly a

sin

gle

term

ov

er a

bra

cket

(int

eger

co

effic

ient

s)

sim

plify

or t

rans

form

lin

ear e

xpre

ssio

ns b

y co

llect

ing

like

term

s;m

ultip

ly a

sin

gle

term

ov

er a

bra

cket

sim

plify

or t

rans

form

al

gebr

aic

expr

essi

ons

by ta

king

out

sin

gle-

term

com

mon

fact

ors;

add

sim

ple

alge

brai

c fr

actio

ns

deve

lop

fluen

cy in

tr

ansf

orm

ing

linea

r ex

pres

sion

s; ex

pand

the

prod

uct o

f tw

o lin

ear e

xpre

ssio

ns

of th

e fo

rm x

± n

and

fact

oris

e si

mpl

e qu

adra

tic e

xpre

ssio

ns;

esta

blis

h id

entit

ies s

uch

as th

e di

ffer

ence

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

;si

mpl

ify o

r tra

nsfo

rm

alge

brai

c fr

actio

ns,

by

fact

oris

ing

and

e.g.

canc

ellin

g co

mm

on

fact

ors;

com

pare

and

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 th

is b

ytr

ansf

orm

atio

n

21

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titi

es (c

onti

nued

)3.

1 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

cons

truc

t and

so

lve

sim

ple

linea

r eq

uatio

ns w

ith in

tege

r co

effic

ient

s (u

nkno

wn

on o

ne s

ide

only

) usi

ng

an a

ppro

pria

te m

etho

d in

vers

e op

erat

ions

) (e

.g.

cons

truc

t and

sol

ve

linea

r equ

atio

ns w

ithin

tege

r coe

ffic

ient

s(u

nkno

wn

on e

ither

or

both

sid

es, w

ithou

t and

w

ith b

rack

ets)

usi

ng

appr

opria

te m

etho

ds

inve

rse

oper

atio

ns,

(e.g

.tr

ansf

orm

ing

both

si

des

in s

ame

way

)

use

grap

hs a

nd s

et

up e

quat

ions

to s

olve

si

mpl

e pr

oble

ms

invo

lvin

g di

rect

prop

ortio

n

cons

truc

t and

sol

ve

linea

r equ

atio

ns w

ithin

tege

r coe

ffic

ient

s(w

ith a

nd w

ithou

t br

acke

ts, n

egat

ive

sign

s an

ywhe

re in

the

equa

tion,

pos

itive

or

nega

tive

solu

tion)

use

alge

brai

c m

etho

ds

to s

olve

pro

blem

s in

volv

ing

dire

ctpr

opor

tion;

rela

teal

gebr

aic

solu

tions

to

gra

phs

of th

e 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 s

ituat

ions

or

mat

hem

atic

alpr

oble

ms;

solv

e lin

ear e

quat

ions

an

d in

equa

litie

s,

repr

esen

ting

the

solu

tion

in th

e co

ntex

t of

the

prob

lem

cons

truc

t sim

ple

quad

ratic

equ

atio

ns

to re

pres

ent r

eal-

life

situ

atio

ns o

r m

athe

mat

ical

pro

blem

s an

d so

lve

them

us

ing

fact

oris

atio

n,

grap

hica

l or t

rial a

ndim

prov

emen

t met

hods

;ju

stify

the

num

ber

of s

olut

ions

usi

ng

alge

brai

c or

gra

phic

al

argu

men

ts a

nd s

elec

tap

prop

riate

sol

utio

ns,

inte

rpre

ting

thei

r ac

cura

cy

repr

esen

t rea

l-lif

e si

tuat

ions

or

mat

hem

atic

al p

robl

ems

invo

lvin

g:

t� m

ore

com

plex

qu

adra

tic

equa

tions

,ch

oosi

ng a

n ap

prop

riate

m

etho

d of

solu

tion

incl

udin

g co

mpl

etin

g th

e sq

uare

and

use

of

the

form

ula

t�

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

titi

es (c

onti

nued

)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

ap

prox

imat

e so

lutio

ns

to e

quat

ions

such

as

x 2 +

x =

20

expl

ore

way

s of

cons

truc

ting

mod

els

of re

al-li

fe si

tuat

ions

by d

raw

ing

grap

hsan

d co

nstr

uctin

g al

gebr

aic

equa

tions

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 si

mul

tane

ous

linea

r eq

uatio

ns to

repr

esen

t re

al-li

fe s

ituat

ions

or

mat

hem

atic

al

prob

lem

s; ex

amin

e an

d co

mpa

re

alge

brai

c m

etho

ds o

f so

lutio

n; u

se g

raph

ical

re

pres

enta

tion

toex

plai

n w

hy th

e in

ters

ectio

n of

two

lines

giv

es th

e co

mm

on

solu

tion

and

why

som

e ca

ses

have

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 s

imul

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

fr

om 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

line

ar a

nd th

e ot

her i

s qu

adra

tic o

r of

the

form

x 2 +

y 2 =

r2

23

Equ

atio

ns, f

orm

ulae

, exp

ress

ions

and

iden

titi

es (c

onti

nued

)3.

1 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

use

sim

ple

form

ulae

fr

om m

athe

mat

ics

and

othe

r sub

ject

s; su

bstit

ute

posi

tive

inte

gers

into

line

ar

expr

essi

ons

and

form

ulae

and

, in

sim

ple

case

s, d

eriv

e a

form

ula

use

form

ulae

from

m

athe

mat

ics

and

othe

r 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;

sub

stitu

tepo

sitiv

e in

tege

rs in

toex

pres

sion

s in

volv

ing

smal

l pow

ers,

e.g

. 3x

2 + 4

or 2

x3 ; der

ive

sim

ple

form

ulae

use

form

ulae

from

m

athe

mat

ics

and

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, c

hang

e its

su

bjec

t

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 fr

om 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

sion

s;m

anip

ulat

e fo

rmul

ae

in o

rder

to re

ach

a so

lutio

n, s

how

insi

ght

into

the

mat

hem

atic

al

conn

ectio

ns, e

.g. u

sing

th

e co

ntex

t and

the

form

ulae

to e

xpla

in th

e pr

opor

tiona

l eff

ect 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

of a

sim

ple

sequ

ence

, giv

en a

ru

le (e

.g. f

indi

ng a

te

rm fr

om th

e pr

evio

us

term

, fin

ding

a te

rm

give

n its

pos

ition

in

the

sequ

ence

)

gene

rate

term

s of

a

linea

r seq

uenc

e us

ing

term

-to-

term

and

posi

tion-

to-t

erm

ru

les,

on

pape

r and

us

ing

a sp

read

shee

t or

grap

hics

cal

cula

tor

gene

rate

term

s of

a

sequ

ence

usi

ng te

rm

to-t

erm

and

pos

ition

to

-ter

m ru

les,

on

pape

r an

d us

ing

ICT

gene

rate

seq

uenc

es

from

pat

tern

s or

pr

actic

al c

onte

xts

and

desc

ribe

the

gene

ral

term

in s

impl

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

seq

uenc

es;

inte

rpre

t, de

duce

and

ju

stify

gen

eral

isat

ions

fo

r the

nth

term

of

linea

r and

qua

drat

ic

sequ

ence

s, in

clud

ing

the

prop

ertie

s of

sq

uare

and

tria

ngul

arnu

mbe

rs

use

linea

r exp

ress

ions

to d

escr

ibe

the

nth

term

of a

sim

ple

arith

met

icse

quen

ce, j

ustif

ying

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

seq

uenc

es

from

pra

ctic

al c

onte

xts

and

writ

e an

d ju

stify

an

expr

essi

on to

des

crib

e th

e nt

h te

rm o

f an

arith

met

ic s

eque

nce

25

Seq

uenc

es, f

unct

ions

and

gra

phs

(con

tinu

ed)

3.2 Ye

ar 7

Ye

ar 8

Ye

ar 9

Ye

ar 1

0 Ye

ar 1

1 Ex

tens

ion

expr

ess

sim

ple

expr

ess

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

usi

ng s

ymbo

ls;

and

repr

esen

t the

m

geom

etric

al

repr

esen

t the

m in

in

map

ping

s or

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:

t�

rota

ting

the

line

y =

mx

+ c

thro

ugh

90° a

bout

any

po

int

t�

refle

ctin

g th

e lin

ey

= m

x +

c in

the

line

y =

x

deriv

e pr

oper

ties

of

perp

endi

cula

r lin

es a

nd

of th

e in

vers

e fu

nctio

n

gene

rate

coo

rdin

ate

gene

rate

poi

nts

in a

ll 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

sat

isfy

a

four

qua

dran

ts a

nd p

lot

plot

gra

phs o

f lin

ear

func

tions

of t

he fo

rm

betw

een

the

form

of

expo

nent

ial a

nd

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

thei

r re

sulti

ng g

raph

s of

fu

nctio

ns a

nd re

cogn

ise

the

grap

hs o

f sim

ple

func

tions

, whe

re y

isgi

ven

impl

icitl

y in

term

slin

ear f

unct

ions

, whe

regi

ven

expl

icitl

y in

term

s of

x (e

.g. a

y +

bx =

0,

char

acte

ristic

sha

pes;

quad

ratic

and

cub

ic

thei

r cha

ract

eris

ticy

is g

iven

exp

licitl

y of

x, o

n pa

per a

nd

y +

bx +

c =

0), o

n pa

per

vary

the

valu

es o

f a, b

fu

nctio

ns s

uch

as:

shap

es; a

pply

to th

ein

term

s of

x, o

n us

ing

ICT;

reco

gnis

e an

d us

ing

ICT;

find

the

and

c in

func

tions

such

gr

aph

y =

f(x) t

het�

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

= a

x 2 + c

, tr

ansf

orm

atio

nst�

y =

(x –

2)(x

2 + 7

x + 12

)re

cogn

ise

stra

ight

-line

the

form

y =

mx

+ c

by e

quat

ions

of t

hey

= ax

3 + c

, y

= f(x

) + a

, y =

af(x

), gr

aphs

par

alle

l to

the

co

rres

pond

to s

trai

ght-

form

y =

mx

+ c,

give

n y

= (x

+ b

)2 usi

ng a

y

= f(x

+ a

), y

= f(a

x) fo

rt�

y =

x 2 – 2

x +

1 x-

axis

or

y-ax

is

line

grap

hs

valu

es fo

r m a

nd c

gr

aph

plot

ter t

o ex

plai

n lin

ear,

quad

ratic

, sin

et�

y =

x 3 + 3

how

this

tran

sfor

ms

an

d co

sine

func

tions

;th

e gr

aph

use

a gr

aph

plot

ter t

oin

clud

e fe

atur

es s

uch

ex

plai

n th

e ef

fect

of

as ro

ots

of th

e eq

uatio

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

(con

tinu

ed)

