mathematics: senior phase · exemplar lesson plans to assist teachers in developing, managing and...
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1MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Mathematics: Senior Phase
Preamble
The Multi-grade toolkit for Mathematics is designed to assist teachers in multi-grade schools to teach
effectively and cover the curriculum adequately. In the past, teachers in the multi-grade schools
were expected to use the mono-grade curriculum and adapt it to meet the needs and demands of
learners in a multi-grade class. Some teachers did not have the requisite skills to carry out this
demanding task and it resulted in content being either too scantly covered or completely ignored, thus
compromising the content knowledge and skills that learners needed to acquire at all levels of
schooling. To obviate such situations in our multi-grade schools and ensure that learners in the multi-
grade schools receive quality education, the Department of Basic Education developed a multi-grade
toolkit for teachers.
The toolkit compromises of the following documents:
A generic manual which provides literature about multi-grade teaching, methodologies,
management, etc.
A Multi-grade Annual Teaching Plan (MATP) informed by and aligned to the Curriculum and
Assessment Policy Statement (CAPS) to support teachers with lesson planning;
Exemplar lesson plans to assist teachers in developing, managing and adapting lessons to
suit their own individual context. The activities in the lesson plans were taken from the
o DBE workbooks for Grades 4 to 9 (the same worksheets numbers are used in the
lesson plans).
o Sasol-Inzalo workbooks for Grades 7 – 9 (chapters were cited).
More activities can be added by the individual teacher using a book of his or her own choice.
Exemplar Assessment tasks to assist teachers in designing their own individual tasks to suit
their own specific context.
The MATP was developed and designed according to topics that could lend themselves for robust
discussion in the Mathematics class without compromising the content thereof. Each topic was
carefully chosen and linked to each other in each grade. A common thread was carefully identified
across the topics before they were aligned together, for example the topic Numeric and Geometric Patterns cuts across all the grades in the Intermediate Phase. A teacher can introduce a concept to all the learners and then pitch to different levels as dictated to by CAPS. .
2 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Through the “Whole Class” teaching methodology, the teacher can easily introduce each topic and
engage learners in discussion before embarking on teaching specific grades in a multi-grade class
through small group teaching.
3MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
M
ulti-
Gra
de A
nnua
l Tea
chin
g Pl
ans
4 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
TER
M 1
TI
ME
(HO
UR
S)
GR
ADE
7
GR
ADE
8
GR
ADE
9
9
Who
le n
umbe
rs (
9)
Pr
oper
ties
C
alcu
latio
ns
C
alcu
latio
n te
chni
ques
Mul
tiple
s an
d fa
ctor
s
Solv
ing
prob
lem
s
Rat
io a
nd ra
te
Fi
nanc
ial m
athe
mat
ics
Who
le n
umbe
rs(6
)
Prop
ertie
s
Cal
cula
tions
Cal
cula
tion
tech
niqu
es
M
ultip
les
and
fact
ors
So
lvin
g pr
oble
ms
R
atio
and
rate
Fina
ncia
l mat
hem
atic
s
Who
le n
umbe
rs (4
.5)
Pr
oper
ties
C
alcu
latio
ns
C
alcu
latio
n te
chni
ques
Mul
tiple
s an
d fa
ctor
s
Solv
ing
prob
lem
s
Rat
io, r
ate
& pr
opor
tion
Fi
nanc
ial m
athe
mat
ics
9
* Int
eger
s (9
)
Cou
ntin
g, o
rder
ing
and
com
parin
g
Cal
cula
tions
Prop
ertie
s
Solv
ing
prob
lem
s
Inte
gers
( 9)
C
ount
ing,
ord
erin
g an
d co
mpa
ring
C
alcu
latio
ns
Pr
oper
ties
So
lvin
g pr
oble
ms
Inte
gers
(4.5
)
Cou
ntin
g, o
rder
ing
and
com
parin
g (R
evis
ion)
Cal
cula
tions
(Rev
isio
n)
Pr
oper
ties
(Rev
isio
n)
So
lvin
g pr
oble
ms
9
Expo
nent
s (9
)
Com
parin
g an
d re
pres
entin
g
Cal
cula
tions
Solv
ing
prob
lem
s
Expo
nent
s (9
)
Com
parin
g an
d re
pres
entin
g
Cal
cula
tions
(inc
lude
law
s of
exp
onen
ts)
So
lvin
g pr
oble
ms
Expo
nent
s (5
)
Com
parin
g an
d re
pres
entin
g
Cal
cula
tions
(inc
lude
law
s of
exp
onen
ts)
So
lvin
g pr
oble
ms
1
Form
al S
BA
Task
: Ass
ignm
ent
Form
al S
BA
Task
: Ass
ignm
ent
Form
al S
BA
Task
: Ass
ignm
ent
6 N
umer
ic a
nd g
eom
etric
pat
tern
s (6
)
Inve
stig
ate
and
exte
nd
Num
eric
and
geo
met
ric p
atte
rns
(6)
In
vest
igat
e an
d ex
tend
N
umer
ic a
nd g
eom
etric
pat
tern
s (6
)
Inve
stig
ate
and
exte
nd
IM
PORT
ANT
NO
TE: A
lway
s co
nsul
t the
CAP
S to
est
ablis
h a
deta
iled
prog
ress
ion
acro
ss th
e gr
ades
and
the
dept
hs w
ithin
the
grad
e re
gard
ing
topi
cs, c
once
pts
and
skills
.
