3 The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
The
Fram
ewor
k fo
r sec
onda
ry m
athe
mat
ics:
ove
rvie
w a
nd le
arni
ng o
bjec
tives
Ove
rvie
w o
f str
ands
Stra
nds
Sub-
stra
nds
Stra
nds
Sub-
stra
nds
1 M
athe
mat
ical
pro
cess
es a
nd a
pplic
atio
ns
3 A
lgeb
ra
1.1
Repr
esen
ting
3.1
Equa
tions
, for
mul
ae, e
xpre
ssio
ns a
nd id
entit
ies
1.2
Anal
ysin
g –
use
mat
hem
atic
al re
ason
ing
3.2
Sequ
ence
s, fu
nctio
ns a
nd g
raph
s
1.3
Anal
ysin
g –
use
appr
opria
te m
athe
mat
ical
pro
cedu
res
4 G
eom
etry
and
mea
sure
s
1.4
Inte
rpre
ting
and
eval
uatin
g 4.
1 G
eom
etric
al re
ason
ing
1.5
Com
mun
icat
ing
and
refle
ctin
g 4.
2 Tr
ansf
orm
atio
ns a
nd c
oord
inat
es
2 N
umbe
r 4.
3 Co
nstr
uctio
n an
d lo
ci
2.1
Plac
e va
lue,
ord
erin
g an
d ro
undi
ng
4.4
Mea
sure
s and
men
sura
tion
2.2
Inte
gers
, pow
ers a
nd ro
ots
5 St
atis
tics
2.3
Frac
tions
, dec
imal
s, pe
rcen
tage
s, ra
tio a
nd p
ropo
rtio
n 5.
1 Sp
ecify
ing
a pr
oble
m, p
lann
ing
and
colle
ctin
g da
ta
2.4
Num
ber o
pera
tions
5.
2 Pr
oces
sing
and
repr
esen
ting
data
2.5
Men
tal c
alcu
latio
n m
etho
ds
5.3
Inte
rpre
ting
and
disc
ussin
g re
sults
2.6
Writ
ten
calc
ulat
ion
met
hods
5.
4 Pr
obab
ility
2.7
Calc
ulat
or m
etho
ds
2.8
Chec
king
resu
lts
© Crown copyright 2009 01061-2009DOM-EN
5
Lear
ning
obj
ectiv
es
1 M
athe
mat
ical
pro
cess
es a
nd a
pplic
atio
nsSo
lve
prob
lem
s, e
xplo
re a
nd in
vest
igat
e in
a ra
nge
of co
ntex
ts
Incr
ease
the
chal
leng
e an
d bu
ild p
rogr
essi
on a
cros
s the
key
stag
e, a
nd fo
r gro
ups o
f pup
ils b
y:
incr
easin
g th
e co
mpl
exit
y of
the
appl
icat
ion,
e.g
. non
-rout
ine,
mul
ti-st
ep p
robl
ems,
exte
nded
enq
uirie
s
redu
cing
the
fam
iliar
ity
of th
e co
ntex
t, e.
g. n
ew c
onte
xts i
n m
athe
mat
ics,
cont
exts
dra
wn
from
oth
er su
bjec
ts, o
ther
asp
ects
of p
upils
’ liv
es
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
incr
easin
g th
e te
chni
cal d
eman
d of
the
mat
hem
atic
s req
uire
d, e
.g. m
ore
adva
nced
con
cept
s, m
ore
diff
icul
t pro
cedu
res
incr
easin
g th
e de
gree
of i
ndep
ende
nce
and
auto
nom
y in
pro
blem
-sol
ving
and
inve
stig
atio
n
Rep
rese
ntin
g 1.
1 iden
tify
the
nece
ssar
y in
form
atio
n to
un
ders
tand
or s
impl
ifya
cont
ext o
r pro
blem
; re
pres
ent p
robl
ems,
mak
ing
corr
ect u
se
of sy
mbo
ls, w
ords
, di
agra
ms,
tabl
es a
nd
grap
hs; u
se a
ppro
pria
te
proc
edur
es a
nd to
ols,
incl
udin
g IC
T
Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
iden
tify
the
mat
hem
atic
al fe
atur
esof
a c
onte
xt o
r pr
oble
m; t
ry o
ut a
nd
com
pare
mat
hem
atic
al
repr
esen
tatio
ns; s
elec
tap
prop
riate
pro
cedu
res
and
tool
s, in
clud
ing
ICT
brea
k do
wn
subs
tant
ial
task
s to
mak
e th
em
mor
e m
anag
eabl
e;
repr
esen
t pro
blem
s an
d sy
nthe
sise
info
rmat
ion
in
alge
brai
c, g
eom
etric
al
or g
raph
ical
form
; m
ove
from
one
form
to
ano
ther
to g
ain
a di
ffere
nt p
ersp
ectiv
eon
the
prob
lem
com
pare
and
eva
luat
ere
pres
enta
tions
;ex
plai
n th
e fe
atur
es
sele
cted
and
just
ify
the
choi
ce o
f re
pres
enta
tion
in
rela
tion
to th
e co
ntex
t
choo
se a
nd c
ombi
ne
repr
esen
tatio
ns fr
om a
ra
nge
of p
ersp
ectiv
es;
intr
oduc
e an
d us
e a
rang
e of
mat
hem
atic
alte
chni
ques
, the
mos
tef
ficie
nt fo
r ana
lysis
and
mos
t effe
ctiv
e fo
r co
mm
unic
atio
n
syst
emat
ical
ly m
odel
co
ntex
ts o
r pro
blem
s th
roug
h pr
ecise
an
d co
nsist
ent u
se
of sy
mbo
ls an
d re
pres
enta
tions
, and
su
stai
n th
is th
roug
hout
th
e w
ork
6
Ana
lysi
ng –
use
mat
hem
atic
al re
ason
ing
1.2 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
clas
sify
and
visu
alise
pr
oper
ties a
nd
patt
erns
; gen
eral
ise
in si
mpl
e ca
ses b
y w
orki
ng lo
gica
lly; d
raw
simpl
e co
nclu
sions
an
d ex
plai
n re
ason
ing;
unde
rsta
nd th
e sig
nific
ance
of a
co
unte
r-exa
mpl
e;ta
ke a
ccou
nt o
f fe
edba
ck a
nd le
arn
from
mist
akes
visu
alise
and
m
anip
ulat
e dy
nam
ic
imag
es; c
onje
ctur
e an
d ge
nera
lise;
mov
ebe
twee
n th
e ge
nera
l an
d th
e pa
rtic
ular
tote
st th
e lo
gic
of a
n ar
gum
ent;
iden
tify
exce
ptio
nal c
ases
or
coun
ter-e
xam
ples
;m
ake
conn
ectio
ns w
ith
rela
ted
cont
exts
use
conn
ectio
ns w
ith
rela
ted
cont
exts
toim
prov
e th
e an
alys
is of
a
situa
tion
or p
robl
em;
pose
que
stio
ns a
nd
mak
e co
nvin
cing
ar
gum
ents
to ju
stify
ge
nera
lisat
ions
or
solu
tions
; rec
ogni
se th
e im
pact
of c
onst
rain
ts o
r as
sum
ptio
ns
iden
tify
a ra
nge
of st
rate
gies
and
ap
prec
iate
that
mor
eth
an o
ne a
ppro
ach
may
be
nece
ssar
y;
expl
ore
the
effe
cts
of v
aryi
ng v
alue
s and
lo
ok fo
r inv
aria
nce
and
cova
rianc
e in
mod
els
and
repr
esen
tatio
ns;
exam
ine
and
refin
ear
gum
ents
, con
clus
ions
an
d ge
nera
lisat
ions
;pr
oduc
e sim
ple
proo
fs
mak
e pr
ogre
ss b
yex
plor
ing
mat
hem
atic
al
task
s, de
velo
ping
an
d fo
llow
ing
alte
rnat
ive
appr
oach
es;
exam
ine
and
exte
ndge
nera
lisat
ions
; sup
port
as
sum
ptio
ns b
y cl
ear
argu
men
t and
follo
wth
roug
h a
sust
aine
dch
ain
of re
ason
ing,
incl
udin
g pr
oof
pres
ent r
igor
ous a
nd
sust
aine
d ar
gum
ents
;re
ason
indu
ctiv
ely,
dedu
ce a
nd p
rove
; ex
plai
n an
d ju
stify
as
sum
ptio
ns a
nd
cons
trai
nts
Ana
lysi
ng –
use
app
ropr
iate
mat
hem
atic
al p
roce
dure
s 1.
3W
ithin
the
appr
opria
te ra
nge
and
cont
ent:
mak
e ac
cura
te m
athe
mat
ical
dia
gram
s, gr
aphs
and
con
stru
ctio
ns o
n pa
per a
nd o
n sc
reen
; cal
cula
te a
ccur
atel
y, se
lect
ing
men
tal m
etho
ds o
r cal
cula
ting
devi
ces a
s app
ropr
iate
; man
ipul
ate
num
bers
, alg
ebra
ic e
xpre
ssio
ns a
nd e
quat
ions
, and
app
ly ro
utin
e al
gorit
hms;
use
accu
rate
not
atio
n, in
clud
ing
corr
ect
synt
ax w
hen
usin
g IC
T; re
cord
met
hods
, sol
utio
ns a
nd c
oncl
usio
ns; e
stim
ate,
app
roxi
mat
e an
d ch
eck
wor
king
7
Inte
rpre
ting
and
eval
uatin
g 1.
4 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
inte
rpre
t inf
orm
atio
n fro
m a
mat
hem
atic
al
repr
esen
tatio
n or
co
ntex
t; re
late
find
ings
to
the
orig
inal
con
text
; ch
eck
the
accu
racy
of
the
solu
tion;
exp
lain
an
d ju
stify
met
hods
an
d co
nclu
sions
;co
mpa
re a
nd e
valu
ate
appr
oach
es
use
logi
cal a
rgum
ent
to in
terp
ret t
he
mat
hem
atic
s in
a gi
ven
cont
ext o
r to
esta
blish
the
trut
h of
a st
atem
ent;
give
accu
rate
solu
tions
ap
prop
riate
to th
e co
ntex
t or p
robl
em;
eval
uate
the
effic
ienc
yof
alte
rnat
ive
stra
tegi
es
and
appr
oach
es
Com
mun
icat
ing
and
refle
ctin
g 1.
5
just
ify th
e m
athe
mat
ical
feat
ures
draw
n fro
m a
con
text
an
d th
e ch
oice
of
appr
oach
; gen
erat
efu
ller s
olut
ions
by
pres
entin
g a
conc
ise,
reas
oned
arg
umen
t us
ing
sym
bols,
di
agra
ms,
grap
hs a
nd
rela
ted
expl
anat
ions
revi
ew a
nd re
fine
own
findi
ngs a
nd
appr
oach
es o
n th
eba
sis o
f disc
ussio
ns
with
oth
ers;
look
for
and
refle
ct o
n ot
her
appr
oach
es a
nd b
uild
on
pre
viou
s exp
erie
nce
of si
mila
r situ
atio
ns
and
outc
omes
refin
e ow
n fin
ding
s an
d ap
proa
ches
on
the
basis
of d
iscus
sions
w
ith o
ther
s; re
cogn
ise
effic
ienc
y in
an
appr
oach
; rel
ate
the
curr
ent p
robl
em a
nd
stru
ctur
e to
pre
viou
ssit
uatio
ns
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 effe
ctiv
ely,
oral
ly a
nd in
writ
ing,
and
disc
uss a
nd
com
pare
app
roac
hes
and
resu
lts w
ith o
ther
s; re
cogn
ise e
quiv
alen
t ap
proa
ches
mak
e se
nse
of, a
nd
judg
e th
e va
lue
of, o
wn
findi
ngs a
nd th
ose
pres
ente
d by
oth
ers;
judg
e th
e st
reng
th o
f em
piric
al e
vide
nce
and
dist
ingu
ish b
etw
een
evid
ence
and
pro
of;
just
ify g
ener
alisa
tions
, ar
gum
ents
or s
olut
ions
use
a ra
nge
of fo
rms
to c
omm
unic
ate
findi
ngs e
ffect
ivel
y to
diffe
rent
aud
ienc
es;
revi
ew fi
ndin
gs a
nd
look
for e
quiv
alen
ce to
diffe
rent
pro
blem
s with
simila
r str
uctu
re
just
ify a
nd e
xpla
in
solu
tions
to p
robl
ems
invo
lvin
g an
unf
amili
ar
cont
ext o
r a n
umbe
r of
feat
ures
or
varia
bles
; com
men
t co
nstr
uctiv
ely
on
reas
onin
g, lo
gic,
proc
ess,
resu
lts a
nd
conc
lusio
ns
use
mat
hem
atic
alla
ngua
ge a
ndsy
mbo
ls ef
fect
ivel
y in
pr
esen
ting
conv
inci
ng
conc
lusio
ns o
r fin
ding
s; cr
itica
lly re
flect
on
own
lines
of e
nqui
ry
whe
n ex
plor
ing;
sear
chfo
r and
app
reci
ate
mor
e el
egan
t for
ms
of c
omm
unic
atin
g ap
proa
ches
and
so
lutio
ns; c
onsid
er th
e ef
ficie
ncy
of a
ltern
ativ
e lin
es o
f enq
uiry
or
proc
edur
es
show
insig
ht in
to
the
mat
hem
atic
al
conn
ectio
ns in
the
cont
ext o
r pro
blem
; cr
itica
lly e
xam
ine
stra
tegi
es a
dopt
ed
and
argu
men
ts
pres
ente
d; c
onsid
er
the
assu
mpt
ions
in th
e m
odel
and
reco
gnise
lim
itatio
ns in
the
accu
racy
of r
esul
ts
and
conc
lusio
ns
Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
rout
inel
y re
view
and
re
fine
findi
ngs a
nd
appr
oach
es; i
dent
ifyho
w o
ther
con
text
s w
ere
diffe
rent
from
, or
simila
r to,
the
curr
ent
situa
tion
and
expl
ain
how
and
why
the
sam
e or
diff
eren
t str
ateg
ies
wer
e us
ed
9
2 N
umbe
r
Pla
ce v
alue
, ord
erin
g an
d ro
undi
ng
2.1
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 b
y 10
, 10
0, 1
000,
and
exp
lain
th
e ef
fect
com
pare
and
ord
er
deci
mal
s in
diffe
rent
co
ntex
ts; k
now
that
whe
n co
mpa
ring
mea
sure
men
ts th
e un
its m
ust b
e th
e sa
me
roun
d po
sitiv
e w
hole
nu
mbe
rs to
the
near
est
10, 1
00 o
r 100
0, a
nd
deci
mal
s to
the
near
est
who
le n
umbe
r or o
ne
deci
mal
pla
ce
read
and
writ
e po
sitiv
e in
tege
r pow
ers o
f 10;
m
ultip
ly a
nd d
ivid
e in
tege
rs a
nd d
ecim
als
by 0
.1 a
nd 0
.01
orde
r dec
imal
s
roun
d po
sitiv
e nu
mbe
rsto
any
giv
en p
ower
of
10;
roun
d de
cim
als
to th
e ne
ares
t who
le
num
ber o
r to
one
or
two
deci
mal
pla
ces
exte
nd k
now
ledg
e of
inte
ger p
ower
s of
10;
reco
gnise
the
equi
vale
nce
of 0
.1, 101
and
10–1
; mul
tiply
and
di
vide
by
any
inte
ger
pow
er o
f 10
use
roun
ding
to m
ake
estim
ates
and
to g
ive
solu
tions
to p
robl
ems
to a
n ap
prop
riate
de
gree
of a
ccur
acy
conv
ert b
etw
een
ordi
nary
and
st
anda
rd in
dex
form
re
pres
enta
tions
, usin
g sig
nific
ant f
igur
es a
s ap
prop
riate
; jus
tify
the
repr
esen
tatio
n us
edan
d ch
oice
of a
ccur
acy
in re
latio
n to
the
prob
lem
and
aud
ienc
e fo
r the
solu
tion
enga
ge in
mat
hem
atic
al ta
sks
whe
re u
sing
num
bers
in
stan
dard
form
is es
sent
ial t
o th
e ca
lcul
atio
ns in
volv
ed;
criti
cally
exa
min
e th
e ef
fect
of n
umer
ical
re
pres
enta
tions
on
the
accu
racy
of
the
solu
tion,
e.g
.un
ders
tand
how
err
ors
can
be c
ompo
unde
d
in c
alcu
latio
ns
com
mun
icat
e th
e so
lutio
n to
a p
robl
em,
expl
aini
ng th
e lim
itatio
ns o
f acc
urac
y, us
ing
uppe
r and
lo
wer
bou
nds
10
Inte
gers
, pow
ers a
nd ro
ots
2.2 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1
unde
rsta
nd n
egat
ive
num
bers
as p
ositi
ons
on a
num
ber l
ine;
or
der,
add
and
subt
ract
inte
gers
in c
onte
xt
reco
gnise
and
use
mul
tiple
s, fa
ctor
s,pr
imes
(les
s tha
n 10
0),
com
mon
fact
ors,
high
est c
omm
onfa
ctor
s and
low
est
com
mon
mul
tiple
s in
simpl
e ca
ses;
use
simpl
ete
sts o
f div
isibi
lity
reco
gnise
the
first
few
tr
iang
ular
num
bers
;re
cogn
ise th
e sq
uare
s of
num
bers
to a
tle
ast 1
2!"!
12 a
nd th
e co
rres
pond
ing
root
s
add,
subt
ract
, mul
tiply
an
d di
vide
inte
gers
use
mul
tiple
s, fa
ctor
s,co
mm
on fa
ctor
s, hi
ghes
t com
mon
fa
ctor
s, lo
wes
t co
mm
on m
ultip
les a
nd
prim
es; f
ind
the
prim
e fa
ctor
dec
ompo
sitio
n of
a n
umbe
r, e.
g.
8000!=!2
6 !"!5
3
use
squa
res,
posit
ive
and
nega
tive
squa
re
root
s, cu
bes a
nd
cube
root
s, an
d in
dex
nota
tion
for s
mal
l po
sitiv
e in
tege
r pow
ers
use
the
prim
e fa
ctor
de
com
posit
ion
of
a nu
mbe
r
use
ICT
to e
stim
ate
squa
re ro
ots a
nd
cube
root
s
use
inde
x no
tatio
n fo
r in
tege
r pow
ers;
know
an
d us
e th
e in
dex
law
s fo
r mul
tiplic
atio
n an
d di
visio
n of
pos
itive
in
tege
r pow
ers
exam
ine
and
exte
nd
the
inde
x la
ws t
o es
tabl
ish th
e m
eani
ng
of n
egat
ive,
frac
tiona
l an
d ze
ro p
ower
s, in
clud
ing
use
of
surd
not
atio
n
exam
ine
and
exte
nd
the
inde
x la
ws t
o es
tabl
ish th
e m
eani
ng
of in
vers
e op
erat
ions
in
rela
tion
to in
dice
s, i.e
. th
e in
vers
e op
erat
ion
of ra
ising
a p
ositi
ve
num
ber t
o po
wer
n is
raisi
ng th
e re
sult
of th
is op
erat
ion
to p
ower
1 n
Exte
nsio
n
solv
e a
prob
lem
us
ing
ratio
nal a
nd
irrat
iona
l num
bers
, in
clud
ing
surd
s
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
11
Fra
ctio
ns, d
ecim
als,
per
cent
ages
, rat
io a
nd p
ropo
rtio
n 2.
3 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
expr
ess a
smal
ler w
hole
nu
mbe
r as a
frac
tion
of
a la
rger
one
; sim
plify
fra
ctio
ns b
y ca
ncel
ling
all c
omm
on fa
ctor
s an
d id
entif
y eq
uiva
lent
fra
ctio
ns; c
onve
rt
term
inat
ing
deci
mal
s to
frac
tions
, e.g
. 23
=0.
2310
0 ; u
se
diag
ram
s to
com
pare
tw
o or
mor
e sim
ple
fract
ions
add
and
subt
ract
sim
ple
fract
ions
and
th
ose
with
com
mon
deno
min
ator
s;ca
lcul
ate
simpl
e fra
ctio
ns o
f qua
ntiti
es
and
mea
sure
men
ts(w
hole
-num
ber
answ
ers);
mul
tiply
a
fract
ion
by a
n in
tege
r
reco
gnise
that
a
recu
rrin
g de
cim
al is
a
fract
ion;
use
div
ision
to
conv
ert a
frac
tion
to a
de
cim
al; o
rder
frac
tions
by
writ
ing
them
with
a
com
mon
den
omin
ator
or
by
conv
ertin
g th
em
to d
ecim
als
unde
rsta
nd th
e eq
uiva
lenc
e of
sim
ple
alge
brai
c fra
ctio
ns;
know
that
a re
curr
ing
deci
mal
is a
n ex
act
fract
ion
expl
ain
the
patt
erns
fo
und
in re
curr
ing
deci
mal
s; ju
stify
w
hy d
ecim
als r
ecur
or
term
inat
e by
co
nsid
erin
g fa
ctor
s of
the
deno
min
ator
expl
ore
the
hist
oric
al
and
cultu
ral r
oots
of
the
num
ber s
yste
m
and
use
alge
bra
toju
stify
and
pro
ve so
me
of it
s fea
ture
s, e.
g. th
at
all r
ecur
ring
deci
mal
s ca
n be
exp
ress
ed a
s a
fract
ion
show
insig
ht in
to th
e in
finite
den
sity
of th
e nu
mbe
r lin
e; m
ake
sens
e of
the
proo
f tha
t #2
is ir
ratio
nal
add
and
subt
ract
fra
ctio
ns b
y w
ritin
g th
em w
ith a
com
mon
de
nom
inat
or; c
alcu
late
fra
ctio
ns o
f qua
ntiti
es
(frac
tion
answ
ers);
m
ultip
ly a
nd d
ivid
e an
in
tege
r by
a fra
ctio
n
use
effic
ient
met
hods
toad
d, su
btra
ct, m
ultip
lyan
d di
vide
frac
tions
,in
terp
retin
g di
visio
n as
a m
ultip
licat
ive
inve
rse;
canc
el co
mm
on fa
ctor
sbe
fore
mul
tiply
ing
or d
ivid
ing
unde
rsta
nd a
nd a
pply
ef
ficie
nt m
etho
ds to
add,
subt
ract
, mul
tiply
an
d di
vide
frac
tions
, in
terp
retin
g re
cipr
ocal
s as
mul
tiplic
ativ
ein
vers
es
12
Fra
ctio
ns, d
ecim
als,
per
cent
ages
, rat
io a
nd p
ropo
rtio
n (c
ontin
ued)
2.
3 Year
7
Year
8
unde
rsta
nd p
erce
ntag
e as
the
‘num
ber o
f par
ts
per 1
00’; c
alcu
late
sim
ple
perc
enta
ges
and
use
perc
enta
ges
to c
ompa
re si
mpl
e pr
opor
tions
reco
gnise
the
equi
vale
nce
of
perc
enta
ges,
fract
ions
an
d de
cim
als
unde
rsta
nd th
e re
latio
nshi
p be
twee
n ra
tio a
nd p
ropo
rtio
n;
use
dire
ct p
ropo
rtio
n in
sim
ple
cont
exts
; use
ra
tio n
otat
ion,
sim
plify
ra
tios a
nd d
ivid
e a
quan
tity
into
two
part
s in
a gi
ven
ratio
; so
lve
simpl
e pr
oble
ms
invo
lvin
g ra
tio a
nd
prop
ortio
n us
ing
info
rmal
stra
tegi
es
inte
rpre
t per
cent
age
as th
e op
erat
or ‘s
o m
any
hund
redt
hsof
’ and
exp
ress
one
gi
ven
num
ber a
s a
perc
enta
ge o
f ano
ther
;ca
lcul
ate
perc
enta
ges
and
find
the
outc
ome
of a
giv
en p
erce
ntag
e in
crea
se o
r dec
reas
e
use
the
equi
vale
nce
of fr
actio
ns, d
ecim
als
and
perc
enta
ges t
oco
mpa
re p
ropo
rtio
ns
appl
y un
ders
tand
ing
of th
e re
latio
nshi
p be
twee
n ra
tio a
nd
prop
ortio
n; si
mpl
ifyra
tios,
incl
udin
g th
ose
expr
esse
d in
diff
eren
t un
its, r
ecog
nisin
g lin
ks
with
frac
tion
nota
tion;
di
vide
a q
uant
ity in
totw
o or
mor
e pa
rts i
n a
give
n ra
tio; u
se th
e un
itary
met
hod
to
solv
e sim
ple
prob
lem
sin
volv
ing
ratio
and
di
rect
pro
port
ion
Year
9
Year
10
Year
11
Exte
nsio
n
reco
gnise
whe
n fra
ctio
ns o
r pe
rcen
tage
s are
ne
eded
to c
ompa
repr
opor
tions
; sol
ve
prob
lem
s inv
olvi
ng
perc
enta
ge c
hang
es
use
prop
ortio
nal
reas
onin
g to
solv
e pr
oble
ms,
choo
sing
the
corr
ect n
umbe
rs
to ta
ke a
s 100
%, o
r as
a w
hole
; com
pare
two
ratio
s; in
terp
ret a
nd
use
ratio
in a
rang
e
of c
onte
xts
iden
tify
whe
n a
prob
lem
in n
umbe
r, al
gebr
a, g
eom
etry
or st
atist
ics i
nvol
ves
prop
ortio
nalit
y; u
se
mul
tiplic
ativ
e m
etho
ds
fluen
tly in
the
solu
tion,
in
clud
ing
inve
rse
calc
ulat
ions
, e.g
. w
ith p
erce
ntag
es
mod
el re
al c
onte
xts
whe
re q
uant
ities
var
yin
dire
ct p
ropo
rtio
n,
incl
udin
g re
peat
edpr
opor
tiona
l cha
nge,
e.
g. g
row
th/d
ecay
; use
al
gebr
aic
met
hods
w
here
app
ropr
iate
and
co
nsid
er li
mita
tions
of
the
mod
el
(as i
n 2.