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

isin

g fr

om re

al-li

fe

grap

hs th

at m

odel

real

-m

athe

mat

ical

func

tions

m

odel

of a

real

-life

fu

nctio

ns a

risin

g fr

om

real

-life

pro

blem

s pr

oble

ms

and

plot

thei

r lif

e si

tuat

ions

, inc

ludi

ng

to p

robl

ems

invo

lvin

g:co

ntex

t or p

robl

em,

real

-life

situ

atio

ns, e

.g.

and

plot

thei

r co

rres

pond

ing

grap

hs;

thos

e ge

nera

ted

iden

tifyi

ng th

e t�

optim

isat

ion,

co

nver

sion

gra

phs

corr

espo

ndin

g gr

aphs

; in

terp

ret g

raph

s ar

isin

g fr

om o

ther

subj

ects

varia

bles

and

thei

rus

ing

num

eric

al,

disc

uss

and

inte

rpre

t fr

om re

al s

ituat

ions

, e.g

. su

ch a

s sc

ienc

e;fu

nctio

nal r

elat

ions

hip;

al

gebr

aic

and

grap

hs a

risin

g fr

om

time

serie

s gr

aphs

us

e m

athe

mat

ical

use

grap

hs a

nd

grap

hica

l, re

al s

ituat

ions

, e.g

. ar

gum

ent t

o ju

stify

sk

etch

es to

exp

lain

te

chni

ques

,di

stan

ce–t

ime

grap

hs

feat

ures

of t

heir

shap

es

the

beha

viou

r of t

he

incl

udin

g m

axim

a va

riabl

es a

nd to

exp

lain

an

d m

inim

a or

just

ify th

e ef

fect

of

t�

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 sc

ienc

e ex

perim

ent

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

rela

ted

to fe

atur

es

of th

e eq

uatio

n,

par

alle

l and

e.

g.pe

rpen

dicu

lar l

ines

27

Seq

uenc

es, f

unct

ions

and

gra

phs

(con

tinu

ed)

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

var

ious

poi

nts a

nd

sect

ions

of s

trai

ght-

line

grap

hs, i

nclu

ding

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

rica

l rea

soni

ng4.

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

sha

pes

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

vert

ical

ly

oppo

site

ang

les

iden

tify

alte

rnat

e an

gles

and

corr

espo

ndin

g an

gles

; un

ders

tand

a p

roof

th

at:

t�

the

angl

e su

m o

f a

tria

ngle

is 1

80° a

nd

of a

qua

drila

tera

l is

360

°

t�

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

posi

te a

ngle

s

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

dist

ingu

ish

betw

een

conv

entio

ns,

defin

ition

s an

d de

rived

prop

ertie

s

expl

ain

how

to fi

nd,

calc

ulat

e an

d us

e:

t�

the

sum

s of

the

inte

rior a

nd

exte

rior a

ngle

s of

qua

drila

tera

ls,

pent

agon

s an

d he

xago

ns

t�

the

inte

rior a

nd

exte

rior a

ngle

s of

re

gula

r pol

ygon

s

know

the

defin

ition

of

a ci

rcle

and

the

nam

es

of it

s pa

rts;

expl

ain

why

insc

ribed

regu

lar

poly

gons

can

be

cons

truc

ted

by e

qual

di

visi

ons

of a

circ

le

exam

ine

and

exam

ine

and

crea

te

pres

ent r

igor

ous

and

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

sol

utio

ns

in th

e so

lutio

n of

ge

omet

rical

pro

blem

s,to

mor

e co

mpl

ex

geom

etric

al p

robl

ems;

dist

ingu

ishi

ng b

etw

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

sim

ple

proo

fs

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

sol

ve

pres

ent a

pro

of th

at th

e to

exp

lain

and

just

ify

prob

lem

s st

anda

rd c

onst

ruct

ions

si

mpl

e ci

rcle

pro

pert

ies

are

exac

t an

d th

eore

ms

30

Geo

met

rica

l rea

soni

ng (c

onti

nued

)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

tria

ngle

s an

d qu

adril

ater

als;

expl

ore

geom

etric

al

prob

lem

s in

volv

ing

thes

e pr

oper

ties,

ex

plai

ning

reas

onin

g or

ally

, usi

ng s

tep

by-s

tep

dedu

ctio

n su

ppor

ted

by d

iagr

ams

solv

e ge

omet

rical

pr

oble

ms

usin

g si

de

and

angl

e pr

oper

ties

of e

quila

tera

l, is

osce

les

and

right

-an

gled

tria

ngle

s an

d sp

ecia

l qua

drila

tera

ls,

expl

aini

ng re

ason

ing

with

dia

gram

s an

d te

xt;

