The mathematics overview and learning objectives...
Transcript of 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
© Crown copyright 2009 01061-2009DOM-EN
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
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
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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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
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)
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
13
Num
ber o
pera
tion
s2.
4 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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)
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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)
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
19
3 A
lgeb
ra
Equ
atio
ns, f
orm
ulae
, exp
ress
ions
and
iden
titi
es
3.1
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
37
5 St
atis
tics
5.1S
peci
fyin
g a
prob
lem
, pla
nnin
g an
d co
llect
ing
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
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)
39 The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
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
© Crown copyright 2009 01061-2009DOM-EN
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
40 The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
41
Inte
rpre
ting
and
dis
cuss
ing
resu
lts
5.3
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
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
42 The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
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
01061-2009DOM-EN © Crown copyright 2009
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
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN