The mathematics overview and learning objectives...

Transcript of The mathematics overview and learning objectives...

3The 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

Stra

nds

Sub

-str

ands

1 M

athe

mat

ical

pro

cess

es a

nd a

pplic

atio

ns3

Alg

ebra

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

s4

Geo

met

ry a

nd m

easu

res

1.4

Inte

rpre

ting

and

eval

uatin

g4.

1 G

eom

etric

al re

ason

ing

1.5

Com

mun

icat

ing

and

refle

ctin

g4.

2 Tr

ansf

orm

atio

ns a

nd c

oord

inat

es

2 N

umbe

r4.

3 Co

nstr

uctio

n an

d lo

ci

2.1

Plac

e va

lue,

ord

erin

g an

d ro

undi

ng4.

4 M

easu

res

and

men

sura

tion

2.2

Inte

gers

, pow

ers

and

root

s5

Sta

tist

ics

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

Proc

essi

ng a

nd re

pres

entin

g da

ta

2.5

Men

tal c

alcu

latio

n m

etho

ds5.

3 In

terp

retin

g an

d di

scus

sing

resu

lts

2.6

Writ

ten

calc

ulat

ion

met

hods

5.4

Prob

abili

ty

2.7

Calc

ulat

or m

etho

ds

2.8

Chec

king

resu

lts

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:

•in

crea

sing

the

com

plex

ity

of th

e ap

plic

atio

n, e

.g. n

on-r

outin

e, m

ulti-

step

pro

blem

s, e

xten

ded

enqu

iries

•re

duci

ng th

e fa

mili

arit

y of

the

cont

ext,

e.g.

new

con

text

s in

mat

hem

atic

s, c

onte

xts

draw

n fr

om o

ther

subj

ects

, oth

er a

spec

ts o

f pup

ils’ l

ives

•in

crea

sing

the

tech

nica

l dem

and

of th

e m

athe

mat

ics

requ

ired,

e.g

. mor

e ad

vanc

ed c

once

pts,

mor

e di

ffic

ult p

roce

dure

s

•in

crea

sing

the

degr

ee o

f ind

epen

denc

e an

d au

tono

my

in p

robl

em-s

olvi

ng a

nd in

vest

igat

ion

1.1

Repr

esen

ting

Year

7Ye

ar 8

Year

9

Year

10

Year

11

Exte

nsio

n

iden

tify

the

nece

ssar

y in

form

atio

n to

un

ders

tand

or s

impl

ify

a co

ntex

t or p

robl

em;

repr

esen

t pro

blem

s,

mak

ing

corr

ect u

se

of s

ymbo

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,

in

clud

ing

ICT

iden

tify

the

mat

hem

atic

al fe

atur

es

of a

con

text

or

prob

lem

; try

out

and

co

mpa

re m

athe

mat

ical

re

pres

enta

tions

; sel

ect

appr

opria

te p

roce

dure

s an

d to

ols,

incl

udin

g IC

T

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

e re

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

al

tech

niqu

es, t

he m

ost

effic

ient

for a

naly

sis

and

mos

t eff

ectiv

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

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

1.2

Ana

lysi

ng –

use

mat

hem

atic

al re

ason

ing

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

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

w

sim

ple

conc

lusi

ons

and

expl

ain

reas

onin

g;

unde

rsta

nd th

e si

gnifi

canc

e of

a

coun

ter-

exam

ple;

ta

ke a

ccou

nt o

f fe

edba

ck a

nd le

arn

from

mis

take

s

visu

alis

e an

d m

anip

ulat

e dy

nam

ic

imag

es; c

onje

ctur

e an

d ge

nera

lise;

mov

e be

twee

n th

e ge

nera

l an

d th

e pa

rtic

ular

to

test

the

logi

c of

an

argu

men

t; id

entif

y ex

cept

iona

l cas

es o

r co

unte

r-ex

ampl

es;

mak

e co

nnec

tions

with

re

late

d co

ntex

ts

use

conn

ectio

ns w

ith

rela

ted

cont

exts

to

impr

ove

the

anal

ysis

of

a si

tuat

ion

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 s

trat

egie

s an

d ap

prec

iate

that

mor

e th

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

e ar

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

y ex

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

nd

gene

ralis

atio

ns; s

uppo

rt

assu

mpt

ions

by

clea

r ar

gum

ent a

nd fo

llow

th

roug

h a

sust

aine

d ch

ain

of re

ason

ing,

in

clud

ing

proo

f

pres

ent r

igor

ous

and

sust

aine

d ar

gum

ents

; re

ason

indu

ctiv

ely,

de

duce

and

pro

ve;

expl

ain

and

just

ify

assu

mpt

ions

and

co

nstr

aint

s

1.3

Ana

lysi

ng –

use

app

ropr

iate

mat

hem

atic

al p

roce

dure

sW

ithin

the

appr

opria

te ra

nge

and

cont

ent:

mak

e ac

cura

te m

athe

mat

ical

dia

gram

s, g

raph

s an

d co

nstr

uctio

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

t sy

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

1.4

Inte

rpre

ting

and

eva

luat

ing

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

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

ve

accu

rate

sol

utio

ns

appr

opria

te to

the

cont

ext o

r pro

blem

; ev

alua

te th

e ef

ficie

ncy

of a

ltern

ativ

e st

rate

gies

an

d ap

proa

ches

just

ify th

e m

athe

mat

ical

feat

ures

dr

awn

from

a c

onte

xt

and

the

choi

ce o

f ap

proa

ch; g

ener

ate

fulle

r sol

utio

ns b

y pr

esen

ting

a co

ncis

e,

reas

oned

arg

umen

t us

ing

sym

bols

, di

agra

ms,

gra

phs

and

rela

ted

expl

anat

ions

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

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

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,

pr

oces

s, re

sults

and

co

nclu

sion

s

1.5

Com

mun

icat

ing

and

refl

ecti

ng

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

com

mun

icat

e ow

n fin

ding

s ef

fect

ivel

y,

oral

ly a

nd in

writ

ing,

an

d di

scus

s an

d co

mpa

re a

ppro

ache

s an

d re

sults

with

oth

ers;

reco

gnis

e eq

uiva

lent

ap

proa

ches

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

s si

tuat

ions

revi

ew a

nd re

fine

own

findi

ngs

and

appr

oach

es o

n th

e ba

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

use

a ra

nge

of fo

rms

to c

omm

unic

ate

findi

ngs

effe

ctiv

ely

to

diff

eren

t aud

ienc

es;