4)
unde
rsta
nd a
nd u
se
dire
ct a
nd in
vers
e pr
opor
tion;
solv
e pr
oble
ms i
nvol
ving
in
vers
e pr
opor
tion
(incl
udin
g y ?
1/x
2 )
(as i
n 2.
4)
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
13
Num
ber o
pera
tions
2.
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 a
nd u
se
mod
el re
al c
onte
xts
unde
rsta
nd a
nd u
se
rule
s of a
rithm
etic
and
ru
les o
f arit
hmet
ic a
nd
of m
ultip
lyin
g an
d re
cipr
ocal
s as a
w
here
qua
ntiti
es v
ary
dire
ct a
nd in
vers
e in
vers
e op
erat
ions
in
inve
rse
oper
atio
ns in
di
vidi
ng b
y nu
mbe
rs
mul
tiplic
ativ
e in
vers
e in
dire
ct p
ropo
rtio
n,
prop
ortio
n; so
lve
the
cont
ext o
f pos
itive
the
cont
ext o
f int
eger
s be
twee
n 0
and
1;
in c
onte
xts s
uch
asin
clud
ing
repe
ated
prob
lem
s inv
olvi
ng
inte
gers
and
dec
imal
s an
d fra
ctio
ns
cons
olid
ate
use
of th
e en
larg
emen
t; ex
plor
e pr
opor
tiona
l cha
nge,
in
vers
e pr
opor
tion
rule
s of a
rithm
etic
and
th
e be
havi
our o
f the
g
row
th/d
ecay
; use
e.
g.(in
clud
ing
y ?!1
/x 2 )
inve
rse
oper
atio
ns
reci
proc
al fu
nctio
n
alge
brai
c m
etho
ds
(as i
n 2.
3)
(y =!"
/x) f
or la
rge
and
whe
re a
ppro
pria
te a
nd
smal
l val
ues o
f x
cons
ider
lim
itatio
ns o
f th
e m
odel
(as i
n 2.
3)
use
the
orde
r of
use
the
orde
r of
unde
rsta
nd th
e or
der
oper
atio
ns, i
nclu
ding
op
erat
ions
, inc
ludi
ng
of p
rece
denc
e of
br
acke
ts
brac
kets
, with
mor
eop
erat
ions
, inc
ludi
ng
com
plex
cal
cula
tions
po
wer
s
14
Men
tal c
alcu
latio
n m
etho
ds
2.5 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
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, a
nd q
uick
ly
deriv
e as
soci
ated
divi
sion
fact
s
stre
ngth
en a
nd e
xten
d m
enta
l met
hods
of
calc
ulat
ion
to in
clud
e de
cim
als,
fract
ions
an
d pe
rcen
tage
s, ac
com
pani
ed w
here
ap
prop
riate
by
suita
ble
jott
ings
; sol
ve si
mpl
e pr
oble
ms m
enta
lly
mak
e an
d ju
stify
es
timat
es a
nd
appr
oxim
atio
ns o
f ca
lcul
atio
ns
reca
ll eq
uiva
lent
fra
ctio
ns, d
ecim
als
and
perc
enta
ges;
use
know
n fa
cts t
o de
rive
unkn
own
fact
s, in
clud
ing
prod
ucts
invo
lvin
g nu
mbe
rs
such
as 0
.7 a
nd 6
, and
an
d 8
0.03
stre
ngth
en a
nd e
xten
d m
enta
l met
hods
of
calc
ulat
ion,
wor
king
w
ith d
ecim
als,
fract
ions
, per
cent
ages
, sq
uare
s and
squa
re
root
s, cu
bes a
nd c
ube
root
s; so
lve
prob
lem
sm
enta
lly
use
know
n fa
cts t
o de
rive
unkn
own
fact
s; ex
tend
men
tal
met
hods
of c
alcu
latio
n,
wor
king
with
dec
imal
s, fra
ctio
ns, p
erce
ntag
es,
fact
ors,
pow
ers a
nd
root
s; so
lve
prob
lem
sm
enta
lly
sele
ct m
enta
l or
writ
ten
stra
tegi
es
and
calc
ulat
ing
devi
ces a
ppro
pria
te
to th
e st
age
of th
e pr
oble
m; c
alcu
late
accu
rate
ly w
ithre
cipr
ocal
s, po
wer
s, tr
igon
omet
rical
func
tions
and
num
bers
in
stan
dard
form
(as i
n 2.
6 an
d 2.
7)
sele
ct a
nd ju
stify
an
app
ropr
iate
and
ef
ficie
nt c
ombi
natio
n of
met
hods
of
calc
ulat
ion,
i.e.
men
tal,
writ
ten,
ICT
or c
alcu
lato
r to
solv
e pr
oble
ms
(as i
n 2.
6)
appr
ecia
te w
hen
resu
lts
of c
alcu
latio
ns c
an b
e m
ore
eleg
antly
and
ex
actly
com
mun
icat
edus
ing
surd
s and
!,
ratio
nalis
ing
a de
nom
inat
or w
here
ap
prop
riate
, e.g
. a
trig
onom
etric
also
lutio
n
(as i
n 2.
6)
mak
e an
d ju
stify
m
ake
and
just
ify
exam
ine
and
refin
ees
timat
es a
nd
estim
ates
and
es
timat
es a
nd
appr
oxim
atio
ns o
f ap
prox
imat
ions
of
appr
oxim
atio
ns o
f ca
lcul
atio
ns
calc
ulat
ions
ca
lcul
atio
ns in
volv
ing
roun
ding
15
Writ
ten
calc
ulat
ion
met
hods
2.
6 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
use
effic
ient
writ
ten
met
hods
to a
dd
and
subt
ract
who
le
num
bers
and
dec
imal
sw
ith u
p to
two
plac
es
mul
tiply
and
div
ide
thre
e-di
git b
y tw
o-di
git w
hole
num
bers
; ex
tend
to m
ultip
lyin
g an
d di
vidi
ng d
ecim
als
with
one
or t
wo
plac
es
by si
ngle
-dig
it w
hole
nu
mbe
rs
use
effic
ient
writ
ten
met
hods
to a
dd a
nd
subt
ract
inte
gers
and
de
cim
als o
f any
size
, in
clud
ing
num
bers
w
ith d
iffer
ing
num
bers
of
dec
imal
pla
ces
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
visio
n of
inte
gers
and
de
cim
als,
incl
udin
g by
de
cim
als s
uch
as 0
.6 o
r0.
06; u
nder
stan
d w
here
to
pos
ition
the
deci
mal
po
int b
y co
nsid
erin
g eq
uiva
lent
cal
cula
tions
use
effic
ient
writ
ten
met
hods
to a
dd a
nd
subt
ract
inte
gers
and
de
cim
als o
f any
size
; m
ultip
ly b
y de
cim
als;
divi
de b
y de
cim
als
by tr
ansf
orm
ing
to
divi
sion
by a
n in
tege
r
sele
ct m
enta
l or
writ
ten
stra
tegi
es
and
calc
ulat
ing
devi
ces a
ppro
pria
te
to th
e st
age
of th
e pr
oble
m; c
alcu
late
accu
rate
ly w
ithre
cipr
ocal
s, po
wer
s, tr
igon
omet
rical
func
tions
and
num
bers
in
stan
dard
form
(as i
n 2.
5 an
d 2.
7)
sele
ct a
nd ju
stify
an
app
ropr
iate
and
ef
ficie
nt c
ombi
natio
n of
met
hods
of
calc
ulat
ion,
i.e.
men
tal,
writ
ten,
ICT
or c
alcu
lato
r to
solv
e pr
oble
ms
(as i
n 2.
5)
appr
ecia
te w
hen
resu
lts
of c
alcu
latio
ns c
an b
e m
ore
eleg
antly
and
ex
actly
com
mun
icat
edus
ing
surd
s and
!,
ratio
nalis
ing
a de
nom
inat
or w
here
ap
prop
riate
, e.g
. a
trig
onom
etric
also
lutio
n
(as i
n 2.
5)
16
Cal
cula
tor m
etho
ds
2.7 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
carr
y ou
t cal
cula
tions
w
ith m
ore
than
one
step
usin
g br
acke
ts a
nd
the
mem
ory;
use
the
squa
re ro
ot a
nd si
gn
chan
ge k
eys
ente
r num
bers
and
in
terp
ret t
he d
ispla
y in
diff
eren
t con
text
s(d
ecim
als,
perc
enta
ges,
mon
ey, m
etric
m
easu
res)
carr
y ou
t mor
e di
ffic
ult
calc
ulat
ions
effe
ctiv
ely
and
effic
ient
ly u
sing
the
func
tion
keys
for
sign
chan
ge, p
ower
s, ro
ots a
nd fr
actio
ns;
use
brac
kets
and
th
e m
emor
y
use
a ca
lcul
ator
ef
ficie
ntly
and
ap
prop
riate
ly to
perf
orm
com
plex
ca
lcul
atio
ns w
ith
num
bers
of a
ny si
ze,
know
ing
not t
o ro
und
durin
g in
term
edia
te
step
s of a
cal
cula
tion;
us
e th
e co
nsta
nt, !
and
sign
chan
ge k
eys;
use
the
func
tion
keys
fo
r pow
ers,
root
s and
fra
ctio
ns; u
se b
rack
ets
and
the
mem
ory
sele
ct m
enta
l or
writ
ten
stra
tegi
es
and
calc
ulat
ing
devi
ces a
ppro
pria
te
to th
e st
age
of th
e pr
oble
m; c
alcu
late
accu
rate
ly w
ithre
cipr
ocal
s, po
wer
s, tr
igon
omet
rical
func
tions
and
num
bers
in
stan
dard
form
(as i
n 2.
5 an
d 2.
6)
criti
cally
exa
min
e al
tern
ativ
e m
etho
ds,
com
pare
stra
tegi
es fo
r:
calc
ulat
ing
(incl
udin
gca
lcul
atin
g de
vice
s)
chec
king
reco
gnise
the
limita
tions
of s
ome
appr
oach
es
(as i
n 2.
8)
refle
ct o
n a
solu
tion
to
a pr
oble
m c
omm
entin
g co
nstr
uctiv
ely
on th
e ch
oice
of c
alcu
latin
g st
rate
gies
(as i
n 2.
8)
ente
r num
bers
and
in
terp
ret t
he d
ispla
y in
diff
eren
t con
text
s(e
xten
d to
neg
ativ
enu
mbe
rs, f
ract
ions
, tim
e)
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
17
Che
ckin
g re
sults
2.
8 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
chec
k re
sults
by
cons
ider
ing
whe
ther
th
ey a
re o
f the
righ
t or
der o
f mag
nitu
dean
d by
wor
king
pr
oble
ms b
ackw
ards
sele
ct fr
om a
rang
e ch
eck
resu
lts u
sing
iden
tify
a ra
nge
of
of c
heck
ing
met
hods
, ap
prop
riate
met
hods
ch
ecki
ng st
rate
gies
and
in
clud
ing
estim
atin
g ap
prec
iate
that
mor
ein
con
text
and
usin
g th
an o
ne w
ay m
ay
inve
rse
oper
atio
ns
be n
eces
sary
in th
e co
ntex
t of t
he p
robl
em
criti
cally
exa
min
e al
tern
ativ
e m
etho
ds,
com
pare
stra
tegi
es fo
r:
calc
ulat
ing
(incl
udin
gca
lcul
atin
g de
vice
s)
chec
king
reco
gnise
the
limita
tions
of s
ome
appr
oach
es
(as i
n 2.
7)
refle
ct o
n a
solu
tion
to
a pr
oble
m c
omm
entin
g co
nstr
uctiv
ely
on th
e ch
oice
of c
heck
ing
stra
tegi
es
(as i
n 2.