clas

sify

qua

drila

tera

ls

by th

eir g

eom

etric

al

prop

ertie

s

know

that

if tw

o 2-

D

shap

es a

re c

ongr

uent

, co

rres

pond

ing

side

s an

d an

gles

are

equ

al

solv

e pr

oble

ms

usin

g pr

oper

ties

of

angl

es, o

f par

alle

l and

in

ters

ectin

g lin

es a

nd

of tr

iang

les

and

othe

rpo

lygo

ns, j

ustif

ying

in

fere

nces

and

ex

plai

ning

reas

onin

g w

ith d

iagr

ams

and

text

unde

rsta

ndco

ngru

ence

and

ex

plor

e si

mila

rity

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

usin

g pr

oper

ties

of li

nes,

an

gles

, pol

ygon

san

d ci

rcle

s; ju

stify

ar

gum

ents

and

so

lutio

ns u

sing

de

duct

ive

reas

onin

g

draw

infe

renc

es a

bout

pr

oper

ties

of s

imila

r 2-

D s

hape

s an

d us

e pr

opor

tiona

l rea

soni

ng

to s

olve

geo

met

rical

an

d tr

igon

omet

rical

prob

lem

s

visu

alis

e an

d m

anip

ulat

edy

nam

ic im

ages

and

use

scal

e dr

awin

g to

inve

stig

ate

area

s of

squa

res o

n si

des o

frig

ht-a

ngle

d an

d no

nrig

ht-a

ngle

d tr

iang

les,

rela

ting

findi

ngs t

oPy

thag

oras

’ the

orem

;us

e Py

thag

oras

’ the

orem

to so

lve

prob

lem

s in

2-D

and

sim

ple

3-D

cas

es

Year

11

form

alis

e ex

istin

gkn

owle

dge

of li

nes,

angl

es a

nd p

olyg

ons b

y:

t�

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

t� ex

plai

ning

why

stan

dard

cons

truc

tions

wor

k,e.

g. o

bser

ving

that

lines

join

ing

poin

tsw

here

com

pass

arc

sm

eet a

re si

des o

f arh

ombu

s

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

of

Pyth

agor

as’ t

heor

em;

use

Pyth

agor

as’

theo

rem

to s

olve

mor

e co

mpl

ex 3

-D p

robl

ems

pres

ent a

nd ju

stify

a

form

al p

roof

of

Pyth

agor

as’ t

heor

em

31

Geo

met

rica

l rea

soni

ng (c

onti

nued

)4.

1 Year

7

Year

8

Year

9

Year

10

Year

11

use

2-D

repr

esen

tatio

ns

visu

alis

e 3-

D s

hape

s to

vis

ualis

e 3-

D s

hape

s fr

om th

eir n

ets;

use

and

dedu

ce s

ome

of

geom

etric

al p

rope

rtie

s th

eir p

rope

rtie

s of

cub

oids

and

sha

pes

mad

e fr

om c

uboi

ds;

use

sim

ple

plan

s an

d el

evat

ions

visu

alis

e an

d us

e 2-

D

repr

esen

tatio

ns o

f 3-D

ob

ject

s; an

alys

e 3-

D

shap

es th

roug

h 2-

D

proj

ectio

ns, i

nclu

ding

pl

ans

and

elev

atio

ns

visu

alis

e an

d de

scrib

e pr

oper

ties

of p

oint

s,

lines

and

pla

nes

in 3

-D

spac

e, in

clud

ing

cros

s se

ctio

ns c

reat

ed b

y sl

icin

g a

3-D

sha

pe

visu

alis

e an

d m

anip

ulat

e im

ages

to

est

ablis

h tr

igon

omet

rical

rela

tions

hips

by:

t�

gene

ratin

g tr

iang

les

usin

g a

rota

ting

unit

radi

us(c

ircle

, cen

tre

the

orig

in)

t�

iden

tifyi

ng th

e pr

oper

ties

of

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 s

olve

si

mpl

e pr

oble

ms

in 2

-D,

incl

udin

g be

arin

gs

deriv

e th

e fo

rmul

a ½

ab s

inC

for t

hear

ea o

f a tr

iang

le;

use

trig

onom

etric

al

rela

tions

hips

toso

lve

mor

e co

mpl

ex

2-D

pro

blem

s an

d pr

oble

ms

in 3

-D, s

uch

as th

e an

gle

betw

een

a lin

e an

d a

plan

e

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 si

ne a

nd c

osin

e ru

les

and

use

them

to s

olve

2-

D a

nd 3

-D p

robl

ems

in a

rang

e of

con

text

s

32

Tra

nsfo

rmat

ions

and

coo

rdin

ates

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

gnis

e an

d vi

sual

ise

the

sym

met

ries

of a

2-D

sh

ape

tran

sfor

m 2

-D s

hape

s by

: t�

refle

ctin

g in

giv

en

mirr

or li

nes

t�

rota

ting

abou

t a

give

n po

int

t�

tran

slat

ing

expl

ore

thes

e tr

ansf

orm

atio

ns a

nd

sym

met

ries

usin

g IC

T

iden

tify

all t

he

sym

met

ries

of 2

-D

shap

es

tran

sfor

m 2

-D s

hape

s by

rota

tion,

refle

ctio

nan

d tr

ansl

atio

n, o

n pa

per a

nd u

sing

ICT

try

out m

athe

mat

ical

re

pres

enta

tions

of

sim

ple

com

bina

tions

of

thes

e tr

ansf

orm

atio

ns

iden

tify

refle

ctio

n sy

mm

etry

in 3

-D

shap

es

reco

gnis

e th

attr

ansl

atio

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

ansl

atio

ns, r

otat

ions

an

d re

flect

ions

of 2

-D

shap

es, o

n pa

per a

nd

usin

g IC

T

devi

se in

stru

ctio

ns fo

r a

com

pute

r to

gene

rate

and

tran

sfor

m s

hape

s

use

prec

ise

lang

uage

an

d no

tatio

n to

des

crib

e an

d ge

nera

lise

the

resu

lts o

f com

bini

ng

tran

sfor

mat

ions

of 2

-D

shap

es o

n pa

per a

nd

usin

g IC

T, in

clud

ing:

t�

rota

tions

abo

ut

any

poin

t

t�

refle

ctio

ns in

any

lin

e

t�

tran

slat

ions

usi

ng

vect

or n

otat

ion

t�

a tr

ansf

orm

atio

n an

d its

inve

rse

gene

rate

and

ana

lyse

pa

tter

ns, e

.g. I

slam

ic

desi

gns

expl

ain

and

dem

onst

rate

grap

hica

lly th

e ef

fect

s of

com

bini

ng

tran

slat

ions

, usi

ng

vect

or n

otat

ion,

in

clud

ing:

t� th

e ru

le fo

rad

ditio

n of

vec

tors

t�

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

com

bini

ng

tran

slat

ions

, usi

ng

vect

or n

otat

ion,

in

clud

ing:

t�

the

diff

eren

ce o

f tw

o ve

ctor

s

t�

the

resu

ltant

of

two

vect

ors

t�

the

com

mut

ativ

e an

d as

soci

ativ

epr

oper

ties

of

vect

or a

dditi

on

solv

e si

mpl

e ge

omet

rical

pro

blem

s in

2-D

usi

ng v

ecto

rs

33

Tra

nsfo

rmat

ions

and

coo

rdin

ates

(con

tinu

ed)

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

po

ints

det

erm

ined

by g

eom

etric

al

info

rmat

ion

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 s

hape

s, g

iven

a

cent

re o

f enl

arge

men

t an

d a

posi

tive

inte

ger

scal

e fa

ctor

; exp

lore

en

larg

emen

t usi

ng IC

T

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 a

nd B

enla

rge

2-D

sha

pes,

gi

ven

a ce

ntre

of

enla

rgem

ent a

nd a

po

sitiv

e in

tege

r sca

le

fact

or, o

n pa

per a

nd

usin

g IC

T; id

entif

yth

e sc

ale

fact

or o

f an

enl

arge

men

t as

the

ratio

of t

he

leng

ths

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

sha

pes

usin

g po

sitiv

e,

frac

tiona

l and

neg

ativ

e sc

ale

fact

ors,

on

pape

r and

usi

ng

ICT;

use

reci

proc

als

as a

mul

tiplic

ativ

ein

vers

e in

the

cont

ext

of e

nlar

gem

ent;

reco

gnis

e th

e si

mila

rity

of re

sulti

ng s

hape

s an

d ex

plai

n th

e ef

fect

of

enl

arge

men

t on

perim

eter

use

and

inte

rpre

t map

s an

d sc

ale

draw

ings

in th

e co

ntex

t of

mat

hem

atic

s an

d ot

her

subj

ects

use

the

coor

dina

te

grid

to s

olve

pro

blem

s in

volv

ing

tran

slat

ions

, ro

tatio

ns, r

efle

ctio

ns

and

enla

rgem

ents

appl

y th

e pr

oper

ties

of s

imila

r tria

ngle

s an

dPy

thag

oras

’ the

orem

to

sol

ving

pro

blem

s pr

esen

ted

on a

2-D

co

ordi

nate

grid

; us

e a

3-D

coo

rdin

ate

grid

to re

pres

ent

sim

ple

shap

es

enla

rge

3-D

sha

pes;

iden

tify

and

expl

ain

the

effe

cts

of e

nlar

gem

ent

on a

reas

and

vol

umes

of

sim

ilar s

hape

s an

d so

lids;

rela

te th

is

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

:

t�

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

t�

cons

truc

t a

tria

ngle

, giv

en

two

side

s an

d th

e in

clud

ed a

ngle

(S

AS)

or t

wo

angl

es

and

the

incl

uded

si

de (A

SA)

use

stra

ight

edg

e an

d co

mpa

sses

to c

onst

ruct

:

t�

the

mid

poin

t and

pe

rpen

dicu

lar

bise

ctor

of a

line

se

gmen

t

t�

the

bise

ctor

of

an a

ngle

t�

the

perp

endi

cula

r fr

om a

poi

nt to

a

line

t�

the

perp

endi

cula

r fr

om a

poi

nt o

n

a lin

e

t�

a tr

iang

le, g

iven

th

ree

side

s (S

SS)

use

stra

ight

edg

e an

d co

mpa

sses

to c

onst

ruct

tr

iang

les,

giv

en ri

ght

angl

e, h

ypot

enus

e an

d si

de (R

HS)

use

ICT

to e

xplo

re

cons

truc

tions

of

tria

ngle

s an

d ot

her

2-D

sha

pes

find

the

locu

s of

a p

oint

th

at m

oves

acc

ordi

ng

to a

sim

ple

rule

, bot

h by

reas

onin

g an

d by

us

ing

ICT

use

prop

ertie

s of

2-

D a

nd 3

-D s

hape

s to

mak

e ac

cura

te

cons

truc

tions

on

pape

r an

d us

ing

ICT;

incl

udin

g co

nstr

uctin

g tr

iang

les

from

com

bina

tions

of s

ide

and

angl

efa

cts,

revi

ewin

g an

d ge

nera

lisin

g fin

ding

sto

iden

tify

whi

ch o

f th

ese

cond

ition

s de

fine

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 s

hape

s,

cub

oid,

regu

lar

e.g.