revi

ew fi

ndin

gs a

nd

look

for e

quiv

alen

ce to

di

ffer

ent p

robl

ems

with

si

mila

r str

uctu

re

rout

inel

y re

view

and

re

fine

findi

ngs

and

appr

oach

es; i

dent

ify

how

oth

er c

onte

xts

wer

e di

ffer

ent f

rom

, or

sim

ilar t

o, th

e cu

rren

t si

tuat

ion

and

expl

ain

how

and

why

the

sam

e or

diff

eren

t str

ateg

ies

wer

e us

ed

use

mat

hem

atic

al

lang

uage

and

sy

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

h fo

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

2 N

umbe

r

2.1

Plac

e va

lue,

ord

erin

g an

d ro

undi

ng

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

rs

and

deci

mal

s by

10,

10

0, 1

000,

and

exp

lain

th

e ef

fect

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

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

, 1 10

and

10–1

; mul

tiply

and

di

vide

by

any

inte

ger

pow

er o

f 10

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

ed

and

choi

ce o

f acc

urac

y in

rela

tion

to th

e pr

oble

m a

nd a

udie

nce

for t

he s

olut

ion

enga

ge in

m

athe

mat

ical

task

s w

here

usi

ng n

umbe

rs

in s

tand

ard

form

is

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

com

pare

and

ord

er

deci

mal

s in

diff

eren

t co

ntex

ts; k

now

that

w

hen

com

parin

g m

easu

rem

ents

the

units

mus

t be

the

sam

e

orde

r dec

imal

s

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

roun

d po

sitiv

e nu

mbe

rs

to a

ny g

iven

pow

er

of 1

0; ro

und

deci

mal

s to

the

near

est w

hole

nu

mbe

r or t

o on

e or

tw

o de

cim

al p

lace

s

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

2.2

Inte

gers

, pow

ers

and

root

s

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

ct

inte

gers

in c

onte

xt

add,

subt

ract

, mul

tiply

an

d di

vide

inte

gers

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

is

rais

ing

the

resu

lt of

this

op

erat

ion

to p

ower

1 n

solv

e a

prob

lem

us

ing

ratio

nal a

nd

irrat

iona

l num

bers

, in

clud

ing

surd

s

reco

gnis

e an

d us

e m

ultip

les,

fact

ors,

prim

es (l

ess t

han

100)

, co

mm

on fa

ctor

s, hi

ghes

t com

mon

fa

ctor

s and

low

est

com

mon

mul

tiple

s in

sim

ple

case

s; us

e si

mpl

e te

sts o

f div

isibi

lity

use

mul

tiple

s, fa

ctor

s,

com

mon

fact

ors,

hi

ghes

t com

mon

fa

ctor

s, lo

wes

t co

mm

on m

ultip

les

and

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

the

prim

e fa

ctor

de

com

posi

tion

of

a nu

mbe

r

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

t le

ast 1

2 ×

12 a

nd th

e co

rres

pond

ing

root

s

use

squa

res,

pos

itive

an

d ne

gativ

e sq

uare

ro

ots,

cub

es a

nd

cube

root

s, a

nd in

dex

nota

tion

for s

mal

l po

sitiv

e in

tege

r pow

ers

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

2.3

Frac

tion

s, d

ecim

als,

per

cent

ages

, rat

io a

nd p

ropo

rtio

n

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

. 0.

23 =

23 10

0; u

se

diag

ram

s to

com

pare

tw

o or

mor

e si

mpl

e fr

actio

ns

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

to

just

ify a

nd p

rove

som

e of

its

feat

ures

, e.g

. tha

t al

l rec

urrin

g de

cim

als

can

be e

xpre

ssed

as

a

frac

tion

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

si

mpl

e fr

actio

ns a

nd

thos

e w

ith c

omm

on

deno

min

ator

s; ca

lcul

ate

sim

ple

frac

tions

of q

uant

ities

an

d m

easu

rem

ents

(w

hole

-num

ber

answ

ers)

; mul

tiply

a

frac

tion

by a

n in

tege

r

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

to

add,

subt

ract

, mul

tiply

an

d di

vide

frac

tions

, in

terp

retin

g di

visi

on a

s a

mul

tiplic

ativ

e in

vers

e;

canc

el c

omm

on fa

ctor

s be

fore

mul

tiply

ing

or

div

idin

g

unde

rsta

nd a

pply

ef

ficie

nt m

etho

ds to

ad

d, su

btra

ct, m

ultip

ly

and

divi

de fr

actio

ns,

inte

rpre

ting

reci

proc

als

as m

ultip

licat

ive

inve

rses

2.3

Frac

tion

s, d

ecim

als,

per

cent

ages

, rat

io a

nd p

ropo

rtio

n (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

inte

rpre

t per

cent

age

as th

e op

erat

or ‘s

o m

any

hund

redt

hs

of’ a

nd e

xpre

ss o

ne

give

n nu

mbe

r as

a pe

rcen

tage

of a

noth

er;

calc

ulat

e pe

rcen

tage

s an

d fin

d th

e ou

tcom

e of

a g

iven

per

cent

age

incr

ease

or d

ecre

ase

reco

gnis

e w

hen

frac

tions

or

perc

enta

ges

are

need

ed to

com

pare

pr

opor

tions

; sol

ve

prob

lem

s in

volv

ing

perc

enta

ge c

hang

es

reco

gnis

e th

e eq

uiva

lenc

e of

pe

rcen

tage

s, fr

actio

ns

and

deci

mal

s

use

the

equi

vale

nce

of fr

actio

ns, d

ecim

als

and

perc

enta

ges

to

com

pare

pro

port

ions

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

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

ify

ratio

s, in

clud

ing

thos

e ex

pres

sed

in d

iffer

ent

units

, rec

ogni

sing

link

s w

ith fr

actio

n no

tatio

n;

divi

de a

qua

ntity

into

tw

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

s in

volv

ing

ratio

and

di

rect

pro

port

ion

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

sta

tistic

s in

volv

es

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

y in

dire

ct p

ropo

rtio

n,

incl

udin

g re

peat

ed

prop

ortio

nal c

hang

e,

e.g.

gro

wth

/dec

ay; u

se

alge

brai

c m

etho

ds

whe

re a

ppro

pria

te a

nd

cons

ider

lim

itatio

ns o

f th

e m

odel

(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)

2.4

Num

ber o

pera

tion

s

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

unde

rsta

nd a

nd u

se th

e ru

les

of a

rithm

etic

and

in

vers

e op

erat

ions

in

the

cont

ext o

f pos

itive

in

tege

rs a

nd d

ecim

als

unde

rsta

nd a

nd u

se th

e ru

les

of a

rithm

etic

and

in

vers

e op

erat

ions

in

the

cont

ext o

f int

eger

s an

d fr

actio

ns

unde

rsta

nd th

e ef

fect

s of

mul

tiply

ing

and

divi

ding

by

num

bers

be

twee

n 0

and

1;

cons

olid

ate

use

of th

e ru

les

of a

rithm

etic

and

in

vers

e op

erat

ions

reco

gnis

e an

d us

e re

cipr

ocal

s as

a

mul

tiplic

ativ

e in

vers

e in

con

text

s su

ch a

s en

larg

emen

t; ex

plor

e th

e be

havi

our o

f the

re

cipr

ocal

func

tion

(y

= 1

/x) f

or la

rge

and

smal

l val

ues

of x

mod

el re

al c

onte

xts

whe

re q

uant

ities

var

y in

dire

ct p

ropo

rtio

n,

incl

udin

g re

peat

ed

prop

ortio

nal c

hang

e,

e.g.

gro

wth

/dec

ay; u

se

alge

brai

c m

etho

ds

whe

re a

ppro

pria

te a

nd

cons

ider

lim

itatio

ns o

f th

e m

odel

(as i

n 2.

3)

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.

3)

use

the

orde

r of

oper

atio

ns, i

nclu

ding

br

acke

ts

use

the

orde

r of

oper

atio

ns, i

nclu

ding

br

acke

ts, w

ith m

ore

com

plex

cal

cula

tions

unde

rsta

nd th

e or

der

of p

rece

denc

e of

op

erat

ions

, inc

ludi

ng

pow

ers

2.5

Men

tal c

alcu

lati

on m

etho

ds

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

reca

ll nu

mbe

r fa

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

ed

divi

sion

fact

s

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

in

volv

ing

num

bers

su

ch a

s 0.

7 an

d 6,

and

0.

03 a

nd 8

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

ac

cura

tely

with

re

cipr

ocal

s, p

ower

s,

trig

onom

etric

al

func

tions

and

num

bers

in

sta

ndar

d fo

rm

(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.

m

enta

l, w

ritte

n, IC

T or

cal

cula

tor t

o so

lve

prob

lem

s

(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

ed

usin

g su

rds

and

π, ra

tiona

lisin

g a

deno

min

ator

whe

re

appr

opria

te, e

.g.

a tr

igon

omet

rical

so

lutio

n

(as i

n 2.

6)

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

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

e ro

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

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

mak

e an

d ju

stify

es

timat

es a

nd

appr

oxim

atio

ns o

f ca

lcul

atio

ns

mak

e an

d ju

stify

es

timat

es a

nd

appr

oxim

atio

ns o

f ca

lcul

atio

ns

exam

ine

and

refin

e es

timat

es a

nd

appr

oxim

atio

ns o

f ca

lcul

atio

ns in

volv

ing

roun

ding

2.6

Wri

tten

cal

cula

tion

met

hods

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

effic

ient

writ

ten

met

hods

to a

dd

and

subt

ract

who

le

num

bers

and

dec

imal

s w

ith u

p to

two

plac

es

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

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

ac

cura

tely

with

re

cipr

ocal

s, p

ower

s,

trig

onom

etric

al

func

tions

and

num

bers

in

sta

ndar

d fo

rm

(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.

m

enta

l, w

ritte

n, IC

T or

cal

cula

tor t

o so

lve

prob

lem

s

(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

ed

usin

g su

rds

and

π, ra

tiona

lisin

g a

deno

min

ator

whe

re

appr

opria

te, e

.g.

a tr

igon

omet

rical

so

lutio

n

(as i

n 2.

5)

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

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

or

0.06

; und

erst

and

whe

re

to p

ositi

on th

e de

cim

al

poin

t by

cons

ider

ing

equi

vale

nt c

alcu

latio

ns

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

2.7

Calc

ulat

or m

etho

ds

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

carr

y ou

t cal

cula

tions

w

ith m

ore

than

one

st

ep u

sing

bra

cket

s an

d th

e m

emor

y; u

se th

e sq

uare

root

and

sig

n ch

ange

key

s

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

pe

rfor

m c

ompl

ex

calc

ulat

ions

with

nu

mbe

rs o

f any

siz

e,

know

ing

not t

o ro

und

durin

g in

term

edia

te

step

s of

a c

alcu

latio

n;

use

the

cons

tant

, 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

ac

cura

tely

with

re

cipr

ocal

s, p

ower

s,

trig

onom

etric

al

func

tions

and

num

bers

in

sta

ndar

d fo

rm

(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:

•ca

lcul

atin

g (in

clud

ing

calc

ulat

ing

devi

ces)

•ch

ecki

ng

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

(dec

imal

s, p

erce

ntag

es,

mon

ey, m

etric

m

easu

res)

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)

2.8

Chec

king

resu

lts

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

de

and

by w

orki

ng

prob

lem

s ba

ckw

ards

sele

ct fr

om a

rang

e of

che

ckin

g m

etho

ds,

incl

udin

g es

timat

ing

in c

onte

xt a

nd u

sing

in

vers

e op

erat

ions

chec

k re

sults

usi

ng

appr

opria

te m

etho

dsid

entif

y a

rang

e of

ch

ecki

ng s

trat

egie

s an

d ap

prec

iate

that

mor

e th

an o

ne w

ay m

ay

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:

•ca

lcul

atin

g (in

clud

ing

calc

ulat

ing

devi

ces)

•ch

ecki

ng

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)

3 A

lgeb

ra

3.1

Equa

tion

s, fo

rmul

ae, e

xpre

ssio

ns a

nd id

enti

ties

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

lett

er s

ymbo

ls to

re

pres

ent u

nkno

wn

num

bers

or v

aria

bles

; kn

ow th

e m

eani

ngs

of th

e w

ords

term

, ex

pres

sion

and

equa

tion

reco

gnis

e th

at le

tter

sy

mbo

ls p

lay

diff

eren

t ro

les

in e

quat

ions

, fo

rmul

ae a

nd fu

nctio

ns;

know

the

mea

ning

s of

the

wor

ds fo

rmul

a an

d fu

nctio

n

dist

ingu

ish

the

diff

eren

t rol

es p

laye

d by

lett

er s

ymbo

ls in

eq

uatio

ns, i

dent

ities

, fo

rmul

ae a

nd fu

nctio

ns

pres

ent c

onvi

ncin

g al

gebr

aic

argu

men

ts to

ju

stify

gen

eral

isat

ions

or

sol

utio

ns; r

elat

e ar

gum

ents

to th

e st

ruct

ure

of th

e co

ntex

t or

pro

blem

; pro

duce

si

mpl

e pr

oofs

exam

ine

and

refin

e al

gebr

aic

argu

men

ts

pres

ente

d to

exp

lain

ge

omet

rical

and

nu

mer

ical

pro

pert

ies;

choo

se a

nd c

ombi

ne

repr

esen

tatio

ns to

pr

esen

t a c

onvi

ncin

g pr

oof

use

sym

bols

and

re

pres

enta

tions

co

nsis

tent

ly to

pre

sent

a

form

al p

roof

, e.g

. de

rivin

g th

e fo

rmul

a fo

r sol

ving

qua

drat

ic

equa

tions

unde

rsta

nd th

at

alge

brai

c op

erat

ions

fo

llow

the

rule

s of

ar

ithm

etic

unde

rsta

nd th

at

alge

brai

c op

erat

ions

, in

clud

ing

the

use

of

brac

kets

, fol

low

the

rule

s of

arit

hmet

ic; u

se

inde

x no

tatio

n fo

r sm

all

posi

tive

inte

ger p

ower

s

use

inde

x no

tatio

n fo

r in

tege

r pow

ers

and

sim

ple

inst

ance

s of

th

e in

dex

law

s

use

alge

brai

c re

pres

enta

tion

to

synt

hesi

se k

now

n ru

les

of a

rithm

etic

, inc

ludi

ng

the

com

mut

ativ

e an

d di

strib

utiv

e la

ws;

just

ify

thes

e ge

nera

lisat

ions

, e.

g. u

sing

spa

tial

repr

esen

tatio

ns; u

se

alge

brai

c ar

gum

ent t

o ge

nera

lise

the

inde

x la

ws

for m

ultip

licat

ion

and

divi

sion

to in

clud

e ze

ro, n

egat

ive

and

frac

tiona

l pow

ers

appr

ecia

te th

e ge

nera

lity

of th

e fo

rms

a +

b =

c an

d ab

= c

, w

here

eac

h te

rm c

an

itsel

f be

an e

xpre

ssio

n;

use

this

insi

ght i

nto

stru

ctur

e to

dev

elop

flu

ency

in tr

ansf

orm

ing

mor

e co

mpl

ex

equa

tions

3.1

Equa

tion

s, fo

rmul

ae, e

xpre

ssio

ns a

nd id

enti

ties

(con

tinu

ed)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

th

e pr

oduc

t of t

wo

linea

r exp

ress

ions

of

the

form

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,

e.g.