7)
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
19
3 Al
gebr
a
Equ
atio
ns, f
orm
ulae
, exp
ress
ions
and
iden
titie
s 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 sy
mbo
ls to
reco
gnise
that
lett
er
dist
ingu
ish th
e pr
esen
t con
vinc
ing
exam
ine
and
refin
eus
e sy
mbo
ls an
d re
pres
ent u
nkno
wn
sym
bols
play
diff
eren
t di
ffere
nt ro
les p
laye
d al
gebr
aic
argu
men
ts to
alge
brai
c ar
gum
ents
re
pres
enta
tions
nu
mbe
rs o
r var
iabl
es;
role
s in
equa
tions
,by
lett
er sy
mbo
ls in
ju
stify
gen
eral
isatio
ns
pres
ente
d to
exp
lain
cons
isten
tly to
pre
sent
know
the
mea
ning
s fo
rmul
ae a
nd fu
nctio
ns;
equa
tions
, ide
ntiti
es,
or so
lutio
ns; r
elat
e ge
omet
rical
and
a
form
al p
roof
, e.g
. of
the
wor
ds te
rm,
know
the
mea
ning
s fo
rmul
ae a
nd fu
nctio
ns
argu
men
ts to
the
num
eric
al p
rope
rtie
s; de
rivin
g th
e fo
rmul
a ex
pres
sion
and
equa
tion
of th
e w
ords
form
ula
stru
ctur
e of
the
cont
ext
choo
se a
nd c
ombi
ne
for s
olvi
ng q
uadr
atic
an
d fu
nctio
n or
pro
blem
; pro
duce
re
pres
enta
tions
toeq
uatio
ns
simpl
e pr
oofs
pr
esen
t a c
onvi
ncin
g pr
oof
unde
rsta
nd th
atun
ders
tand
that
use
inde
x no
tatio
n fo
r us
e al
gebr
aic
appr
ecia
te th
e al
gebr
aic
oper
atio
nsal
gebr
aic
oper
atio
ns,
inte
ger p
ower
s and
re
pres
enta
tion
toge
nera
lity
of th
e fo
rms
follo
w th
e ru
les o
f in
clud
ing
the
use
of
simpl
e in
stan
ces o
f sy
nthe
sise
know
n ru
les
a +
b =
c an
d ab
= c
, ar
ithm
etic
br
acke
ts, f
ollo
w th
e th
e in
dex
law
s of
arit
hmet
ic, i
nclu
ding
w
here
eac
h te
rm c
an
rule
s of a
rithm
etic
; use
th
e co
mm
utat
ive
and
itsel
f be
an e
xpre
ssio
n;
inde
x no
tatio
n fo
r sm
all
dist
ribut
ive
law
s; ju
stify
us
e th
is in
sight
into
posit
ive
inte
ger p
ower
s th
ese
gene
ralis
atio
ns,
stru
ctur
e to
dev
elop
us
ing
spat
ial
e.g.
flu
ency
in tr
ansf
orm
ing
repr
esen
tatio
ns; u
se
mor
e co
mpl
ex
alge
brai
c ar
gum
ent t
oeq
uatio
ns
gene
ralis
e th
e in
dex
law
s for
mul
tiplic
atio
n an
d di
visio
n to
incl
ude
zero
, neg
ativ
e an
d fra
ctio
nal p
ower
s
20
Equ
atio
ns, f
orm
ulae
, exp
ress
ions
and
iden
titie
s (co
ntin
ued)
3.
1 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
simpl
ify li
near
alg
ebra
ic
expr
essio
ns b
y co
llect
ing
like
term
s;m
ultip
ly a
sing
le te
rm
over
a b
rack
et (i
nteg
er
coef
ficie
nts)
simpl
ify o
r tra
nsfo
rm
linea
r exp
ress
ions
by
colle
ctin
g lik
e te
rms;
mul
tiply
a si
ngle
term
ov
er a
bra
cket
simpl
ify o
r tra
nsfo
rm
alge
brai
c ex
pres
sions
by
taki
ng o
ut si
ngle
-te
rm c
omm
on fa
ctor
s;ad
d sim
ple
alge
brai
c fra
ctio
ns
deve
lop
fluen
cy in
tr
ansf
orm
ing
linea
r ex
pres
sions
; exp
and
the
prod
uct o
f tw
o lin
ear e
xpre
ssio
ns
of th
e fo
rm x
± n
and
fact
orise
sim
ple
quad
ratic
exp
ress
ions
;es
tabl
ish id
entit
ies s
uch
as th
e di
ffere
nce
of
two
squa
res;
com
pare
an
d ev
alua
te d
iffer
ent
repr
esen
tatio
ns o
f the
sa
me
cont
ext;
iden
tify
equi
vale
nt e
xpre
ssio
ns
and
conf
irm b
y tr
ansf
orm
atio
n
expa
nd a
nd fa
ctor
ise
quad
ratic
exp
ress
ions
;sim
plify
or t
rans
form
al
gebr
aic
fract
ions
, b
y fa
ctor
ising
and
e.
g.ca
ncel
ling
com
mon
fa
ctor
s; co
mpa
re a
nd
eval
uate
diff
eren
t re
pres
enta
tions
of t
he
sam
e co
ntex
t; id
entif
yeq
uiva
lent
exp
ress
ions
an
d co
nfirm
this
bytr
ansf
orm
atio
n
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
titie
s (co
ntin
ued)
3.
1 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
cons
truc
t and
so
lve
simpl
e lin
ear
equa
tions
with
inte
ger
coef
ficie
nts (
unkn
own
on o
ne si
de o
nly)
usin
g an
app
ropr
iate
met
hod
inve
rse
oper
atio
ns)
(e.g
.
cons
truc
t and
solv
e lin
ear e
quat
ions
with
inte
ger c
oeff
icie
nts
(unk
now
n on
eith
er o
r bo
th si
des,
with
out a
nd
with
bra
cket
s) u
sing
appr
opria
te m
etho
ds
inve
rse
oper
atio
ns,
(e.g
.tr
ansf
orm
ing
both
sid
es in
sam
e w
ay)
use
grap
hs a
nd se
t up
equ
atio
ns to
solv
e sim
ple
prob
lem
s in
volv
ing
dire
ctpr
opor
tion
cons
truc
t and
solv
e lin
ear e
quat
ions
with
inte
ger c
oeff
icie
nts
(with
and
with
out
brac
kets
, neg
ativ
e sig
ns a
nyw
here
in th
e eq
uatio
n, p
ositi
ve o
r ne
gativ
e so
lutio
n)
use
alge
brai
c m
etho
ds
to so
lve
prob
lem
s in
volv
ing
dire
ctpr
opor
tion;
rela
teal
gebr
aic
solu
tions
to
gra
phs o
f the
eq
uatio
ns; u
se IC
T
as a
ppro
pria
te
cons
truc
t lin
ear
equa
tions
and
sim
ple
linea
r ine
qual
ities
(one
va
riabl
e) to
repr
esen
t re
al-li
fe si
tuat
ions
or
mat
hem
atic
alpr
oble
ms;
solv
e lin
ear e
quat
ions
an
d in
equa
litie
s, re
pres
entin
g th
e so
lutio
n in
the
cont
ext
of th
e pr
oble
m
cons
truc
t sim
ple
quad
ratic
equ
atio
ns
to re
pres
ent r
eal-
life
situa
tions
or
mat
hem
atic
al p
robl
ems
and
solv
e th
em
usin
g fa
ctor
isatio
n,
grap
hica
l or t
rial a
ndim
prov
emen
t met
hods
;ju
stify
the
num
ber
of so
lutio
ns u
sing
alge
brai
c or
gra
phic
al
argu
men
ts a
nd se
lect
appr
opria
te so
lutio
ns,
inte
rpre
ting
thei
r ac
cura
cy
repr
esen
t rea
l-lif
e sit
uatio
ns o
r m
athe
mat
ical
pro
blem
s in
volv
ing:
mor
e co
mpl
ex
quad
ratic
eq
uatio
ns,
choo
sing
an
appr
opria
te
met
hod
ofso
lutio
n in
clud
ing
com
plet
ing
the
squa
re a
nd u
se o
f th
e fo
rmul
a
dire
ct o
r inv
erse
pr
opor
tion,
in
clud
ing
y ?
x 2 , y
?
1/x 2
rela
te a
lgeb
raic
so
lutio
ns to
gra
phic
al
repr
esen
tatio
n of
the
func
tions
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
titie
s (co
ntin
ued)
3.
1 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
use
syst
emat
ic tr
ial a
nd
impr
ovem
ent m
etho
ds
and
ICT
tool
s to
find
appr
oxim
ate
solu
tions
to
equ
atio
ns su
ch a
s x 2 +
x =
20
expl
ore
way
s of
cons
truct
ing
mod
els
of re
al-li
fe si
tuat
ions
by d
raw
ing
grap
hsan
d co
nstru
ctin
g al
gebr
aic e
quat
ions
and
in
equa
litie
s
(See
obj
ectiv
e ab
ove
(S
ee o
bjec
tive
abov
e
(See
obj
ectiv
e ab
ove
fo
r pro
gres
sion)
fo
r pro
gres
sion)
fo
r pro
gres
sion)
cons
truc
t a p
air o
f sim
ulta
neou
s lin
ear
equa
tions
to re
pres
ent
real
-life
situ
atio
ns
or m
athe
mat
ical
pr
oble
ms;
exam
ine
and
com
pare
al
gebr
aic
met
hods
of
solu
tion;
use
gra
phic
al
repr
esen
tatio
n to
expl
ain
why
the
inte
rsec
tion
of tw
o lin
es g
ives
the
com
mon
so
lutio
n an
d w
hy so
me
case
s hav
e no
com
mon
so
lutio
n an
d ot
hers
have
an
infin
ite n
umbe
r
sele
ct a
nd ju
stify
op
timum
met
hods
fo
r sol
ving
a p
air
of si
mul
tane
ous
linea
r equ
atio
ns in
a
varie
ty o
f con
text
s; co
nstr
uct s
ever
al
linea
r ine
qual
ities
in
one
and
two
varia
bles
to re
pres
ent
real
-life
situ
atio
ns
or m
athe
mat
ical
prob
lem
s; so
lve
the
ineq
ualit
ies
grap
hica
lly, i
dent
ifyin
g an
d in
terp
retin
g th
e so
lutio
n se
t in
the
cont
ext o
f the
pro
blem
solv
e m
ore
com
plex
pa
irs o
f sim
ulta
neou
s eq
uatio
ns g
ener
ated
fro
m re
al-li
fe c
onte
xts
or g
eom
etric
al
inve
stig
atio
ns,
incl
udin
g pa
irs w
here
on
e is
linea
r and
the
othe
r is q
uadr
atic
or o
f th
e fo
rm x
2 + y
2 = r2
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
titie
s (co
ntin
ued)
3.
1 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
use
simpl
e fo
rmul
ae
from
mat
hem
atic
s an
d ot
her s
ubje
cts;
subs
titut
e po
sitiv
ein
tege
rs in
to li
near
ex
pres
sions
and
fo
rmul
ae a
nd, i
n sim
ple
case
s, de
rive
a fo
rmul
a
use
form
ulae
from
m
athe
mat
ics a
nd o
ther
su
bjec
ts; s
ubst
itute
inte
gers
into
sim
ple
form
ulae
, inc
ludi
ng
exam
ples
that
lead
to
an
equa
tion
toso
lve;
subs
titut
epo
sitiv
e in
tege
rs in
toex
pres
sions
invo
lvin
g sm
all p
ower
s, e.
g.
3x 2 +
4 o
r 2x3 ; d
eriv
e sim
ple
form
ulae
use
form
ulae
from
m
athe
mat
ics a
nd
othe
r sub
ject
s;su
bstit
ute
num
bers
in
to e
xpre
ssio
ns a
nd
form
ulae
; der
ive
a fo
rmul
a an
d, in
sim
ple
case
s, ch
ange
its
subj
ect
deriv
e fo
rmul
ae, e
.g.