tetr

ahed

ron,

squ

are-

base

d py

ram

id,

tria

ngul

ar p

rism

find

sim

ple

loci

, bot

hby

reas

onin

g an

d by

us

ing

ICT,

to p

rodu

ce

shap

es a

nd p

aths

, e.g

. an

equ

ilate

ral t

riang

le

visu

alis

e an

d de

scrib

e th

e lo

cus

of a

poi

nt

that

mov

es a

ccor

ding

to

a m

ore

com

plex

ru

le; e

xpla

in th

e pa

th u

sing

acc

urat

e ge

omet

rical

voc

abul

ary

and

nota

tion

and

use

a va

riety

of m

edia

, in

clud

ing

dyna

mic

ge

omet

ry s

oftw

are,

sk

etch

es a

nd g

raph

s

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

of a

poi

nt

35

Mea

sure

s an

d m

ensu

rati

on

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

eve

ryda

yco

ntex

ts; c

onve

rt o

ne

met

ric u

nit t

o an

othe

r, g

ram

s to

kilo

gram

s; e.

g.re

ad a

nd in

terp

ret s

cale

s on

a ra

nge

of m

easu

ring

inst

rum

ents

dist

ingu

ish

betw

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

co

mm

on u

se, s

uch

as

mile

s, p

ound

s (lb

) an

d pi

nts

use

bear

ings

to s

peci

fy

dire

ctio

n

solv

e pr

oble

ms

invo

lvin

g 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,

incl

udin

g fr

om o

ther

su

bjec

ts a

nd re

al li

fe;

solv

e pr

oble

ms

invo

lvin

g ra

tes;

conv

ert b

etw

een

com

poun

d m

easu

res,

ch

oosi

ng u

nits

mos

tsu

ited

to th

e so

lutio

n

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

ffec

ton

the

accu

racy

of t

he

solu

tion,

e.g

. und

erst

and

how

err

ors

can

beco

mpo

unde

d

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

up

per a

nd

low

er b

ound

s

36

Mea

sure

s an

d m

ensu

rati

on (c

onti

nued

)4.

4 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

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 s

hape

s m

ade

from

ca

lcul

ate

area

s of

rect

angl

es

com

poun

d sh

apes

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

ndfr

ustu

ms o

f con

es

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

cub

oids

and

sh

apes

mad

e fr

om

cubo

ids

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:

t�

leng

ths

of c

ircul

ar

arcs

t�

area

s of

sec

tors

of

a ci

rcle

t�

surf

ace

area

of

a cy

linde

r

t�

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

py

ram

ids

and

cone

s;ex

plor

e co

nnec

tions

be

twee

n:

t�

form

ulae

for

the

volu

me

of a

py

ram

id a

nd th

e re

late

d cu

boid

t� fo

rmul

ae fo

rth

e su

rfac

e ar

ea

and

volu

me

of a

sp

here

and

the

circ

umsc

ribed

and

in

scrib

ed c

ubes

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

ossi

ble

answ

ers,

giv

en a

qu

estio

n th

at c

an b

ead

dres

sed

by s

tatis

tical

m

etho

ds

disc

uss

a pr

oble

m th

at

can

be a

ddre

ssed

by

stat

istic

al m

etho

ds

and

iden

tify

rela

ted

ques

tions

to e

xplo

re

sugg

est a

pro

blem

toex

plor

e us

ing

stat

istic

al

met

hods

, fra

me

ques

tions

and

rais

e co

njec

ture

s

inde

pend

ently

dev

ise

a su

itabl

e pl

an fo

r a m

ore

com

plex

sta

tistic

al

proj

ect,

sele

ctin

g su

itabl

e hy

poth

eses

to

addr

ess

the

prob

lem

eval

uate

pos

sibl

e di

ffic

ultie

s w

ith

plan

ned

appr

oach

es;

adju

st th

e pr

ojec

t pl

an a

ccor

ding

ly,

incl

udin

g re

cons

ider

ing

hypo

thes

es

deci

de w

hich

dat

a w

ould

be

rele

vant

to a

n en

quiry

and

pos

sibl

e 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 s

ourc

es;

cons

ider

app

ropr

iate

sa

mpl

e si

ze

disc

uss

how

diff

eren

t se

ts o

f dat

a re

late

to

the

prob

lem

; ide

ntify

poss

ible

prim

ary

or

seco

ndar

y so

urce

s;de

term

ine

the

sam

ple

size

and

mos

t ap

prop

riate

deg

ree

of

accu

racy

just

ify th

e sa

mpl

ing

met

hod

sele

cted

, id

entif

y po

ssib

le

sour

ces

of b

ias

and

plan

how

to m

inim

ise

it

iden

tify

prac

tical

prob

lem

s su

ch a

sno

n-re

spon

se o

r m

issi

ng d

ata

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

tinu

ed)

Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

plan

how

to c

olle

ct a

nd

orga

nise

sm

all s

ets

of

data

from

sur

veys

and

ex

perim

ents

:

t�

desi

gn d

ata

colle

ctio

n sh

eets

or q

uest

ionn

aire

s to

use

in a

sim

ple

surv

ey

t�

cons

truc

tfr

eque

ncy

tabl

es

for g

athe

ring

disc

rete

dat

a,

grou

ped

whe

reap

prop

riate

in

equ

al c

lass

in

terv

als

plan

how

to c

olle

ct

the

data

; con

stru

ct

freq

uenc

y ta

bles

with

equa

l cla

ss in

terv

als

for

gath

erin

g co

ntin

uous

da

ta a

nd tw

o-w

ay

tabl

es fo

r rec

ordi

ng

disc

rete

dat

a

desi

gn 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;

desi

gn, t

rial

and

if ne

cess

ary

refin

eda

ta c

olle

ctio

n sh

eets

;co

nstr

uct t

able

s fo

r ga

ther

ing

larg

e di

scre

tean

d co

ntin

uous

set

s of

raw

dat

a, c

hoos

ing

suita

ble

clas

s in

terv

als;

desi

gn a

nd u

se tw

o-w

ay ta

bles

gath

er d

ata

from

sp

ecifi

ed s

econ

dary

so

urce

s, in

clud

ing

prin

ted

tabl

es a

nd li

sts,

an

d IC

T-ba

sed

sour

ces,

in

clud

ing

the

inte

rnet

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 s

ourc

es:

prim

ary

(obs

erva

tion,

con

trol

led

expe

rimen

t, da

talo

ggin

g) a

nd s

econ

dary

(s

prea

dshe

et d

ata,

pr

inte

d ta

bles

, lis

ts)