by

fact

oris

ing

and

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

y tr

ansf

orm

atio

n

3.1

Equa

tion

s, fo

rmul

ae, e

xpre

ssio

ns a

nd id

enti

ties

(con

tinu

ed)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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 (e

.g. i

nver

se o

pera

tions

)

cons

truc

t and

sol

ve

linea

r equ

atio

ns w

ith

inte

ger c

oeff

icie

nts

(unk

now

n on

eith

er o

r bo

th s

ides

, with

out a

nd

with

bra

cket

s) u

sing

ap

prop

riate

met

hods

(e

.g. i

nver

se o

pera

tions

, tr

ansf

orm

ing

both

si

des

in s

ame

way

)

cons

truc

t and

sol

ve

linea

r equ

atio

ns w

ith

inte

ger c

oeff

icie

nts

(with

and

with

out

brac

kets

, neg

ativ

e si

gns

anyw

here

in th

e eq

uatio

n, p

ositi

ve o

r ne

gativ

e so

lutio

n)

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

al

prob

lem

s; so

lve

linea

r equ

atio

ns

and

ineq

ualit

ies,

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

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

nd

impr

ovem

ent m

etho

ds;

just

ify th

e nu

mbe

r of

sol

utio

ns u

sing

al

gebr

aic

or g

raph

ical

ar

gum

ents

and

sel

ect

appr

opria

te s

olut

ions

, in

terp

retin

g th

eir

accu

racy

repr

esen

t rea

l-lif

e si

tuat

ions

or

mat

hem

atic

al p

robl

ems

invo

lvin

g:

•m

ore

com

plex

qu

adra

tic

equa

tions

, ch

oosi

ng a

n ap

prop

riate

m

etho

d of

so

lutio

n in

clud

ing

com

plet

ing

the

squa

re a

nd u

se o

f th

e fo

rmul

a

•di

rect

or i

nver

se

prop

ortio

n,

incl

udin

g 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

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

pr

opor

tion

use

alge

brai

c m

etho

ds

to s

olve

pro

blem

s in

volv

ing

dire

ct

prop

ortio

n; re

late

al

gebr

aic

solu

tions

to

gra

phs

of th

e eq

uatio

ns; u

se IC

T

as a

ppro

pria

te

3.1

Equa

tion

s, fo

rmul

ae, e

xpre

ssio

ns a

nd id

enti

ties

(con

tinu

ed)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

x2 +

x =

20

(See

obj

ectiv

e ab

ove

fo

r pro

gres

sion)

(See

obj

ectiv

e ab

ove

fo

r pro

gres

sion)

(See

obj

ectiv

e ab

ove

fo

r pro

gres

sion)

expl

ore

way

s of

cons

truc

ting

mod

els

of re

al-li

fe si

tuat

ions

by

dra

win

g gr

aphs

an

d co

nstr

uctin

g al

gebr

aic

equa

tions

and

in

equa

litie

s

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 si

tuat

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

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 s

ome

case

s ha

ve n

o co

mm

on

solu

tion

and

othe

rs

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

pr

oble

ms;

solv

e th

e in

equa

litie

s gr

aphi

cally

, ide

ntify

ing

and

inte

rpre

ting

the

solu

tion

set i

n th

e co

ntex

t of t

he p

robl

em

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

x2 +

y2 =

r2

3.1

Equa

tion

s, fo

rmul

ae, e

xpre

ssio

ns a

nd id

enti

ties

(con

tinu

ed)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

in

tege

rs in

to s

impl

e fo

rmul

ae, i

nclu

ding

ex

ampl

es th

at le

ad

to a

n eq

uatio

n to

so

lve;

subs

titut

e po

sitiv

e in

tege

rs in

to

expr

essi

ons

invo

lvin

g sm

all p

ower

s, e

.g.

3x2 +

4 o

r 2x3 ; d

eriv

e si

mpl

e fo

rmul

ae

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

dr

awn

from

real

-life

co

ntex

ts a

nd o

ther

su

bjec

ts, r

elat

ing

the

varia

bles

to th

e co

ntex

t and

des

crib

ing

thei

r beh

avio

ur;

solv

e pr

oble

ms

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

3.2

Sequ

ence

s, fu

ncti

ons

and

grap

hs

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

desc

ribe

inte

ger

sequ

ence

s; ge

nera

te

term

s of

a s

impl

e se

quen

ce, g

iven

a

rule

(e.g

. fin

ding

a

term

from

the

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

linea

r seq

uenc

e us

ing

term

-to-

term

an

d po

sitio

n-to

-ter

m

rule

s, o

n pa

per a

nd

usin

g a

spre

adsh

eet o

r gr

aphi

cs c

alcu

lato

r

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

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

ar

num

bers

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

use

linea

r exp

ress

ions

to

des

crib

e th

e nt

h te

rm

of a

sim

ple

arith

met

ic

sequ

ence

, jus

tifyi

ng

its fo

rm b

y re

ferr

ing

to

the

activ

ity o

r pra

ctic

al

cont

ext f

rom

whi

ch it

w

as g

ener

ated

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

3.2

Sequ

ence

s, fu

ncti

ons

and

grap

hs (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

expr

ess

sim

ple

func

tions

in w

ords

, th

en u

sing

sym

bols

; re

pres

ent t

hem

in

map

ping

s

expr

ess

sim

ple

func

tions

alg

ebra

ical

ly

and

repr

esen

t the

m

in m

appi

ngs

or o

n a

spre

adsh

eet

find

the

inve

rse

of a

lin

ear f

unct

ion

com

pare

gra

phic

al,

alge

brai

c an

d ge

omet

rical

re

pres

enta

tions

, in

clud

ing

map

ping

di

agra

ms,

to e

xpla

in

the

effe

ct o

f:

•ro

tatin

g th

e lin

e

y =

mx

+ c

thro

ugh

90° a

bout

any

po

int

•re

flect

ing

the

line

y

= 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

pairs

that

sat

isfy

a

sim

ple

linea

r rul

e; p

lot

the

grap

hs o

f sim

ple

linea

r fun

ctio

ns, w

here

y

is g

iven

exp

licitl

y in

term

s of

x, o

n pa

per a

nd u

sing

ICT;

re

cogn

ise

stra

ight

-line

gr

aphs

par

alle

l to

the

x-

axis

or

y-ax

is

gene

rate

poi

nts

in a

ll fo

ur q

uadr

ants

and

plo

t th

e gr

aphs

of l

inea

r fu

nctio

ns, w

here

y is

gi

ven

expl

icitl

y in

term

s of

x, o

n pa

per a

nd

usin

g IC

T; re

cogn

ise

that

equ

atio

ns o

f th

e fo

rm y

= m

x +

c co

rres

pond

to s

trai

ght-

line

grap

hs

gene

rate

poi

nts a

nd

plot

gra

phs o

f lin

ear

func

tions

, whe

re y

is

give

n im

plic

itly

in te

rms

of x

(e.g

. ay

+ bx

= 0

, y

+ bx

+ c

= 0

), on

pap

er

and

usin

g IC

T; fi

nd th

e gr

adie

nt o

f lin

es g

iven

by

equ

atio

ns o

f the

fo

rm y

= m

x +

c, gi

ven

valu

es fo

r m a

nd c

expl

ore

grap

hs o

f fu

nctio

ns o

f the

form

y

= xn (n

an

inte

ger)

and

reco

gnis

e th

eir

char

acte

ristic

sha

pes;

vary

the

valu

es o

f a, b

an

d c

in fu

nctio

ns su

ch

as y

= a

x2 + c

, y

= ax

3 + c

, y

= (x

+ b

)2 usi

ng a

gr

aph

plot

ter t

o ex

plai

n ho

w th

is tr

ansf

orm

s

the

grap

h

expl

ore

conn

ectio

ns

betw

een

the

form

of

the

equa

tion

and

the

resu

lting

gra

phs

of

quad

ratic

and

cub

ic

func

tions

such

as:

•y

= (x

+ 2

)(x –

5)

•y

= (x

– 2

)(x2 +

7x +

12)

•y

= x2 –

2x

+ 1

•y

= x3 +

3

incl

ude

feat

ures

suc

h

as ro

ots

of th

e eq

uatio

n,

inte

rcep

ts a

nd tu

rnin

g po

ints

expl

ore

grap

hs o

f ex

pone

ntia

l and

tr

igon

omet

rical

fu

nctio

ns a

nd re

cogn

ise

thei

r cha

ract

eris

tic

shap

es; a

pply

to th

e gr

aph

y =

f(x) t

he

tran

sfor

mat

ions

y

= f(

x) +

a, y

= a

f(x),

y =

f(x +

a),

y =

f(ax)

for

linea

r, qu

adra

tic, s

ine

and

cosi

ne fu

nctio

ns;

use

a gr

aph

plot

ter t

o ex

plai

n th

e ef

fect

of

tran

sfor

mat

ions

on

the

grap

h an

d ge

nera

lise

to

oth

er fu

nctio

ns

3.2

Sequ

ence

s, fu

ncti

ons

and

grap

hs (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

plot

and

inte

rpre

t the

gr

aphs

of s

impl

e lin

ear

func

tions

aris

ing

from

re

al-li

fe s

ituat

ions

, e.g

. co

nver

sion

gra

phs

cons

truc

t lin

ear

func

tions

aris

ing

from

re

al-li

fe p

robl

ems

and

plot

thei

r co

rres

pond

ing

grap

hs;

disc

uss

and

inte

rpre

t gr

aphs

aris

ing

from

re

al s

ituat

ions

, e.g

. di

stan

ce–t

ime

grap

hs

cons

truc

t fun

ctio

ns

aris

ing

from

real

-life

pr

oble

ms

and

plot

thei

r co

rres

pond

ing

grap

hs;

inte

rpre

t gra

phs

aris

ing

from

real

situ

atio

ns, e

.g.

time

serie

s gr

aphs

sket

ch a

nd in

terp

ret

grap

hs th

at m

odel

real

-lif

e si

tuat

ions

, inc

ludi

ng

thos

e ge

nera

ted

from

oth

er su

bjec

ts

such

as

scie

nce;

us

e m

athe

mat

ical

ar

gum

ent t

o ju

stify

fe

atur

es o

f the

ir sh

apes

appl

y kn

owle

dge

of

mat

hem

atic

al fu

nctio

ns

to p

robl

ems

invo

lvin

g:

•op

timis

atio

n,

usin

g nu

mer

ical

, al

gebr

aic

and

grap

hica

l, te

chni

ques

, in

clud

ing

max

ima

and

min

ima

•us

ing

ICT

to fi

t a

curv

e to

dat

a fr

om a

real

con

text

su

ch a

s a

scie

nce

expe

rimen

t

•re

peat

ed

prop

ortio

nal

chan

ge, e

.g.

com

poun

d in

tere

st

set u

p a

mat

hem

atic

al

mod

el o

f a re

al-li

fe

cont

ext o

r pro

blem

, id

entif

ying

the

varia

bles

and

thei

r fu

nctio

nal r

elat

ions

hip;

us

e gr

aphs

and

sk

etch

es to

exp

lain

th

e be

havi

our o

f the

va

riabl

es a

nd to

exp

lain

or

just

ify th

e ef

fect

of

assu

mpt

ions

in

the

mod

el

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,

e.g.

par

alle

l and

pe

rpen

dicu

lar l

ines

3.2

Sequ

ence

s, fu

ncti

ons

and

grap

hs (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

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

in

ters

ectio

ns, e

.g. s

olvi

ng

simul

tane

ous l

inea

r eq

uatio

ns

4 G

eom

etry

and

mea

sure

s

4.1

Geo

met

rica

l rea

soni

ng

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

corr

ectly

the

voca

bula

ry, n

otat

ion

and

labe

lling

co

nven

tions

for l

ines

, an

gles

and

sha

pes

dist

ingu

ish

betw

een

conv

entio

ns,

defin

ition

s an

d de

rived

pr

oper

ties

exam

ine

and

refin

e ar

gum

ents

in

solu

tions

to

geom

etric

al p

robl

ems,

dist

ingu

ishin

g be

twee

n pr

actic

al d

emon

stra

tion

and

proo

f; pr

oduc

e si

mpl

e pr

oofs

exam

ine

and

crea

te

chai

ns o

f ded

uctiv

e re

ason

ing

in s

olut

ions

to

mor

e co

mpl

ex

geom

etric

al p

robl

ems

pres

ent r

igor

ous

and

sust

aine

d ar

gum

ents

in

the

solu

tion

of

geom

etric

al p

robl

ems;

cons

truc

t for

mal

ge

omet

rical

pro

ofs

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

co

rres

pond

ing

angl

es;

unde

rsta

nd a

pro

of

that

:

•th

e an

gle

sum

of a

tr

iang

le is

180

° and

of

a q

uadr

ilate

ral

is 3

60°

•th

e ex

terio

r ang

le

of a

tria

ngle

is

equa

l to

the

sum

of

the

two

inte

rior

oppo

site

ang

les

expl

ain

how

to fi

nd,

calc

ulat

e an

d us

e:

•th

e su

ms

of th

e in

terio

r and

ex

terio

r ang

les

of q

uadr

ilate

rals

, pe

ntag

ons

and

hexa

gons

•th

e in

terio

r and

ex

terio

r ang

les

of

regu

lar p

olyg

ons

use

dyna

mic

imag

es to

de

mon

stra

te in

varia

nt

rela

tions

hips

bet

wee

n ra

dii,

chor

ds a

nd

tang

ents

in c

ircle

s; de

velo

p ar

gum

ents

to

exp

lain

and

just

ify

sim

ple

circ

le p

rope

rtie

s an

d th

eore

ms

exam

ine

and

crea

te

proo

fs o

f the

circ

le

theo

rem

s; us

e ci

rcle

th

eore

ms

to s

olve

pr

oble

ms

exam

ine

the

poin

ts

and

lines

use

d to

cre

ate

stan

dard

con

stru

ctio

ns

and

use

the

cond

ition

s of

con

grue

nce

to

pres

ent a

pro

of th

at th

e st

anda

rd c

onst

ruct

ions

ar

e ex

act

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

4.1

Geo

met

rica

l rea

soni

ng (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

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

r po

lygo

ns, j

ustif

ying

in

fere

nces

and

ex

plai

ning

reas

onin

g w

ith d

iagr

ams

and

text

solv

e ge

omet

rical

pr

oble

ms

usin

g pr

oper

ties

of li

nes,

an

gles

, pol

ygon

s an

d ci

rcle

s; ju

stify

ar

gum

ents

and

so

lutio

ns u

sing

de

duct

ive

reas

onin

g

form

alis

e ex

istin

g kn

owle

dge

of li

nes,

angl

es a

nd p

olyg

ons b

y:

•us

ing

the

cong

ruen

ce

cond

ition

s (SS

S, S

AS,

RH

S, A

SA) t

o de

duce

fa

mili

ar p

rope

rtie

s of

tria

ngle

s and

qu

adril

ater

als,

e.g.

an

isos

cele

s tria

ngle

ha

s tw

o eq

ual

angl

es

•ex

plai

ning

w

hy st

anda

rd

cons

truc

tions

wor

k,

e.g.

obs

ervi

ng th

at

lines

join

ing

poin

ts

whe

re c

ompa

ss a

rcs

mee

t are

side

s of a

rh

ombu

s

(see

obj

ectiv

e ab

ove

for p

rogr

essio

n)

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

unde

rsta

nd

cong

ruen

ce a

nd

expl

ore

sim

ilarit

y

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

pr

oble

ms

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

visu

alis

e an

d m

anip

ulat

e dy

nam

ic im

ages

and

us

e sc

ale

draw

ing

to

inve

stig

ate

area

s of

squa

res o

n si

des o

f rig

ht-a

ngle

d an

d no

n rig

ht-a

ngle

d tr

iang

les,

rela

ting

findi

ngs t

o Py

thag

oras

’ the

orem

; us

e Py

thag

oras

’ the

orem

to

solv

e pr

oble

ms i

n 2-

D

and

sim

ple

3-D

cas

es

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

4.1

Geo

met

rica

l rea

soni

ng (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

2-D

repr

esen

tatio

ns

to v

isua

lise

3-D

sha

pes

and

dedu

ce s

ome

of

thei

r pro

pert

ies

visu

alis

e 3-

D s

hape

s fr

om th

eir n

ets;

use

geom

etric

al 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

re

latio

nshi

ps b

y:

•ge

nera

ting

tria

ngle

s us

ing

a ro

tatin

g un

it ra

dius

(c

ircle

, cen

tre

the

orig

in)

•id

entif

ying

the

prop

ertie

s of

si

mila

r tria

ngle

s fo

rmed

by

enla

rgem

ents

of

the

circ

le

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

he

area

of a

tria

ngle

; us

e tr

igon

omet

rical

re

latio

nshi

ps to

so

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

draw

, ske

tch

and

com

pare

the

grap

hs

of tr

igon

omet

rical

fu

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

4.2

Tran

sfor

mat

ions

and

coo

rdin

ates

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

unde

rsta

nd a

nd u

se th

e la

ngua

ge a

nd n

otat

ion

asso

ciat

ed w

ith

refle

ctio

ns, t

rans

latio

ns

and

rota

tions

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:

•ro

tatio

ns a

bout

an

y po

int

•re

flect

ions

in a

ny

line

•tr

ansl

atio

ns u

sing

ve

ctor

not

atio

n

•a

tran

sfor

mat

ion

and

its in

vers

e

gene

rate

and

ana

lyse

pa

tter

ns, e

.g. I

slam

ic

desi

gns

expl

ain

and

dem

onst

rate

gr

aphi

cally

the

effe

cts

of c

ombi

ning

tr

ansl

atio

ns, u

sing

ve

ctor

not

atio

n,

incl

udin

g:

•th

e ru

le fo

r ad

ditio

n of

vec

tors

•sc

alar

m

ultip

licat

ion

of a

ve

ctor

(rep

eate

d ad

ditio

n)

expl

ain

and

dem

onst

rate

gr

aphi

cally

the

effe

cts

of c

ombi

ning

tr

ansl

atio

ns, u

sing

ve

ctor

not

atio

n,

incl

udin

g:

•th

e di

ffer

ence

of

two

vect

ors

•th

e re

sulta

nt o

f tw

o ve

ctor

s

•th

e co

mm

utat

ive

and

asso

ciat

ive

prop

ertie

s of

ve

ctor

add

ition

solv

e si

mpl

e ge

omet

rical

pro

blem

s in

2-D

usi

ng v

ecto

rs

reco

gnis

e an

d vi

sual

ise

the

sym

met

ries

of a

2-D

sh

ape

iden

tify

all t

he

sym

met

ries

of 2

-D

shap

es

iden

tify

refle

ctio

n sy

mm

etry

in 3

-D

shap

es

tran

sfor

m 2

-D s

hape

s by

: •re

flect

ing

in g

iven

m

irror

line

s

•ro

tatin

g ab

out a

gi

ven

poin

t

•tr

ansl

atin

g

tran

sfor

m 2

-D s

hape

s by

rota

tion,

refle

ctio

n an

d tr

ansl

atio

n, o

n pa

per a

nd u

sing

ICT

reco

gnis

e th

at

tran

slat

ions

, rot

atio

ns

and

refle

ctio

ns

pres

erve

leng

th a

nd

angl

e, a

nd m

ap o

bjec

ts

on to

con

grue

nt

imag

es

expl

ore

thes

e tr

ansf

orm

atio

ns a

nd

sym

met

ries

usin

g IC

T

try

out m

athe

mat

ical

re

pres

enta

tions

of

sim

ple

com

bina

tions

of

thes

e tr

ansf

orm

atio

ns

expl

ore

and

com

pare

m

athe

mat

ical

re

pres

enta

tions

of

com

bina

tions

of

tran

slat

ions

, 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

an

d tr

ansf

orm

sha

pes

4.2

Tran

sfor

mat

ions

and

coo

rdin

ates

(con

tinu

ed)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

unde

rsta

nd a

nd u

se th

e la

ngua

ge a

nd n

otat

ion

asso

ciat

ed w

ith

enla

rgem

ent;

enla

rge

2-D

sha

pes,

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

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

y th

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

ts

pres

erve

ang

le b

ut n

ot

leng

th, a

nd u

nder

stan

d th

e im

plic

atio

ns o

f en

larg

emen

t for

pe

rimet

er

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

e in

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

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

m

ake

scal

e dr

awin

gsus

e an

d in

terp

ret m

aps

and

scal

e dr

awin

gs

in th

e co

ntex

t of

mat

hem

atic

s an

d ot

her

subj

ects

appl

y th

e pr

oper

ties

of s

imila

r tria

ngle

s an

d Py

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

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

geo

met

rical

in

form

atio

n

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

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

4.3

Cons

truc

tion

and

loci

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

a ru

ler a

nd

prot

ract

or to

:

•m

easu

re a

nd d

raw

lin

es to

the

near

est

mill

imet

re a

nd

angl

es, i

nclu

ding

re

flex

angl

es, t

o th

e ne

ares

t deg

ree

•co

nstr

uct a

tr

iang

le, g

iven

tw

o si

des

and

the

incl

uded

ang

le

(SA

S) o

r tw

o an

gles

an

d th

e in

clud

ed

side

(ASA

)

use

stra

ight

edg

e an

d co

mpa

sses

to c

onst

ruct

:

•th

e m

idpo

int a

nd

perp

endi

cula

r bi

sect

or o

f a li

ne

segm

ent

•th

e bi

sect

or o

f an

ang

le

•th

e pe

rpen

dicu

lar

from

a p

oint

to

a lin

e

•th

e pe

rpen

dicu

lar

from

a p

oint

on

a

line

•a

tria

ngle

, giv

en

thre

e si

des

(SSS

)

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

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

sid

e an

d an

gle

fact

s, re

view

ing

and

gene

ralis

ing

findi

ngs

to id

entif

y w

hich

of

thes

e co

nditi

ons

defin

e un

ique

con

stru

ctio

ns

use

ICT

to e

xplo

re

cons

truc

tions

use

ICT

to e

xplo

re

thes

e co

nstr

uctio

nsus

e IC

T to

exp

lore

co

nstr

uctio

ns o

f tr

iang

les

and

othe

r 2-

D s

hape

s

use

rule

r and

pro

trac

tor

to c

onst

ruct

sim

ple

nets

of 3

-D s

hape

s,

e.g.

cub

oid,

regu

lar

tetr

ahed

ron,

squ

are-

base

d py

ram

id,

tria

ngul

ar p

rism

find

sim

ple

loci

, bot

h by

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

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

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

e th

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

4.4

Mea

sure

s an

d m

ensu

rati

on

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

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

y co

ntex

ts; c

onve

rt o

ne

met

ric u

nit t

o an

othe

r, e.

g. g

ram

s to

kilo

gram

s; re

ad a

nd in

terp

ret s

cale

s on

a ra

nge

of m

easu

ring

inst

rum

ents

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

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

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

t su

ited

to th

e so

lutio

n

mak

e co

nnec

tions

be

twee

n th

e co

ntin

uity

of

the

num

ber l

ine

and

cont

inuo

us m

easu

res;

criti

cally

exa

min

e th

e m

easu

rem

ents

use

d in

a

prob

lem

and

thei

r eff

ect

on th

e ac

cura

cy o

f the

so

lutio

n, e

.g. u

nder

stan

d ho

w e

rror

s ca

n be

co

mpo

unde

d

com

mun

icat

e th

e so

lutio

n to

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

dist

ingu

ish

betw

een

and

estim

ate

the

size

of a

cute

, ob

tuse

and

refle

x an

gles

use

bear

ings

to s

peci

fy

dire

ctio

n

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.

4.4

Mea

sure

s an

d m

ensu

rati

on (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

know

and

use

the

form

ula

for t

he a

rea

of

a re

ctan

gle;

cal

cula

te

the

perim

eter

and

are

a of

sha

pes

mad

e fr

om

rect

angl

es

deriv

e an

d us

e fo

rmul

ae

for t

he a

rea

of a

tr

iang

le, p

aral

lelo

gram

an

d tr

apez

ium

; ca

lcul

ate

area

s of

com

poun

d sh

apes

know

and

use

the

form

ulae

for t

he

circ

umfe

renc

e an

d ar

ea

of a

circ

le

pres

ent a

con

cise

re

ason

ed a

rgum

ent t

o de

rive

form

ulae

for:

•le

ngth

s of

circ

ular

ar

cs

•ar

eas

of s

ecto

rs o

f a

circ

le

•su

rfac

e ar

ea o

f a

cylin

der

•vo

lum

e of

a

cylin

der

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

g fo

rmul

ae fo

r the

su

rfac

e ar

eas

of

pyra

mid

s an

d co

nes;

expl

ore

conn

ectio

ns

betw

een:

•fo

rmul

ae fo

r th

e vo

lum

e of

a

pyra

mid

and

the

rela

ted

cubo

id

•fo

rmul

ae fo

r th

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

solv

e pr

oble

ms

invo

lvin

g m

ore

com

plex

shap

es

and

solid

s, in

clud

ing

segm

ents

of c

ircle

s and

fr

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

vo

lum

es a

nd su

rfac

e ar

eas

of c

uboi

ds a

nd

shap

es m

ade

from

cu

boid

s

calc

ulat

e th

e su

rfac

e ar

ea a

nd v

olum

e of

rig

ht p

rism

s

5 St

atis

tics

5.1S

peci

fyin

g a

prob

lem

, pla

nnin

g an

d co

llect

ing

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

sugg

est p

ossi

ble

answ

ers,

giv

en a

qu

estio

n th

at c

an b

e ad

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

to

expl

ore

usin

g st

atis

tical

m

etho

ds, f

ram

e qu

estio

ns a

nd ra

ise

conj

ectu

res

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

ify

poss

ible

sou

rces

; co

nsid

er a

ppro

pria

te

sam

ple

size

disc

uss

how

diff

eren

t se

ts o

f dat

a re

late

to

the

prob

lem

; ide

ntify

po

ssib

le p

rimar

y or

se

cond

ary

sour

ces;

dete

rmin

e th

e sa

mpl

e si

ze a

nd 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

of b

ias

and

plan

how

to m

inim

ise

it

iden

tify

prac

tical

pr

oble

ms

such

as

non-

resp

onse

or

mis

sing

dat

a an

d re

fine

appr

oach

es to

min

imis

e th

eir i

mpa

ct o

n th

e va

lidity

of t

he re

sults

5.1S

peci

fyin

g a

prob

lem

, pla

nnin

g an

d co

llect

ing

(con

tinu

ed)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

plan

how

to c

olle

ct a

nd

orga

nise

sm

all s

ets

of

data

from

surv

eys

and

expe

rimen

ts:

•de

sign

dat

a co

llect

ion

shee

ts

or q

uest

ionn

aire

s to

use

in a

sim

ple

surv

ey

•co

nstr

uct

freq

uenc

y ta

bles

fo

r gat

herin

g di

scre

te d

ata,

gr

oupe

d w

here

ap

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

eq

ual c

lass

inte

rval

s fo

r ga

ther

ing

cont

inuo

us

data

and

two-

way

ta

bles

for r

ecor

ding

di

scre

te d

ata

desi

gn a

surv

ey o

r ex

perim

ent t

o ca

ptur

e th

e ne

cess

ary

data

fr

om o

ne o

r mor

e so

urce

s; de

sign

, tria

l an

d if

nece

ssar

y re

fine

data

col

lect

ion

shee

ts;

cons

truc

t tab

les

for

gath

erin

g la

rge

disc

rete

an

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

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

t ap

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

ta

logg

ing)

and

sec

onda

ry

(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,

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

sa

mpl

ing;

exp

lain

the

effe

ct o

n re

liabi

lity

an

d va

lidity

colle

ct s

mal

l set

s of

da

ta fr

om su

rvey

s an

d ex

perim

ents

, as

plan

ned

colle

ct d

ata

usin

g a

suita

ble

met

hod

(e.g

. ob

serv

atio

n, c

ontr

olle

d ex

perim

ent,

data

lo

ggin

g us

ing

ICT)

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

5.2

Proc

essi

ng a

nd re

pres

enti

ng d

ata

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

calc

ulat

e st

atis

tics

for

smal

l set

s of

dis

cret

e da

ta:

•fin

d th

e m

ode,

m

edia

n an

d ra

nge,

an

d th

e m

odal

cl

ass

for g

roup

ed

data

•ca

lcul

ate

the

mea

n, in

clud

ing

from

a s

impl

e fr

eque

ncy

tabl

e,

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

t ap

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

g th

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

m

ovin

g av

erag

es

5.2

Proc

essi

ng a

nd re

pres

enti

ng d

ata

(con

tinu

ed)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

cons

truc

t, on

pap

er

and

usin

g IC

T, g

raph

s an

d di

agra

ms

to

repr

esen

t dat

a,

incl

udin

g:

•ba

r-lin

e gr

aphs

•fr

eque

ncy

diag

ram

s fo

r gr

oupe

d di

scre

te

data

•si

mpl

e pi

e ch

arts

cons

truc

t gra

phic

al

repr

esen

tatio

ns, o

n pa

per a

nd u

sing

ICT,

an

d id

entif

y w

hich

ar

e m

ost u

sefu

l in

the

cont

ext o

f the

pro

blem

, in

clud

ing:

•pi

e ch

arts

for

cate

goric

al d

ata

•ba

r cha

rts

and

freq

uenc

y di

agra

ms

for

disc

rete

and

co

ntin

uous

dat

a

•si

mpl

e lin

e gr

aphs

fo

r tim

e se

ries

•si

mpl

e sc

atte

r gr

aphs

•st

em-a

nd-le

af

diag

ram

s

sele

ct, c

onst

ruct

an

d m

odify

, on

pape

r and

usi

ng IC

T,

suita

ble

grap

hica

l re

pres

enta

tions

to

prog

ress

an

enqu

iry

and

iden

tify

key

feat

ures

pre

sent

in th

e da

ta. I

nclu

de:

•lin

e gr

aphs

for

time

serie

s

•sc

atte

r gra

phs

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:

•hi

stog

ram

s fo

r gro

uped

co

ntin

uous

dat

a w

ith e

qual

cla

ss

inte

rval

s

•cu

mul

ativ

e fr

eque

ncy

tabl

es

and

diag

ram

s

•bo

x pl

ots

•sc

atte

r gra

phs

and

lines

of b

est f

it

(by

eye)

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

choo

se a

nd c

ombi

ne

suita

ble

grap

hica

l re

pres

enta

tions

to

prog

ress

an

unfa

mili

ar

or n

on-r

outin

e en

quiry

, in

clud

ing

hist

ogra

ms

with

equ

al o

r une

qual

cl

ass

inte

rval

s

use

prec

ise

and

cons

iste

nt g

raph

ical

re

pres

enta

tion

to

prog

ress

an

unfa

mili

ar

and

non-

rout

ine

enqu

iry

wor

k th

roug

h th

e en

tire

hand

ling

data

cyc

le to

ex

plor

e re

latio

nshi

ps

with

in b

i-var

iate

dat

a,

incl

udin

g ap

plic

atio

ns

to g

loba

l citi

zens

hip,

e.g

. ho

w fa

ir is

our s

ocie

ty?

5.3

Inte

rpre

ting

and

dis

cuss

ing

resu

lts

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

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

s an

d si

mpl

e st

atis

tics

for

a si

ngle

dis

trib

utio

n

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

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

find

patt

erns

and

ex

cept

ions

and

exp

lain

an

omal

ies;

incl

udin

g in

terp

reta

tion

of

soci

al s

tatis

tics

and

eval

uatio

n of

the

stre

ngth

of a

ssoc

iatio

n w

ithin

bi-v

aria

te d

ata

(cor

rela

tion,

line

s of

be

st fi

t)

inte

rpre

t and

com

pare

di

strib

utio

ns, i

nclu

ding

cu

mul

ativ

e fr

eque

ncy

diag

ram

s; m

ake

and

disc

uss

infe

renc

es,

usin

g th

e sh

ape

of

the

dist

ribut

ions

and

m

easu

res

of a

vera

ge

and

spre

ad, i

nclu

ding

m

edia

n an

d qu

artil

es

expl

ain

and

just

ify

assu

mpt

ions

and

co

nstr

aint

s; in

clud

e in

terp

reta

tion

and

com

paris

on o

f hi

stog

ram

s w

ith

uneq

ual c

lass

inte

rval

sco

mpa

re tw

o si

mpl

e di

strib

utio

ns u

sing

the

rang

e an

d on

e of

the

mod

e, m

edia

n or

mea

n

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

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

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

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

revi

ew in

terp

reta

tions

an

d re

sults

of a

st

atis

tical

enq

uiry

on

the

basi

s of

dis

cuss

ions

; co

mm

unic

ate

thes

e in

terp

reta

tions

and

re

sults

usi

ng s

elec

ted

tabl

es, g

raph

s an

d di

agra

ms

eval

uate

the

resu

lts

of a

sta

tistic

al

enqu

iry; r

evie

w a

nd

just

ify o

r ref

ine

the

choi

ce o

f sta

tistic

al

repr

esen

tatio

ns a

nd

rela

te su

mm

aris

ed d

ata

to th

e qu

estio

ns b

eing

ex

plor

ed

criti

cally

exa

min

e st

rate

gies

ado

pted

an

d ar

gum

ents

pr

esen

ted,

rela

ting

them

to th

e or

igin

al

hypo

thes

es; r

ecog

nise

th

e lim

itatio

ns o

f any

as

sum

ptio

ns a

nd th

e ef

fect

s th

at v

aryi

ng

assu

mpt

ions

cou

ld

have

on

conc

lusi

ons

draw

n fr

om d

ata

anal

ysis

use

stat

istic

al

anal

ysis

eff

ectiv

ely

in

pres

entin

g co

nvin

cing

co

nclu

sion

s; cr

itica

lly

refle

ct o

n ow

n lin

es

of e

nqui

ry; s

earc

h fo

r and

app

reci

ate

mor

e el

egan

t for

ms

of c

omm

unic

atin

g co

nclu

sion

s

5.4

Prob

abili

ty

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

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

nid

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

d ev

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

pr

obab

ilitie

s

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

pr

obab

ilitie

s as

soci

ated

w

ith in

depe

nden

t an

d m

utua

lly

excl

usiv

e ev

ents

whe

n in

terp

retin

g da

ta

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

5.4

Prob

abili

ty (c

onti

nued

)

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

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;

com

pare

exp

erim

enta

l an

d th

eore

tical

pr

obab

ilitie

s in

sim

ple

cont

exts

com

pare

est

imat

ed

expe

rimen

tal

prob

abili

ties

with

theo

retic

al

prob

abili

ties,

re

cogn

isin

g th

at:

•if

an e

xper

imen

t is

repe

ated

the

outc

ome

may

, and

us

ually

will

, be

diff

eren

t

•in

crea

sing

the

num

ber o

f tim

es

an e

xper

imen

t is

repe

ated

gen

eral

ly

lead

s to

bet

ter

estim

ates

of

prob

abili

ty

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

te

the

diff

eren

ce

betw

een

mat

hem

atic

al

expl

anat

ion

and

expe

rimen

tal e

vide

nce

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

by

mod

ellin

g re

al e

vent

s th

roug

h si

mul

atio

n;

just

ify d

ecis

ions

bas

ed

on e

xper

imen

tal

prob

abili

ty a

nd

com

men

t on

the

effe

ct o

f ass

umpt

ions

an

d sa

mpl

e si

ze

on th

e re

liabi

lity

of

conc

lusi

ons

The mathematics overview and learning objectives...

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Transcript of The mathematics overview and learning objectives...

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