in th
e co
ntex
t of
men
sura
tion;
inte
rpre
t a
rang
e of
form
ulae
draw
n fro
m re
al-li
fe
cont
exts
and
oth
er
subj
ects
, rel
atin
gth
e va
riabl
es to
the
cont
ext a
nd d
escr
ibin
g th
eir b
ehav
iour
; so
lve
prob
lem
s by
man
ipul
atin
g fo
rmul
ae
deriv
e an
d us
e fo
rmul
ae th
at in
volv
e m
ore
varia
bles
or
mor
e co
mpl
ex
alge
brai
c ex
pres
sions
;m
anip
ulat
e fo
rmul
ae
in o
rder
to re
ach
a so
lutio
n, sh
ow in
sight
in
to th
e m
athe
mat
ical
co
nnec
tions
, e.g
. usin
g th
e co
ntex
t and
the
form
ulae
to e
xpla
in th
e pr
opor
tiona
l effe
ct o
f va
ryin
g va
lues
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 o
f a si
mpl
e se
quen
ce, g
iven
a
rule
(e.g
. fin
ding
a
term
from
the
prev
ious
te
rm, f
indi
ng a
term
gi
ven
its p
ositi
on in
th
e se
quen
ce)
gene
rate
term
s of
a lin
ear s
eque
nce
usin
g te
rm-to
-term
and
posit
ion-
to-te
rm
rule
s, on
pap
er a
nd
usin
g a
spre
adsh
eet o
r gr
aphi
cs c
alcu
lato
r
gene
rate
term
s of a
se
quen
ce u
sing
term
-to
-term
and
pos
ition
-to
-term
rule
s, on
pap
er
and
usin
g IC
T
gene
rate
sequ
ence
s fro
m p
atte
rns o
r pr
actic
al c
onte
xts a
nd
desc
ribe
the
gene
ral
term
in si
mpl
e ca
ses
deve
lop,
com
pare
and
ev
alua
te a
lgeb
raic
and
sp
atia
l rep
rese
ntat
ions
of
situ
atio
ns th
at
gene
rate
sequ
ence
s; in
terp
ret,
dedu
ce a
nd
just
ify g
ener
alisa
tions
fo
r the
nth
term
of
linea
r and
qua
drat
ic
sequ
ence
s, in
clud
ing
the
prop
ertie
s of
squa
re a
nd tr
iang
ular
num
bers
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, ju
stify
ing
its fo
rm b
y re
ferr
ing
toth
e ac
tivity
or p
ract
ical
cont
ext f
rom
whi
ch it
was
gen
erat
ed
gene
rate
sequ
ence
s fro
m p
ract
ical
con
text
san
d w
rite
and
just
ify a
n ex
pres
sion
to d
escr
ibe
the
nth
term
of a
n ar
ithm
etic
sequ
ence
25
Seq
uenc
es, f
unct
ions
and
gra
phs (
cont
inue
d)
3.2 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
expr
ess s
impl
e ex
pres
s sim
ple
find
the
inve
rse
of a
co
mpa
re g
raph
ical
, fu
nctio
ns in
wor
ds,
func
tions
alg
ebra
ical
ly
linea
r fun
ctio
n al
gebr
aic
and
then
usin
g sy
mbo
ls;an
d re
pres
ent t
hem
ge
omet
rical
re
pres
ent t
hem
in
in m
appi
ngs o
r on
a re
pres
enta
tions
,m
appi
ngs
spre
adsh
eet
incl
udin
g m
appi
ng
diag
ram
s, to
exp
lain
th
e ef
fect
of:
rota
ting
the
line
y =
mx
+ c
thro
ugh
90° a
bout
any
po
int
refle
ctin
g th
e lin
ey
= m
x +
c in
the
line
y =
x
deriv
e pr
oper
ties o
f pe
rpen
dicu
lar l
ines
and
of
the
inve
rse
func
tion
gene
rate
coo
rdin
ate
gene
rate
poi
nts i
n al
l ge
nera
te p
oint
s and
expl
ore
grap
hs o
f ex
plor
e co
nnec
tions
ex
plor
e gr
aphs
of
pairs
that
satis
fy a
fo
ur q
uadr
ants
and
plo
t pl
ot g
raph
s of l
inea
rfu
nctio
ns o
f the
form
be
twee
n th
e fo
rm o
f ex
pone
ntia
l and
sim
ple
linea
r rul
e; p
lot
the
grap
hs o
f lin
ear
func
tions
, whe
re y
isy
= xn (n
an
inte
ger)
the
equa
tion
and
the
trig
onom
etric
alan
d re
cogn
ise th
eir
resu
lting
gra
phs o
f fu
nctio
ns a
nd re
cogn
iseth
e gr
aphs
of s
impl
e fu
nctio
ns, w
here
y is
give
n im
plic
itly
in te
rms
linea
r fun
ctio
ns, w
here
give
n ex
plic
itly
in te
rms
of x
(e.g
. ay
+ bx
= 0
, ch
arac
teris
tic sh
apes
; qu
adra
tic a
nd c
ubic
th
eir c
hara
cter
istic
y is
give
n ex
plic
itly
of x
, on
pape
r and
y
+ bx
+ c
= 0)
, on
pape
rva
ry th
e va
lues
of a
, b
func
tions
such
as:
shap
es; a
pply
to th
ein
term
s of x
, on
usin
g IC
T; re
cogn
ise
and
usin
g IC
T; fin
d th
ean
d c
in fu
nctio
ns su
ch
grap
h y =
f(x)
the
y =
(x +
2)(x
– 5
) pa
per a
nd u
sing
ICT;
that
equ
atio
ns o
f gr
adie
nt o
f lin
es g
iven
as y
= ax
2 + c,
tr
ansf
orm
atio
ns
y =
(x –
2)(x
2 + 7
x + 12
)re
cogn
ise st
raig
ht-li
neth
e fo
rm y
= m
x +
cby
equ
atio
ns o
f the
y =
ax 3 +
c,
y =
f(x) +
a, y
= a
f(x),
grap
hs p
aral
lel t
o th
e
corr
espo
nd to
stra
ight
-fo
rm y
= m
x +
c, gi
ven
y =
(x +
b)2 u
sing
a y
= f(x
+ a
), y
= f(a
x) fo
r
y =
x 2 – 2
x +
1 x-
axis
or y
-axi
s lin
e gr
aphs
va
lues
for m
and
c gr
aph
plot
ter t
o ex
plai
n lin
ear,
quad
ratic
, sin
e
y =
x 3 + 3
how
this
tran
sfor
ms
and
cosin
e fu
nctio
ns;
the
grap
h us
e a
grap
h pl
otte
r to
incl
ude
feat
ures
such
ex
plai
n th
e ef
fect
of
as ro
ots o
f the
equ
atio
n,
tran
sfor
mat
ions
on
the
inte
rcep
ts a
nd tu
rnin
g gr
aph
and
gene
ralis
epo
ints
to
oth
er fu
nctio
ns
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 (
cont
inue
d)
3.2 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
plot
and
inte
rpre
t the
co
nstr
uct l
inea
r co
nstr
uct f
unct
ions
sk
etch
and
inte
rpre
t ap
ply
know
ledg
e of
se
t up
a m
athe
mat
ical
gr
aphs
of s
impl
e lin
ear
func
tions
aris
ing
from
ar
ising
from
real
-life
gr
aphs
that
mod
el re
al-
mat
hem
atic
al fu
nctio
ns
mod
el o
f a re
al-li
fe
func
tions
aris
ing
from
re
al-li
fe p
robl
ems
prob
lem
s and
plo
t the
ir lif
e sit
uatio
ns, i
nclu
ding
to
pro
blem
s inv
olvi
ng:
cont
ext o
r pro
blem
, re
al-li
fe si
tuat
ions
, e.g
. an
d pl
ot th
eir
corr
espo
ndin
g gr
aphs
; th
ose
gene
rate
d id
entif
ying
the
optim
isat
ion,
co
nver
sion
grap
hs
corr
espo
ndin
g gr
aphs
; in
terp
ret g
raph
s aris
ing
from
oth
er su
bjec
tsva
riabl
es a
nd th
eir
usin
g nu
mer
ical
, di
scus
s and
inte
rpre
t fro
m re
al si
tuat
ions
, e.g
. su
ch a
s sci
ence
;fu
nctio
nal r
elat
ions
hip;
al
gebr
aic
and
grap
hs a
risin
g fro
m
time
serie
s gra
phs
use
mat
hem
atic
alus
e gr
aphs
and
gr
aphi
cal,
real
situ
atio
ns, e
.g.
argu
men
t to
just
ify
sket
ches
to e
xpla
in
tech
niqu
es,
dist
ance
–tim
e gr
aphs
fe
atur
es o
f the
ir sh
apes
th
e be
havi
our o
f the
in
clud
ing
max
ima
varia
bles
and
to e
xpla
in
and
min
ima
or ju
stify
the
effe
ct o
f
usin
g IC
T to
fit
assu
mpt
ions
in
a cu
rve
to d
ata
the
mod
el
from
a re
al c
onte
xt
such
as a
scie
nce
expe
rimen
t re
peat
edpr
opor
tiona
l ch
ange
, e.g
. co
mpo
und
inte
rest
use
ICT
to e
xplo
re th
e gr
aphi
cal r
epre
sent
atio
n of
alg
ebra
ic e
quat
ions
an
d to
inte
rpre
t how
pr
oper
ties o
f the
gra
ph
are r
elat
ed to
feat
ures
of
the
equa
tion,
p
aral
lel a
nd
e.g.
perp
endi
cula
r lin
es
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 (
cont
inue
d)
3.2 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
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
vario
us p
oint
s and
se
ctio
ns o
f stra
ight
-lin
e gr
aphs
, incl
udin
g in
terc
epts
and
inte
rsec
tions
, e.g
. sol
ving
sim
ulta
neou
s lin
ear
equa
tions
29
4 G
eom
etry
and
mea
sure
s
Geo
met
rical
reas
onin
g4.
1 Year
7
use
corr
ectly
the
voca
bula
ry, n
otat
ion
and
labe
lling
co
nven
tions
for l
ines
, an
gles
and
shap
es
iden
tify
para
llel a
nd
perp
endi
cula
r lin
es;
know
the
sum
of a
ngle
s at
a p
oint
, on
a st
raig
ht
line
and
in a
tria
ngle
; re
cogn
ise v
ertic
ally
op
posit
e an
gles
iden
tify
alte
rnat
e an
gles
and
corr
espo
ndin
g an
gles
; un
ders
tand
a p
roof
th
at: th
e an
gle
sum
of a
tr
iang
le is
180
° and
of
a q
uadr
ilate
ral
is 36
0°
the
exte
rior a
ngle
of
a tr
iang
le is
eq
ual t
o th
e su
m
of th
e tw
o in
terio
r op
posit
e an
gles
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 b
etw
een
conv
entio
ns,
defin
ition
s and
der
ived
prop
ertie
s
expl
ain
how
to fi
nd,
calc
ulat
e an
d us
e:
the
sum
s of t
he
inte
rior a
nd
exte
rior a
ngle
s of
qua
drila
tera
ls,
pent
agon
s and
he
xago
ns
the
inte
rior a
nd
exte
rior a
ngle
s of
regu
lar p
olyg
ons
know
the
defin
ition
of
a ci
rcle
and
the
nam
es
of it
s par
ts; e
xpla
in
why
insc
ribed
regu
lar
poly
gons
can
be
cons
truc
ted
by e
qual
di
visio
ns o
f a c
ircle
exam
ine
and
exam
ine
and
crea
te
pres
ent r
igor
ous a
nd
refin
e ar
gum
ents
chai
ns o
f ded
uctiv
e su
stai
ned
argu
men
ts
in so
lutio
ns to
reas
onin
g in
solu
tions
in
the
solu
tion
of
geom
etric
al p
robl
ems,
to m
ore
com
plex
ge
omet
rical
pro
blem
s;di
stin
guish
ing
betw
een
geom
etric
al p
robl
ems
cons
truc
t for
mal
pr
actic
al d
emon
stra
tion
geom
etric
al p
roof
s an
d pr
oof;
prod
uce
simpl
e pr
oofs
use
dyna
mic
imag
es to
ex
amin
e th
e po
ints
dem
onst
rate
inva
riant
an
d lin
es u
sed
to c
reat
e re
latio
nshi
ps b
etw
een
exam
ine
and
crea
te
stan
dard
con
stru
ctio
ns
radi
i, ch
ords
and
pr
oofs
of t
he c
ircle
an
d us
e th
e co
nditi
ons
tang
ents
in c
ircle
s;th
eore
ms;
use
circ
le
of c
ongr
uenc
e to
deve
lop
argu
men
ts
theo
rem
s to
solv
e pr
esen
t a p
roof
that
the
to e
xpla
in a
nd ju
stify
pr
oble
ms
stan
dard
con
stru
ctio
ns
simpl
e ci
rcle
pro
pert
ies
are
exac
t an
d th
eore
ms
30
Geo
met
rical
reas
onin
g (c
ontin
ued)
4.
1
Year
8
Year
9
Year
10
Year
7
iden
tify
and
use
angl
e,
side
and
sym
met
ry
prop
ertie
s of t
riang
les
and
quad
rilat
eral
s; ex
plor
e ge
omet
rical
pr
oble
ms i
nvol
ving
th
ese
prop
ertie
s, ex
plai
ning
reas
onin
g or
ally
, usin
g st
ep-
by-s
tep
dedu
ctio
n su
ppor
ted
by d
iagr
ams
solv
e ge
omet
rical
pr
oble
ms u
sing
side
and
angl
e pr
oper
ties
of e
quila
tera
l, iso
scel
es a
nd ri
ght-
angl
ed tr
iang
les a
nd
spec
ial q
uadr
ilate
rals,
ex
plai
ning
reas
onin
g w
ith d
iagr
ams a
nd te
xt;
clas
sify
quad
rilat
eral
s by
thei
r geo
met
rical
pr
oper
ties
know
that
if tw
o 2-
D
shap
es a
re c
ongr
uent
, co
rres
pond
ing
sides
an
d an
gles
are
equ
al
solv
e pr
oble
ms
usin
g pr
oper
ties o
f an
gles
, of p
aral
lel a
nd
inte
rsec
ting
lines
and
of
tria
ngle
s and
oth
erpo
lygo
ns, j
ustif
ying
in
fere
nces
and
ex
plai
ning
reas
onin
g w
ith d
iagr
ams a
nd te
xt
unde
rsta
ndco
ngru
ence
and
ex
plor
e sim
ilarit
y
inve
stig
ate
Pyth
agor
as’
theo
rem
, usin
g a
varie
ty
of m
edia
, thr
ough
its
hist
oric
and
cultu
ral
root
s, in
clud
ing
‘pic
ture
’ pr
oofs
solv
e ge
omet
rical
pr
oble
ms u
sing
prop
ertie
s of l
ines
, an
gles
, pol
ygon
san
d ci
rcle
s; ju
stify
ar
gum
ents
and
so
lutio
ns u
sing
dedu
ctiv
e re
ason
ing
draw
infe
renc
es a
bout
pr
oper
ties o
f sim
ilar
2-D
shap
es a
nd u
se
prop
ortio
nal r
easo
ning
to
solv
e ge
omet
rical
an
d tr
igon
omet
rical
prob
lem
s
visu
alise
and
man
ipul
ate
dyna
mic
imag
es a
ndus
e sc
ale
draw
ing
toin
vest
igat
e ar
eas o
fsq
uare
s on
sides
of
right
-ang
led
and
non
right
-ang
led
tria
ngle
s,re
latin
g fin
ding
s to
Pyth
agor
as’ t
heor
em;
use
Pyth
agor
as’ t
heor
emto
solv
e pr
oble
ms i
n 2-
Dan
d sim
ple
3-D
cas
es
Year
11
form
alise
exi
stin
gkn
owle
dge
of li
nes,
angl
es a
nd p
olyg
ons b
y:
usin
g th
eco
ngru
ence
cond
ition
s (SS
S, S
AS,
RHS,
ASA
) to
dedu
cefa
mili
ar p
rope
rtie
sof
tria
ngle
s and
quad
rilat
eral
s, e.