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,

iden

tifyi

ng p

oten

tial

barr

iers

and

lim

itatio

ns;

iden

tify

wha

t ext

ra

info

rmat

ion

may

be

requ

ired

to p

ursu

e a

furt

her l

ine

of e

nqui

ry

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 s

trat

ified

sam

plin

g; e

xpla

in th

e ef

fect

on

relia

bilit

y

and

valid

ity

colle

ct s

mal

l set

s of

co

llect

dat

a us

ing

a da

ta fr

om s

urve

ys

suita

ble

met

hod

(e.g

. an

d ex

perim

ents

, as

obse

rvat

ion,

con

trol

led

plan

ned

expe

rimen

t, da

talo

ggin

g us

ing

ICT)

Pro

cess

ing

and

repr

esen

ting

dat

a5.

2 Year

7

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

calc

ulat

e st

atis

tics

for

smal

l set

s of

dis

cret

eda

ta:

t�

find

the

mod

e,

med

ian

and

rang

e,

and

the

mod

al

clas

s fo

r gro

uped

da

ta

t�

calc

ulat

e th

e m

ean,

incl

udin

g fr

om a

sim

ple

freq

uenc

y ta

ble,

usin

g a

calc

ulat

or

for a

larg

er n

umbe

r of

item

s

calc

ulat

e st

atis

tics

for s

ets

of d

iscr

ete

and

cont

inuo

us

data

, inc

ludi

ng w

ith

a ca

lcul

ator

and

sp

read

shee

t; re

cogn

ise

whe

n it

is a

ppro

pria

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

atis

tics

and

sele

ct th

ose

mos

tap

prop

riate

to th

e pr

oble

m o

r whi

ch

addr

ess

the

ques

tions

po

sed

use

an a

ppro

pria

te

rang

e of

sta

tistic

al

met

hods

to e

xplo

re

and

sum

mar

ise

larg

e da

ta s

ets,

just

ifyin

gth

e ch

oice

s m

ade;

in

clud

e gr

oupi

ng

data

, est

imat

ing

and

findi

ng th

e m

ean,

m

edia

n, q

uart

iles

and

inte

rqua

rtile

rang

e

proc

ess

data

dra

wn

from

pro

blem

s in

volv

ing

seas

onal

ity

and

tren

ds in

a ti

me

serie

s; ch

oose

and

co

mbi

ne s

tatis

tical

m

etho

ds to

ana

lyse

th

e pr

oble

m, i

nclu

ding

mov

ing

aver

ages

Pro

cess

ing

and

repr

esen

ting

dat

a (c

onti

nued

)5.

2 Year

7

Year

8

Year

9

Year

10

cons

truc

t, on

pap

er

and

usin

g IC

T, g

raph

san

d di

agra

ms

tore

pres

ent d

ata,

in

clud

ing:

t�

bar-

line

grap

hs

t�

freq

uenc

y di

agra

ms

for

grou

ped

disc

rete

data

t�

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:

t�

pie

char

ts fo

r ca

tego

rical

dat

a

t�

bar c

hart

s an

d fr

eque

ncy

diag

ram

s fo

r di

scre

te a

nd

cont

inuo

us d

ata

t�

sim

ple

line

grap

hs

for t

ime

serie

s

t�

sim

ple

scat

ter

grap

hs

t�

stem

-and

-leaf

di

agra

ms

sele

ct, c

onst

ruct

an

d m

odify

, on

pape

r and

usi

ng 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

:

t�

line

grap

hs fo

r tim

e se

ries

t�

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:

t�

hist

ogra

ms

for g

roup

ed

cont

inuo

us d

ata

with

equ

al c

lass

in

terv

als

t�

cum

ulat

ive

freq

uenc

y ta

bles

an

d di

agra

ms

t�

box

plot

s

t�

scat

ter g

raph

s an

d 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

cyc

le 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 in

terv

als

use

prec

ise

and

cons

iste

nt g

raph

ical

repr

esen

tatio

n to

prog

ress

an

unfa

mili

ar

and

non-

rout

ine

enqu

iry

41

Inte

rpre

ting

and

dis

cuss

ing

resu

lts

5.3

Year

8

Year

9

Year

10

Year

11

Exte

nsio

n

Year

7

inte

rpre

t dia

gram

s an

d gr

aphs

(inc

ludi

ng p

ie

char

ts) a

nd d

raw

sim

ple

conc

lusi

ons

base

d on

th

e sh

ape

of g

raph

san

d si

mpl

e st

atis

tics

for

a si

ngle

dis

trib

utio

n

com

pare

two

sim

ple

dist

ribut

ions

usi

ng th

e ra

nge

and

one

of th

em

ode,

med

ian

or m

ean

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,

gra

phs

and

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,

gr

aphs

and

dia

gram

s fo

r dis

cret

e an

d co

ntin

uous

dat

a,

rela

ting

sum

mar

y st

atis

tics

and

findi

ngs

to th

e qu

estio

ns b

eing

ex

plor

ed

com

pare

two

dist

ribut

ions

usi

ng 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

the

resu

lts o

f a

stat

istic

al e

nqui

ry u

sing

IC

T as

app

ropr

iate

;ju

stify

the

met

hods

us

ed

inte

rpre

t gra

phs

and

diag

ram

s an

d m

ake

infe

renc

es to

supp

ort

or c

ast d

oubt

on

initi

al

conj

ectu

res;