g.an
isos
cele
s tria
ngle
has t
wo
equa
lan
gles
expl
aini
ngw
hy st
anda
rdco
nstr
uctio
ns w
ork,
e.g.
obs
ervi
ng th
atlin
es jo
inin
g po
ints
whe
re co
mpa
ss a
rcs
mee
t are
side
s of a
rhom
bus
Exte
nsio
n
(see
obj
ectiv
e ab
ove
for p
rogr
essio
n)
enga
ge w
ith a
nd
expl
ain
the
stag
es o
f a
varie
ty o
f pro
ofs o
f Py
thag
oras
’ the
orem
;us
e Py
thag
oras
’ th
eore
m to
solv
e m
ore
com
plex
3-D
pro
blem
s
pres
ent a
nd ju
stify
a
form
al p
roof
of
Pyth
agor
as’ t
heor
em
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
31
Geo
met
rical
reas
onin
g (c
ontin
ued)
4.
1 Year
7
Year
8
Year
9
Year
10
Year
11
use
2-D
repr
esen
tatio
ns
visu
alise
3-D
shap
es
to v
isual
ise 3
-D sh
apes
fro
m th
eir n
ets;
use
and
dedu
ce so
me
of
geom
etric
al p
rope
rtie
s th
eir p
rope
rtie
s of
cub
oids
and
shap
es
mad
e fro
m c
uboi
ds;
use
simpl
e pl
ans a
nd
elev
atio
ns
visu
alise
and
use
2-D
re
pres
enta
tions
of 3
-D
obje
cts;
anal
yse
3-D
sh
apes
thro
ugh
2-D
pr
ojec
tions
, inc
ludi
ng
plan
s and
ele
vatio
ns
visu
alise
and
des
crib
e pr
oper
ties o
f poi
nts,
lines
and
pla
nes i
n 3-
D
spac
e, in
clud
ing
cros
s se
ctio
ns c
reat
ed b
y sli
cing
a 3
-D sh
ape
visu
alise
and
m
anip
ulat
e im
ages
to
est
ablis
h tr
igon
omet
rical
rela
tions
hips
by:
gene
ratin
g tr
iang
les u
sing
a ro
tatin
g un
it ra
dius
(circ
le, c
entr
e th
eor
igin
)
iden
tifyi
ng th
e pr
oper
ties o
f sim
ilar t
riang
les
form
ed b
yen
larg
emen
ts o
f th
e ci
rcle
use
trig
onom
etric
al
rela
tions
hips
to so
lve
simpl
e pr
oble
ms i
n 2-
D,
incl
udin
g be
arin
gs
deriv
e th
e fo
rmul
a $
ab si
nC fo
r the
area
of a
tria
ngle
; us
e tr
igon
omet
rical
re
latio
nshi
ps to
solv
e m
ore
com
plex
2-
D p
robl
ems a
nd
prob
lem
s in
3-D
, suc
has
the
angl
e be
twee
n a
line
and
a pl
ane
Exte
nsio
n
draw
, ske
tch
and
com
pare
the
grap
hsof
trig
onom
etric
alfu
nctio
ns a
nd
tran
sfor
mat
ions
of
thes
e gr
aphs
; pro
ve th
e sin
e an
d co
sine
rule
s an
d us
e th
em to
solv
e 2-
D a
nd 3
-D p
robl
ems
in a
rang
e of
con
text
s
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
32
Tra
nsfo
rmat
ions
and
coor
dina
tes
4.2
Year
8
Year
9
Year
10
Year
11
Year
7
unde
rsta
nd a
nd u
se th
e la
ngua
ge a
nd n
otat
ion
asso
ciat
ed w
ithre
flect
ions
, tra
nsla
tions
and
rota
tions
reco
gnise
and
visu
alise
th
e sy
mm
etrie
s of a
2-D
sh
ape
tran
sfor
m 2
-D sh
apes
by
: refle
ctin
g in
giv
en
mirr
or li
nes
rota
ting
abou
t a
give
n po
int
tran
slatin
g
expl
ore
thes
e tr
ansf
orm
atio
ns a
nd
sym
met
ries u
sing
ICT
iden
tify
all t
he
sym
met
ries o
f 2-D
sh
apes
tran
sfor
m 2
-D sh
apes
by
rota
tion,
refle
ctio
nan
d tr
ansla
tion,
on
pape
r and
usin
g IC
T
try
out m
athe
mat
ical
re
pres
enta
tions
of
simpl
e co
mbi
natio
ns o
f th
ese
tran
sfor
mat
ions
iden
tify
refle
ctio
n sy
mm
etry
in 3
-D
shap
es
reco
gnise
that
tran
slatio
ns, r
otat
ions
an
d re
flect
ions
pr
eser
ve le
ngth
and
an
gle,
and
map
obj
ects
on
to c
ongr
uent
im
ages
expl
ore
and
com
pare
m
athe
mat
ical
repr
esen
tatio
ns o
f co
mbi
natio
ns o
f tr
ansla
tions
, rot
atio
ns
and
refle
ctio
ns o
f 2-D
sh
apes
, on
pape
r and
us
ing
ICT
devi
se in
stru
ctio
ns fo
r a
com
pute
r to
gene
rate
and
tran
sfor
m sh
apes
use
prec
ise la
ngua
ge
and
nota
tion
to d
escr
ibe
and
gene
ralis
e th
e re
sults
of c
ombi
ning
tr
ansf
orm
atio
ns o
f 2-D
sh
apes
on
pape
r and
us
ing
ICT,
incl
udin
g:
rota
tions
abo
ut
any
poin
t
refle
ctio
ns in
any
lin
e
tran
slatio
ns u
sing
vect
or n
otat
ion
a tr
ansf
orm
atio
n an
d its
inve
rse
gene
rate
and
ana
lyse
pa
tter
ns, e
.g. I
slam
ic
desig
ns
expl
ain
and
dem
onst
rate
grap
hica
lly th
e ef
fect
s of c
ombi
ning
tr
ansla
tions
, usin
g ve
ctor
not
atio
n,
incl
udin
g: th
e ru
le fo
rad
ditio
n of
vec
tors
scal
ar
mul
tiplic
atio
n of
a
vect
or (r
epea
ted
addi
tion)
Exte
nsio
n
expl
ain
and
dem
onst
rate
grap
hica
lly th
e ef
fect
s of c
ombi
ning
tr
ansla
tions
, usin
g ve
ctor
not
atio
n,
incl
udin
g:
the
diffe
renc
e of
tw
o ve
ctor
s
the
resu
ltant
of
two
vect
ors
the
com
mut
ativ
e an
d as
soci
ativ
epr
oper
ties o
f ve
ctor
add
ition
solv
e sim
ple
geom
etric
al p
robl
ems
in 2
-D u
sing
vect
ors
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
33
Tra
nsfo
rmat
ions
and
coor
dina
tes (
cont
inue
d)
4.2
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
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
poin
ts d
eter
min
edby
geo
met
rical
in
form
atio
n
unde
rsta
nd a
nd u
se th
e la
ngua
ge a
nd n
otat
ion
asso
ciat
ed w
ithen
larg
emen
t; en
larg
e 2-
D sh
apes
, giv
en a
ce
ntre
of e
nlar
gem
ent
and
a po
sitiv
e in
tege
r sc
ale
fact
or; e
xplo
re
enla
rgem
ent u
sing
ICT
mak
e sc
ale
draw
ings
find
the
mid
poin
t of
the
line
segm
ent A
B,
give
n th
e co
ordi
nate
s of
poi
nts A
and
B
enla
rge
2-D
shap
es,
give
n a
cent
re o
f en
larg
emen
t and
a
posit
ive
inte
ger s
cale
fa
ctor
, on
pape
r and
us
ing
ICT;
iden
tify
the
scal
e fa
ctor
of
an e
nlar
gem
ent
as th
e ra
tio o
f the
le
ngth
s of a
ny tw
o co
rres
pond
ing
line
segm
ents
; rec
ogni
se
that
enl
arge
men
tspr
eser
ve a
ngle
but
not
le
ngth
, and
und
erst
and
the
impl
icat
ions
of
enla
rgem
ent f
or
perim
eter
enla
rge
2-D
shap
es
usin
g po
sitiv
e,
fract
iona
l and
neg
ativ
e sc
ale
fact
ors,
on
pape
r and
usin
g IC
T; u
se re
cipr
ocal
s as
a m
ultip
licat
ive
inve
rse
in th
e co
ntex
t of
enl
arge
men
t; re
cogn
ise th
e sim
ilarit
yof
resu
lting
shap
es
and
expl
ain
the
effe
ct
of e
nlar
gem
ent o
n pe
rimet
er
use
and
inte
rpre
t map
s an
d sc
ale
draw
ings
in th
e co
ntex
t of
mat
hem
atic
s and
oth
er
subj
ects
use
the
coor
dina
te
grid
to so
lve
prob
lem
s in
volv
ing
tran
slatio
ns,
rota
tions
, ref
lect
ions
an
d en
larg
emen
ts
appl
y th
e pr
oper
ties
of si
mila
r tria
ngle
s and
Pyth
agor
as’ t
heor
em
to so
lvin
g pr
oble
ms
pres
ente
d on
a 2
-D
coor
dina
te g
rid;
use
a 3-
D c
oord
inat
e gr
id to
repr
esen
t sim
ple
shap
es
enla
rge
3-D
shap
es;
iden
tify
and
expl
ain
the
effe
cts o
f enl
arge
men
t on
are
as a
nd v
olum
es
of si
mila
r sha
pes
and
solid
s; re
late
this
unde
rsta
ndin
g to
pr
actic
al c
onte
xts,
e.g.
in
bio
logy
34
Con
stru
ctio
n an
d lo
ci
4.3 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
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
:
mea
sure
and
dra
w
lines
to th
e ne
ares
tm
illim
etre
and
angl
es, i
nclu
ding
re
flex
angl
es, t
oth
e ne
ares
t deg
ree
cons
truc
t a
tria
ngle
, giv
en
two
sides
and
the
incl
uded
ang
le
(SAS
) or t
wo
angl
es
and
the
incl
uded
sid
e (A
SA)
use
stra
ight
edg
e an
d co
mpa
sses
to co
nstr
uct:
the
mid
poin
t and
pe
rpen
dicu
lar
bise
ctor
of a
line
se
gmen
t
the
bise
ctor
of
an a
ngle
the
perp
endi
cula
r fro
m a
poi
nt to
a
line
the
perp
endi
cula
r fro
m a
poi
nt o
n
a lin
e
a tr
iang
le, g
iven
th
ree
sides
(SSS
)
use
stra
ight
edg
e an
d co
mpa
sses
to c
onst
ruct
tr
iang
les,
give
n rig
ht
angl
e, h
ypot
enus
e an
d sid
e (R
HS)
use
ICT
to e
xplo
re
cons
truc
tions
of
tria
ngle
s and
oth
er2-
D sh
apes
find
the
locu
s of a
poi
nt
that
mov
es a
ccor
ding
to
a si
mpl
e ru
le, b
oth
by re
ason
ing
and
by
usin
g IC
T
use
prop
ertie
s of
2-D
and
3-D
shap
es
to m
ake
accu
rate
co
nstr
uctio
ns o
n pa
per
and
usin
g IC
T; in
clud
ing
cons
truc
ting
tria
ngle
s fro
m c
ombi
natio
nsof
side
and
ang
lefa
cts,
revi
ewin
g an
d ge
nera
lisin
g fin
ding
sto
iden
tify
whi
ch o
f th
ese
cond
ition
s def
ine
uniq
ue c
onst
ruct
ions
use
ICT
to e
xplo
re
use
ICT
to e
xplo
re
cons
truc
tions
th
ese
cons
truc
tions
use
rule
r and
pro
trac
tor
to c
onst
ruct
sim
ple
nets
of 3
-D sh
apes
, c
uboi
d, re
gula
r e.
g.te
trah
edro
n, sq
uare
-ba
sed
pyra
mid
, tr
iang
ular
pris
m
find
simpl
e lo
ci, b
oth
by re
ason
ing
and
by
usin
g IC
T, to
pro
duce
sh
apes
and
pat
hs, e
.g.
an e
quila
tera
l tria
ngle
visu
alise
and
des
crib
e th
e lo
cus o
f a p
oint
th
at m
oves
acc
ordi
ng
to a
mor
e co
mpl
ex
rule
; exp
lain
the
path
usin
g ac
cura
te
geom
etric
al v
ocab
ular
yan
d no
tatio
n an
d us
e a
varie
ty o
f med
ia,
incl
udin
g dy
nam
ic
geom
etry
soft
war
e,
sket
ches
and
gra
phs
crea
te a
cha
in o
f re
ason
ing
to d
educ
eth
e eq
uatio
n of
a
circ
le b
y ap
plyi
ng
Pyth
agor
as’ t
heor
em to
th
e lo
cus o
f a p
oint
35
Mea
sure
s and
men
sura
tion
4.4 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
choo
se a
nd u
se u
nits
of
mea
sure
men
t to
mea
sure
, est
imat
e,
calc
ulat
e an
d so
lve
prob
lem
s in
ever
yday
cont
exts
; con
vert
one
m
etric
uni
t to
anot
her,
gra
ms t
o ki
logr
ams;
e.g.
read
and
inte
rpre
t sca
les
on a
rang
e of
mea
surin
g in
stru
men
ts
dist
ingu
ish b
etw
een
and
estim
ate
the
size
of a
cute
,ob
tuse
and
refle
x an
gles
choo
se a
nd u
se u
nits
of
mea
sure
men
t to
mea
sure
, est
imat
e,
calc
ulat
e an
d so
lve
prob
lem
s in
a ra
nge
of
cont
exts
; kno
w ro
ugh
met
ric e
quiv
alen
ts o
f im
peria
l mea
sure
s in
com
mon
use
, suc
h
as m
iles,
poun
ds (l
b)
and
pint
s
use
bear
ings
to sp
ecify
di
rect
ion
solv
e pr
oble
ms i
nvol
ving
m
easu
rem
ents
in a
va
riety
of c
onte
xts;
conv
ert b
etw
een
area
m
easu
res (
e.g.
mm
2 to
cm2 , c
m2 to
m2 , a
nd v
ice
vers
a) a
nd b
etw
een
volu
me
mea
sure
s (e.
g.
mm
3 to c
m3 , c
m3 to
m3 ,
and
vice
ver
sa)
Inte
rpre
t and
exp
lore
co
mbi
ning
mea
sure
s in
to ra
tes o
f cha
nge
in
ever
yday
cont
exts
(e.g
. km
pe
r hou
r, pe
nce
per m
etre
); us
e co
mpo
und
mea
sure
s to
com
pare
in re
al-li
fe
cont
exts
(e.g
. tra
vel g
raph
s an
d va
lue
for m
oney
), us
ing
ICT
as a
ppro
pria
te.
inte
rpre
t and
use
co
mpo
und
mea
sure
s, in
clud
ing
from
oth
er
subj
ects
and
real
life
; so
lve
prob
lem
s inv
olvi
ng
rate
s; co
nver
t bet
wee
n co
mpo
und
mea
sure
s, ch
oosin
g un
its m
ost
suite
d to
the
solu
tion
mak
e co
nnec
tions
be
twee
n th
e co
ntin
uity
of th
e nu
mbe
r lin
e an
d co
ntin
uous
mea
sure
s; cr
itica
lly e
xam
ine
the
mea
sure
men
ts u
sed
in a
pr
oble
m a
nd th
eir e
ffect
on th
e ac
cura
cy o
f the
so
lutio
n, e
.g. u
nder
stan
d ho
w e
rror
s can
be
com
poun
ded
com
mun
icat
e th
e so
lutio
nto
a p
robl
em
invo
lvin
g m
easu
rem
ent,
expl
aini
ng th
e lim
itatio
ns o
f ac
cura
cy u
sing
uppe
r and
lo
wer
bou
nds
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN
36
Mea
sure
s and
men
sura
tion
(con
tinue
d)
4.4 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
know
and
use
the
deriv
e an
d us
e fo
rmul
ae
form
ula
for t
he a
rea
of
for t
he a
rea
of a
a re
ctan
gle;
cal
cula
tetr
iang
le, p
aral
lelo
gram
th
e pe
rimet
er a
nd a
rea
and
trap
eziu
m;
of sh
apes
mad
e fro
m
calc
ulat
e ar
eas o
fre
ctan
gles
co
mpo
und
shap
es
solv
e pr
oble
ms
invo
lvin
g m
ore
com
plex
shap
esan
d so
lids,
incl
udin
gse
gmen
ts o
f circ
les a
ndfru
stum
s of c
ones
calc
ulat
e th
e su
rfac
e ar
ea o
f cub
es a
nd
cubo
ids
know
and
use
the
form
ula
for t
he v
olum
e of
a c
uboi
d; c
alcu
late
volu
mes
and
surf
ace
area
s of c
uboi
ds a
nd
shap
es m
ade
from
cu
boid
s
know
and
use
the
form
ulae
for t
heci
rcum
fere
nce
and
area
of
a c
ircle
calc
ulat
e th
e su
rfac
e ar
ea a
nd v
olum
e of
rig
ht p
rism
s
pres
ent a
con
cise
re
ason
ed a
rgum
ent t
ode
rive
form
ulae
for:
leng
ths o
f circ
ular
ar
cs
area
s of s
ecto
rs o
f a
circ
le
surf
ace
area
of
a cy
linde
r
volu
me
of a
cy
linde
r
solv
e pr
oble
ms
invo
lvin
g th
e us
e of
th
ese
form
ulae
pres
ent a
con
cise
re
ason
ed a
rgum
ent
whe
n de
rivin
gfo
rmul
ae fo
r the
surf
ace
area
s of
pyra
mid
s and
con
es;
expl
ore
conn
ectio
ns
betw
een:
form
ulae
for
the
volu
me
of a
py
ram
id a
nd th
e re
late
d cu
boid
form
ulae
for
the
surf
ace
area
an
d vo
lum
e of
a
sphe
re a
nd th
eci
rcum
scrib
ed a
nd
insc
ribed
cub
es
solv
e pr
oble
ms
invo
lvin
g th
e us
e of
th
ese
form
ulae
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
ossib
le
answ
ers,
give
n a
ques
tion
that
can
be
addr
esse
d by
stat
istic
al
met
hods
disc
uss a
pro
blem
that
ca
n be
add
ress
ed b
y st
atis
tical
met
hods
an
d id
entif
y re
late
d qu
estio
ns to
exp
lore
sugg
est a
pro
blem
toex
plor
e us
ing
stat
istic
al
met
hods
, fra
me
ques
tions
and
raise
co
njec
ture
s
inde
pend
ently
dev
ise a
su
itabl
e pl
an fo
r a m
ore
com
plex
stat
istic
al
proj
ect,
sele
ctin
g su
itabl
e hy
poth
eses
to
addr
ess t
he p
robl
em
eval
uate
pos
sible
di
ffic
ultie
s with
pl
anne
d ap
proa
ches
; ad
just
the
proj
ect
plan
acc
ordi
ngly
,in
clud
ing
reco
nsid
erin
g hy
poth
eses
deci
de w
hich
dat
a w
ould
be
rele
vant
to a
n en
quiry
and
pos
sible
so
urce
s
deci
de w
hich
dat
a to
col
lect
to a
nsw
er
a qu
estio
n an
d th
e de
gree
of a
ccur
acy
need
ed; i
dent
ifypo
ssib
le so
urce
s;co
nsid
er a
ppro
pria
te
sam
ple
size
disc
uss h
ow d
iffer
ent
sets
of d
ata
rela
te to
th
e pr
oble
m; i
dent
ifypo
ssib
le p
rimar
y or
se
cond
ary
sour
ces;
dete
rmin
e th
e sa
mpl
e siz
e an
d m
ost
appr
opria
te d
egre
e of
ac
cura
cy
just
ify th
e sa
mpl
ing
met
hod
sele
cted
, id
entif
y po
ssib
le
sour
ces o
f bia
s and
pl
an h
ow to
min
imise
it
iden
tify
prac
tical
prob
lem
s suc
h as
non-
resp
onse
or
miss
ing
data
and
refin
e ap
proa
ches
to m
inim
ise
thei
r im
pact
on
the
valid
ity o
f the
resu
lts
38
5.1S
peci
fyin
g a
prob
lem
, pla
nnin
g an
d co
llect
ing
(con
tinue
d)
Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
plan
how
to c
olle
ct a
nd
orga
nise
smal
l set
s of
data
from
surv
eys a
nd
expe
rimen
ts:
desig
n da
ta
colle
ctio
n sh
eets
or q
uest
ionn
aire
s to
use
in a
sim
ple
surv
ey
cons
truc
tfre
quen
cy ta
bles
fo
r gat
herin
g di
scre
te d
ata,
gr
oupe
d w
here
appr
opria
te
in e
qual
cla
ss
inte
rval
s
plan
how
to c
olle
ct
the
data
; con
stru
ct
frequ
ency
tabl
es w
itheq
ual c
lass
inte
rval
s for
ga
ther
ing
cont
inuo
us
data
and
two-
way
ta
bles
for r
ecor
ding
di
scre
te d
ata
desig
n a
surv
ey o
r ex
perim
ent t
o ca
ptur
e th
e ne
cess
ary
data
from
one
or m
ore
sour
ces;
desig
n, tr
ial
and
if ne
cess
ary
refin
eda
ta c
olle
ctio
n sh
eets
;co
nstr
uct t
able
s for
ga
ther
ing
larg
e di
scre
tean
d co
ntin
uous
sets
of
raw
dat
a, c
hoos
ing
suita
ble
clas
s int
erva
ls;de
sign
and
use
two-
way
tabl
es
gath
er d
ata
from
sp
ecifi
ed se
cond
ary
sour
ces,
incl
udin
g pr
inte
d ta
bles
and
list
s, an
d IC
T-ba
sed
sour
ces,
incl
udin
g th
e in
tern
et
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 so
urce
s: p
rimar
y (o
bser
vatio
n, co
ntro
lled
expe
rimen
t, da
talo
ggin
g) a
nd se
cond
ary
(spr
eads
heet
dat
a,
prin
ted
tabl
es, l
ists)
sele
ct, j
ustif
y an
d us
e th
e da
ta-g
athe
ring
tech
niqu
e ap
prop
riate
to
com
plex
and
un
fam
iliar
pro
blem
s, id
entif
ying
pot
entia
l ba
rrie
rs a
nd li
mita
tions
;id
entif
y w
hat e
xtra
in
form
atio
n m
ay b
e re
quire
d to
pur
sue
a fu
rthe
r lin
e of
enq
uiry
sele
ct a
nd c
ritic
ally
ev
alua
te a
sam
plin
g sc
hem
e an
d a
met
hod
to in
vest
igat
e a
popu
latio
n, in
clud
ing
rand
om a
nd st
ratif
ied
sam
plin
g; e
xpla
in th
e ef
fect
on
relia
bilit
y
and
valid
ity
colle
ct sm
all s
ets o
f co
llect
dat
a us
ing
a da
ta fr
om su
rvey
s su
itabl
e m
etho
d (e
.g.
and
expe
rimen
ts, a
sob
serv
atio
n, c
ontr
olle
d pl
anne
d ex
perim
ent,
data
logg
ing
usin
g IC
T)
39 The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
Pro
cess
ing
and
repr
esen
ting
data
5.