have

a

basi

c un

ders

tand

ing

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 s

tatis

tics

revi

ew in

terp

reta

tions

and

resu

lts o

f a

stat

istic

al e

nqui

ry o

n th

e ba

sis

of d

iscu

ssio

ns;

com

mun

icat

e th

ese

inte

rpre

tatio

ns a

nd

resu

lts u

sing

sel

ecte

d ta

bles

, gra

phs

and

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 fr

eque

ncy

cons

trai

nts;

incl

ude

inte

rpre

tatio

n of

di

agra

ms;

mak

e an

d in

terp

reta

tion

and

soci

al s

tatis

tics

and

disc

uss

infe

renc

es,

com

paris

on o

fev

alua

tion

of th

e us

ing

the

shap

e of

hi

stog

ram

s 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

ave

rage

(c

orre

latio

n, li

nes

of

and

spre

ad, i

nclu

ding

be

st fi

t)

med

ian

and

quar

tiles

eval

uate

the

resu

lts

criti

cally

exa

min

e us

e st

atis

tical

of a

sta

tistic

al

stra

tegi

es a

dopt

ed

anal

ysis

eff

ectiv

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

lusi

ons;

criti

cally

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

dat

ath

e lim

itatio

ns o

f any

fo

r and

app

reci

ate

to th

e qu

estio

ns b

eing

as

sum

ptio

ns a

nd th

e m

ore

eleg

ant f

orm

s ex

plor

ed

effe

cts

that

var

ying

of

com

mun

icat

ing

assu

mpt

ions

cou

ld

conc

lusi

ons

have

on

conc

lusi

ons

draw

n fr

om 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

pro

babi

lity,

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

lusi

ve

or in

depe

nden

t; us

e an

d in

terp

ret t

ree

diag

ram

s to

repr

esen

t ou

tcom

es o

f com

bine

dev

ents

and

to in

form

th

e ca

lcul

atio

n of

th

eir p

roba

bilit

ies;

deci

de w

hen

to a

dd

and

whe

n to

mul

tiply

prob

abili

ties

inte

rpre

t the

eff

ect

on p

roba

bilit

y of

co

ntex

ts in

volv

ing

sele

ctio

n w

ith a

nd

with

out r

epla

cem

ent;

choo

se a

nd c

ombi

ne

repr

esen

tatio

ns

to c

omm

unic

ate

prob

abili

ties

as p

art o

f a

solu

tion

to a

pro

blem

reco

gnis

e w

hen

and

how

to w

ork

with

prob

abili

ties

asso

ciat

edw

ith in

depe

nden

t an

d m

utua

llyex

clus

ive

even

ts w

hen

inte

rpre

ting

data

unde

rsta

nd a

nd u

se

the

prob

abili

ty s

cale

fr

om 0

to 1

; fin

d an

d ju

stify

pro

babi

litie

s ba

sed

on e

qual

ly li

kely

ou

tcom

es in

sim

ple

cont

exts

; ide

ntify

all

the

poss

ible

mut

ually

ex

clus

ive

outc

omes

of a

si

ngle

eve

nt

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

; use

di

agra

ms

and

tabl

es to

re

cord

in a

sys

tem

atic

w

ay a

ll po

ssib

le

mut

ually

exc

lusi

ve

outc

omes

for s

ingl

e ev

ents

and

for t

wo

succ

essi

ve e

vent

s

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

all

mut

ually

exc

lusi

ve

outc

omes

is 1

and

us

e th

is w

hen

solv

ing

prob

lem

s

43

Pro

babi

lity

(con

tinu

ed)

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 si

mpl

e ex

perim

ent

and

reco

rdin

g it

in

a fr

eque

ncy

tabl

e;co

mpa

re e

xper

imen

tal

and

theo

retic

al

prob

abili

ties

in s

impl

e co

ntex

ts

com

pare

est

imat

ed

expe

rimen

tal

prob

abili

ties

with

theo

retic

al

prob

abili

ties,

re

cogn

isin

g th

at:

t�

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

ffer

ent

t�

incr

easi

ng th

e nu

mbe

r of t

imes

an

exp

erim

ent i

sre

peat

ed g

ener

ally

le

ads

to b

ette

r 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

ffer

ence

be

twee

n m

athe

mat

ical

ex

plan

atio

n an

d ex

perim

enta

l evi

denc

e

use

rela

tive

freq

uenc

y as

an

estim

ate

of

prob

abili

ty, i

nclu

ding

si

mul

atio

n us

ing

ICT

to g

ener

ate

larg

er

sam

ples

; dis

cuss

its

relia

bilit

y ba

sed

on s

ampl

e si

ze a

nd

use

to in

terp

ret a

nd

com

pare

out

com

es o

f ex

perim

ents

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

sim

ulat

ion;

ju

stify

dec

isio

ns 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

the

relia

bilit

y of

co

nclu

sion

s

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