2 Year
7
Year
8
Year
9
Year
10
Year
11
Exte
nsio
n
calc
ulat
e st
atist
ics f
or
smal
l set
s of d
iscre
teda
ta: fin
d th
e m
ode,
m
edia
n an
d ra
nge,
an
d th
e m
odal
cl
ass f
or g
roup
ed
data
calc
ulat
e th
e m
ean,
incl
udin
g fro
m a
sim
ple
frequ
ency
tabl
e,us
ing
a ca
lcul
ator
fo
r a la
rger
num
ber
of it
ems
calc
ulat
e st
atist
ics
for s
ets o
f disc
rete
an
d co
ntin
uous
da
ta, i
nclu
ding
with
a
calc
ulat
or a
nd
spre
adsh
eet;
reco
gnise
w
hen
it is
appr
opria
te
to u
se th
e ra
nge,
mea
n,
med
ian
and
mod
e an
d,
for g
roup
ed d
ata,
the
mod
al c
lass
calc
ulat
e st
atist
ics
and
sele
ct th
ose
mos
tap
prop
riate
to th
e pr
oble
m o
r whi
ch
addr
ess t
he q
uest
ions
po
sed
use
an a
ppro
pria
te
rang
e of
stat
istic
al
met
hods
to e
xplo
re
and
sum
mar
ise la
rge
data
sets
, jus
tifyi
ngth
e ch
oice
s mad
e;
incl
ude
grou
ping
da
ta, e
stim
atin
g an
d fin
ding
the
mea
n,
med
ian,
qua
rtile
s and
in
terq
uart
ile ra
nge
proc
ess d
ata
draw
n fro
m p
robl
ems
invo
lvin
g se
ason
ality
an
d tr
ends
in a
tim
e se
ries;
choo
se a
nd
com
bine
stat
istic
al
met
hods
to a
naly
se
the
prob
lem
, inc
ludi
ngm
ovin
g av
erag
es
© Crown copyright 2009 01061-2009DOM-EN
Pro
cess
ing
and
repr
esen
ting
data
(con
tinue
d)
5.2 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0
cons
truc
t, on
pap
er
and
usin
g IC
T, g
raph
san
d di
agra
ms t
ore
pres
ent d
ata,
in
clud
ing:
bar-l
ine
grap
hs
freq
uenc
y di
agra
ms f
or
grou
ped
disc
rete
data
sim
ple
pie
char
ts
cons
truc
t gra
phic
al
repr
esen
tatio
ns, o
n pa
per a
nd u
sing
ICT,
and
iden
tify
whi
ch
are
mos
t use
ful i
n th
e co
ntex
t of t
he p
robl
em,
incl
udin
g:
pie
char
ts fo
r ca
tego
rical
dat
a
bar c
hart
s and
fre
quen
cy
diag
ram
s for
di
scre
te a
nd
cont
inuo
us d
ata
sim
ple
line
grap
hs
for t
ime
serie
s
sim
ple
scat
ter
grap
hs
stem
-and
-leaf
di
agra
ms
sele
ct, c
onst
ruct
an
d m
odify
, on
pape
r and
usin
g IC
T,su
itabl
e gr
aphi
cal
repr
esen
tatio
ns to
pr
ogre
ss a
n en
quiry
an
d id
entif
y ke
y fe
atur
es p
rese
nt in
the
data
. Inc
lude
:
line
grap
hs fo
r tim
e se
ries
scat
ter g
raph
s to
deve
lop
furt
her
unde
rsta
ndin
g of
co
rrel
atio
n
cons
truc
t on
pape
r an
d us
ing
ICT
suita
ble
grap
hica
l re
pres
enta
tions
,in
clud
ing:
hist
ogra
ms
for g
roup
ed
cont
inuo
us d
ata
with
equ
al c
lass
in
terv
als
cum
ulat
ive
frequ
ency
tabl
es
and
diag
ram
s
box
plot
s
scat
ter g
raph
s and
lin
es o
f bes
t fit
(b
y ey
e)
just
ify th
eir s
uita
bilit
y w
ith re
fere
nce
to th
e co
ntex
t of t
he p
robl
em
and
the
audi
ence
wor
k th
roug
h th
e en
tire
hand
ling
data
cycl
e to
expl
ore
rela
tions
hips
w
ithin
bi-v
aria
te d
ata,
in
clud
ing
appl
icat
ions
to g
loba
l citi
zens
hip,
e.g
. ho
w fa
ir is
our s
ocie
ty?
Year
11
Exte
nsio
n
choo
se a
nd c
ombi
ne
suita
ble
grap
hica
l re
pres
enta
tions
topr
ogre
ss a
n un
fam
iliar
or
non
-rou
tine
enqu
iry,
incl
udin
g hi
stog
ram
s w
ith e
qual
or u
nequ
al
clas
s int
erva
ls
use
prec
ise a
nd
cons
isten
t gra
phic
alre
pres
enta
tion
topr
ogre
ss a
n un
fam
iliar
an
d no
n-ro
utin
e en
quiry
40 The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
01061-2009DOM-EN © Crown copyright 2009
41
Inte
rpre
ting
and
disc
ussi
ng re
sults
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 and
gr
aphs
(inc
ludi
ng p
ie
char
ts) a
nd d
raw
sim
ple
conc
lusio
ns b
ased
on
the
shap
e of
gra
phs
and
simpl
e st
atist
ics f
or
a sin
gle
dist
ribut
ion
com
pare
two
simpl
e di
strib
utio
ns u
sing
the
rang
e an
d on
e of
the
mod
e, m
edia
n or
mea
n
writ
e a
shor
t rep
ort o
f a
stat
istic
al e
nqui
ry,
incl
udin
g ap
prop
riate
di
agra
ms,
grap
hs a
nd
char
ts, u
sing
ICT
as
appr
opria
te; j
ustif
y th
e ch
oice
of p
rese
ntat
ion
inte
rpre
t tab
les,
grap
hs a
nd d
iagr
ams
for d
iscre
te a
nd
cont
inuo
us d
ata,
re
latin
g su
mm
ary
stat
istic
s and
find
ings
to
the
ques
tions
bei
ng
expl
ored
com
pare
two
dist
ribut
ions
usin
g th
e ra
nge
and
one
or m
ore
of th
e m
ode,
med
ian
and
mea
n
writ
e ab
out a
nd
disc
uss t
he re
sults
of a
st
atis
tical
enq
uiry
usin
g IC
T as
app
ropr
iate
;ju
stify
the
met
hods
us
ed
inte
rpre
t gra
phs a
nd
diag
ram
s and
mak
ein
fere
nces
to su
ppor
t or
cas
t dou
bt o
n in
itial
co
njec
ture
s; ha
ve a
ba
sic u
nder
stan
ding
of
corr
elat
ion
com
pare
two
or
mor
e di
strib
utio
ns
and
mak
e in
fere
nces
, us
ing
the
shap
e of
th
e di
strib
utio
ns a
nd
appr
opria
te st
atist
ics
revi
ew in
terp
reta
tions
and
resu
lts o
f a
stat
istic
al e
nqui
ry o
n th
e ba
sis o
f disc
ussio
ns;
com
mun
icat
e th
ese
inte
rpre
tatio
ns a
nd
resu
lts u
sing
sele
cted
ta
bles
, gra
phs a
nd
diag
ram
s
find
patt
erns
and
in
terp
ret a
nd c
ompa
re
expl
ain
and
just
ify
exce
ptio
ns a
nd e
xpla
in
dist
ribut
ions
, inc
ludi
ng
assu
mpt
ions
and
an
omal
ies;
incl
udin
g cu
mul
ativ
e fre
quen
cy
cons
trai
nts;
incl
ude
inte
rpre
tatio
n of
di
agra
ms;
mak
e an
d in
terp
reta
tion
and
soci
al st
atist
ics a
nd
disc
uss i
nfer
ence
s, co
mpa
rison
of
eval
uatio
n of
the
usin
g th
e sh
ape
of
hist
ogra
ms w
ith
stre
ngth
of a
ssoc
iatio
nth
e di
strib
utio
ns a
nd
uneq
ual c
lass
inte
rval
sw
ithin
bi-v
aria
te d
ata
mea
sure
s of a
vera
ge
(cor
rela
tion,
line
s of
and
spre
ad, i
nclu
ding
be
st fi
t) m
edia
n an
d qu
artil
es
eval
uate
the
resu
lts
criti
cally
exa
min
e us
e st
atist
ical
of a
stat
istic
al
stra
tegi
es a
dopt
ed
anal
ysis
effe
ctiv
ely
in
enqu
iry; r
evie
w a
nd
and
argu
men
ts
pres
entin
g co
nvin
cing
ju
stify
or r
efin
e th
e pr
esen
ted,
rela
ting
conc
lusio
ns; c
ritic
ally
ch
oice
of s
tatis
tical
th
em to
the
orig
inal
re
flect
on
own
lines
re
pres
enta
tions
and
hy
poth
eses
; rec
ogni
se
of e
nqui
ry; s
earc
h re
late
sum
mar
ised
data
the
limita
tions
of a
ny
for a
nd a
ppre
ciat
e to
the
ques
tions
bei
ng
assu
mpt
ions
and
the
mor
e el
egan
t for
ms
expl
ored
ef
fect
s tha
t var
ying
of
com
mun
icat
ing
assu
mpt
ions
cou
ld
conc
lusio
ns
have
on
conc
lusio
ns
draw
n fro
m d
ata
anal
ysis
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 p
roba
bilit
y, dr
awin
g on
exp
erie
nce
inte
rpre
t the
resu
lts
of a
n ex
perim
ent
usin
g th
e la
ngua
ge o
f pr
obab
ility
; app
reci
ate
that
rand
om p
roce
sses
ar
e un
pred
icta
ble
inte
rpre
t res
ults
in
volv
ing
unce
rtai
nty
and
pred
ictio
n id
entif
y w
hen
the
even
ts in
a p
robl
em
are
mut
ually
exc
lusiv
e or
inde
pend
ent;
use
and
inte
rpre
t tre
edi
agra
ms t
o re
pres
ent
outc
omes
of c
ombi
ned
even
ts a
nd to
info
rm
the
calc
ulat
ion
of
thei
r pro
babi
litie
s;de
cide
whe
n to
add
an
d w
hen
to m
ultip
lypr
obab
ilitie
s
inte
rpre
t the
effe
cton
pro
babi
lity
of
cont
exts
invo
lvin
g se
lect
ion
with
and
w
ithou
t rep
lace
men
t; ch
oose
and
com
bine
re
pres
enta
tions
to
com
mun
icat
e pr
obab
ilitie
s as p
art o
f a
solu
tion
to a
pro
blem
reco
gnise
whe
n an
d ho
w to
wor
k w
ithpr
obab
ilitie
s ass
ocia
ted
with
inde
pend
ent
and
mut
ually
excl
usiv
e ev
ents
whe
n in
terp
retin
g da
ta
unde
rsta
nd a
nd u
se
the
prob
abili
ty sc
ale
from
0 to
1; f
ind
and
just
ify p
roba
bilit
ies
base
d on
equ
ally
like
ly
outc
omes
in si
mpl
e co
ntex
ts; i
dent
ify a
ll th
e po
ssib
le m
utua
lly
excl
usiv
e ou
tcom
es o
f a
singl
e ev
ent
know
that
if th
e pr
obab
ility
of a
n ev
ent
occu
rrin
g is
p th
en th
e pr
obab
ility
of i
t not
oc
curr
ing
is 1!%!
p; u
se
diag
ram
s and
tabl
es to
re
cord
in a
syst
emat
ic
way
all
poss
ible
m
utua
lly e
xclu
sive
outc
omes
for s
ingl
e ev
ents
and
for t
wo
succ
essiv
e ev
ents
iden
tify
all t
he m
utua
lly
excl
usiv
e ou
tcom
es
of a
n ex
perim
ent;
know
that
the
sum
of
pro
babi
litie
s of a
ll m
utua
lly e
xclu
sive
outc
omes
is 1
and
us
e th
is w
hen
solv
ing
prob
lem
s
01061-2009DOM-EN © Crown copyright 2009
43
Pro
babi
lity
(con
tinue
d)
5.4 Ye
ar 7
Ye
ar 8
Ye
ar 9
Ye
ar 1
0 Ye
ar 1
1 Ex
tens
ion
estim
ate
prob
abili
ties
by c
olle
ctin
g da
ta fr
om
a sim
ple
expe
rimen
tan
d re
cord
ing
it in
a
frequ
ency
tabl
e;co
mpa
re e
xper
imen
tal
and
theo
retic
al
prob
abili
ties i
n sim
ple
cont
exts
com
pare
est
imat
ed
expe
rimen
tal
prob
abili
ties
with
theo
retic
al
prob
abili
ties,
reco
gnisi
ng th
at:
if an
exp
erim
ent
is re
peat
ed th
e ou
tcom
e m
ay, a
nd
usua
lly w
ill, b
e di
ffere
nt
incr
easin
g th
e nu
mbe
r of t
imes
an
exp
erim
ent i
sre
peat
ed g
ener
ally
le
ads t
o be
tter
es
timat
es o
f pr
obab
ility
com
pare
exp
erim
enta
l an
d th
eore
tical
pr
obab
ilitie
s in
a ra
nge
of c
onte
xts;
appr
ecia
teth
e di
ffere
nce
betw
een
mat
hem
atic
al
expl
anat
ion
and
expe
rimen
tal e
vide
nce
use
rela
tive
frequ
ency
as
an
estim
ate
of
prob
abili
ty, i
nclu
ding
sim
ulat
ion
usin
g IC
T to
gen
erat
e la
rger
sa
mpl
es; d
iscus
s its
relia
bilit
y ba
sed
on sa
mpl
e siz
e an
d us
e to
inte
rpre
t and
co
mpa
re o
utco
mes
of
expe
rimen
ts
expl
ore
a re
leva
nt a
nd
purp
osef
ul p
robl
em
invo
lvin
g un
cert
aint
y;
estim
ate
risk
bym
odel
ling
real
eve
nts
thro
ugh
simul
atio
n;
just
ify d
ecisi
ons b
ased
on
exp
erim
enta
lpr
obab
ility
and
co
mm
ent o
n th
e ef
fect
of a
ssum
ptio
ns
and
sam
ple
size
on th
e re
liabi
lity
of
conc
lusio
ns
The National Strategies | Secondary Secondary mathematics subject leader development materials: Spring 2010
© Crown copyright 2009 01061-2009DOM-EN