4 Fu
nctio
ns a
nd re
latio
nshi
ps (3
)
Inpu
t and
out
put v
alue
s
Equi
vale
nt fo
rms
Func
tions
and
rela
tions
hips
(3)
In
put a
nd o
utpu
t val
ues
Eq
uiva
lent
form
s
Func
tions
and
rela
tions
hips
(4)
In
put a
nd o
utpu
t val
ues
Equi
vale
nt fo
rms
4 G
raph
s (4
)
Inte
rpre
ting
G
raph
s( 4
)
Inte
rpre
ting
G
raph
s ( 4
)
Inte
rpre
ting
5MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
D
raw
ing
Dra
win
g
Dra
win
g
2 Fo
rmal
SB
A Ta
sk: T
est
Form
al S
BA
Task
: Tes
t Fo
rmal
SB
A Ta
sk: T
est
6 R
evis
ion
Rev
isio
n R
evis
ion
50
IM
PORT
ANT
NO
TE: A
lway
s co
nsul
t the
CAP
S to
est
ablis
h a
deta
iled
prog
ress
ion
acro
ss th
e gr
ades
and
the
dept
hs w
ithin
the
grad
e re
gard
ing
mat
hs to
pics
, co
ncep
ts a
nd s
kills
.
6 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
TER
M 2
TI
ME
(HO
UR
S)
GR
ADE
7
GR
ADE
8
GR
ADE
9
5 C
onst
ruct
ion
of g
eom
etric
figu
res
(5)
M
easu
ring
angl
es
C
onst
ruct
ion
Con
stru
ctio
n of
geo
met
ric fi
gure
s (4
)
Con
stru
ctio
n
Prop
ertie
s
Con
stru
ctio
n of
geo
met
ric fi
gure
s (4
)
Con
stru
ctio
n
Prop
ertie
s
10
Geo
met
ry o
f 2D
sha
pes
(10)
Cla
ssify
Sim
ilarit
y an
d co
ngru
ency
Solv
e pr
oble
ms
Geo
met
ry o
f 2D
sha
pes(
8)
C
lass
ify
Si
mila
rity
and
cong
ruen
cy
So
lve
prob
lem
s
Geo
met
ry o
f 2D
sha
pes(
9)
C
lass
ify
Si
mila
rity
and
cong
ruen
cy
So
lve
prob
lem
s
1 Fo
rmal
SB
A Ta
sk: I
nves
tigat
ion
Form
al S
BA
Task
: Inv
estig
atio
n Fo
rmal
SB
A Ta
sk: I
nves
tigat
ion
9 G
eom
etry
of s
trai
ght l
ines
(2)
D
efin
ition
of l
ines
and
line
rela
tions
hips
G
eom
etry
of s
trai
ght l
ines
(9)
An
gle
rela
tions
hips
Solv
ing
prob
lem
s
Geo
met
ry o
f str
aigh
t lin
es (9
)
Angl
e re
latio
nshi
ps
So
lvin
g pr
oble
ms
9
Alge
brai
c ex
pres
sion
s (3
)
Alge
brai
c la
ngua
ge
Alge
brai
c ex
pres
sion
s (9
)
Alge
brai
c la
ngua
ge
Ex
pand
and
sim
plify
Alge
brai
c ex
pres
sion
s( 9
)
Alge
brai
c la
ngua
ge
Ex
pand
and
sim
plify
Fact
oriz
atio
n 1
Form
al S
BA
Task
: Tes
t Fo
rmal
SB
A Ta
sk: T
est
Form
al S
BA
Task
: Tes
t
8 Al
gebr
aic
equa
tions
(3)
N
umbe
r sen
tenc
es
Alge
brai
c eq
uatio
ns (3
)
Equa
tions
Al
gebr
aic
equa
tions
(8)
Eq
uatio
ns
2 Fo
rmal
SB
A Ta
sk: E
xam
inat
ion
Form
al S
BA
Task
: Exa
min
atio
n Fo
rmal
SB
A Ta
sk: E
xam
inat
ion
5 R
evis
ion
Rev
isio
n R
evis
ion
50
IMPO
RTAN
T N
OTE
: Alw
ays
cons
ult t
he C
APS
to e
stab
lish
a de
taile
d pr
ogre
ssio
n ac
ross
the
grad
es a
nd th
e de
pths
with
in th
e gr
ade
rega
rdin
g m
aths
topi
cs,
conc
epts
and
ski
lls.
7MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
TER
M 3
TI
ME
(HO
UR
S)
GR
ADE
7
GR
ADE
8
GR
ADE
9
8
Com
mon
frac
tions
Ord
erin
g, c
ompa
ring
and
sim
plify
ing
C
alcu
latio
ns w
ith fr
actio
ns
C
alcu
latio
n te
chni
ques
Solv
ing
prob
lem
s
Pe
rcen
tage
s
Equi
vale
nt fo
rm
Com
mon
frac
tions
Cal
cula
tions
with
frac
tions
Cal
cula
tion
tech
niqu
es
So
lvin
g pr
oble
ms
Perc
enta
ges
Eq
uiva
lent
form
s
Com
mon
frac
tions
Cal
cula
tions
with
frac
tions
Cal
cula
tion
tech
niqu
es
So
lvin
g pr
oble
ms
Eq
uiva
lent
form
s
8
Dec
imal
frac
tions
Ord
erin
g an
d co
mpa
ring
C
alcu
latio
ns w
ith d
ecim
al fr
actio
ns
C
alcu
latio
ns te
chni
ques
Solv
ing
Prob
lem
s
Eq
uiva
lent
form
s
Dec
imal
frac
tions
Ord
erin
g an
d co
mpa
ring
C
alcu
latio
ns w
ith d
ecim
al fr
actio
ns
C
alcu
latio
ns te
chni
ques
Solv
ing
Prob
lem
s
Eq
uiva
lent
form
s
Dec
imal
frac
tions
Ord
erin
g an
d co
mpa
ring
C
alcu
latio
ns te
chni
ques
Solv
ing
Prob
lem
s
Eq
uiva
lent
form
s
7
Dat
a ha
ndlin
g
Col
lect
dat
a
Org
anis
e an
d su
mm
aris
e da
ta
R
epre
sent
dat
a
Inte
rpre
t dat
a
Anal
yse
data
Rep
ort d
ata
Dat
a ha
ndlin
g
Col
lect
dat
a
Org
anis
e an
d su
mm
aris
e da
ta
R
epre
sent
dat
a
Inte
rpre
t dat
a
Anal
yse
data
Rep
ort d
ata
Dat
a ha
ndlin
g
Col
lect
dat
a
Org
anis
e an
d su
mm
aris
e da
ta
R
epre
sent
dat
a
Inte
rpre
t dat
a
Anal
yse
data
Rep
ort d
ata
1 Fo
rmal
SB
A Ta
sk: P
roje
ct
Form
al S
BA
Task
: Pro
ject
Fo
rmal
SB
A Ta
sk: P
roje
ct
5
2D G
eom
etry
Cla
ssify
ing
2D s
hape
s
Area
of s
quar
e, re
ctan
gle
and
trian
gle
Theo
rem
of P
ytha
gora
s
D
evel
op a
nd u
se T
heor
em o
f Pyt
hago
ras
Theo
rem
of P
ytha
gora
s
So
lve
prob
lem
s us
ing
the
theo
rem
of
Pyth
agor
as
8 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
7 Ar
ea a
nd p
erim
eter
of 2
D s
hape
s
Area
and
per
imet
er
C
alcu
latio
ns a
nd s
olvi
ng p
robl
ems
Area
and
per
imet
er o
f 2D
sha
pes
Ar
ea a
nd p
erim
eter
Cal
cula
tions
and
sol
ving
pro
blem
s
Area
and
per
imet
er o
f 2D
sha
pes
Ar
ea a
nd p
erim
eter
1 Fo
rmal
SB
A Ta
sk: A
ssig
nmen
t Fo
rmal
SB
A Ta
sk: A
ssig
nmen
t Fo
rmal
SB
A Ta
sk: A
ssig
nmen
t
8 Su
rfac
e ar
ea a
nd v
olum
e of
3D
obj
ects
Surfa
ce a
rea
and
volu
me
C
alcu
latio
ns a
nd s
olvi
ng p
robl
ems
Surf
ace
area
and
vol
ume
of 3
D o
bjec
ts
Su
rface
are
a an
d vo
lum
e
Cal
cula
tions
and
sol
ving
pro
blem
s
Surf
ace
area
and
vol
ume
of 3
D o
bjec
ts
Su
rface
are
a an
d vo
lum
e
2 Fo
rmal
SB
A Ta
sk: T
est
Form
al S
BA
Task
: Tes
t Fo
rmal
SB
A Ta
sk: T
est
3 R
evis
ion
Rev
isio
n R
evis
ion
50
IMPO
RTAN
T NO
TE: A
lway
s co
nsul
t the
CAP
S to
est
ablis
h a
deta
iled
prog
ress
ion
acro
ss th
e gr
ades
and
the
dept
hs w
ithin
the
grad
e re
gard
ing
mat
hs to
pics
, con
cept
s an
d sk
ills.
9MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
TER
M 4
TI
ME
(HO
UR
S)
GR
ADE
7
GR
ADE
8
GR
ADE
9
6 Fu
nctio
ns a
nd re
latio
nshi
ps
In
put a
nd o
utpu
t val
ues
Eq
uiva
lent
form
s
Func
tions
and
rela
tions
hips
Inpu
t and
out
put v
alue
s
Equi
vale
nt fo
rms
Func
tions
and
rela
tions
hips
Inpu
t and
out
put v
alue
s
Equi
vale
nt fo
rms
2 G
raph
s
Inte
rpre
t gr
aphs
Dra
wl g
raph
s
Gra
phs
In
terp
ret
grap
hs
D
raw
gra
phs
Gra
phs
In
terp
ret l
gra
phs
D
raw
gra
phs
1 Fo
rmal
SB
A Ta
sk: I
nves
tigat
ion
Form
al S
BA
Task
: Inv
estig
atio
n Fo
rmal
SB
A Ta
sk: I
nves
tigat
ion
5
Con
stru
ctio
n of
geo
met
ric s
hape
s
Mea
surin
g of
ang
les
Con
stru
ctio
ns
Con
stru
ctio
n of
geo
met
ric s
hape
s
Con
stru
ctio
ns
In
vest
igat
ing
prop
ertie
s of
geo
met
ric
figur
es
Con
stru
ctio
n of
geo
met
ric s
hape
s
Con
stru
ctio
ns
In
vest
igat
ing
prop
ertie
s of
geo
met
ric
figur
es
9
Tran
sfor
mat
ion
geo
met
ry
Tr
ansf
orm
atio
ns
En
larg
emen
t and
Red
uctio
n
Tran
sfor
mat
ion
geo
met
ry
Tr
ansf
orm
atio
ns
En
larg
emen
t and
Red
uctio
n of
geo
met
ric
figur
es.
Tran
sfor
mat
ion
geo
met
ry
Tr
ansf
orm
atio
ns
En
larg
emen
t and
Red
uctio
n
1 Fo
rmal
SB
A Ta
sk: A
ssig
nmen
t Fo
rmal
SB
A Ta
sk: A
ssig
nmen
t Fo
rmal
SB
A Ta
sk: A
ssig
nmen
t
9 G
eom
etry
of 3
D o
bjec
ts
C
lass
ifyin
g 3-
D o
bjec
ts
Bu
ildin
g 3-
D m
odel
s
Geo
met
ry o
f 3D
obj
ects
Cla
ssify
ing
3-D
obj
ects
Build
ing
3-D
mod
els
Geo
met
ry o
f 3D
obj
ects
Cla
ssify
ing
3-D
obj
ects
Build
ing
3-D
mod
els
4.
5
Prob
abili
ty
Pro
babi
lity
P
roba
bilit
y 4
Rev
isio
n R
evis
ion
Rev
isio
n 3
Form
al T
ask:
End
of y
ear E
xam
inat
ion
Form
al T
ask:
End
of y
ear E
xam
inat
ion
Form
al T
ask:
End
of y
ear E
xam
inat
ion
IMPO
RTAN
T NO
TE: A
lway
s co
nsul
t the
CAP
S to
est
ablis
h a
deta
iled
prog
ress
ion
acro
ss th
e gr
ades
and
the
dept
hs w
ithin
the
grad
e re
gard
ing
mat
hs to
pics
, con
cept
s an
d sk
ills.
10 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LESS
ON
PLA
NS:
TER
M 1
11MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
1
TER
M 1
Dat
e : f
rom
....
to ..
....
T
ime:
9 ho
urs
G
rade
7
8 9
Topi
c W
HO
LE N
UM
BER
S C
APS
page
s 40
- 42
75
- 77
11
9 - 1
21
Skills
and
Kn
owle
dge
Ord
erin
g an
d co
mpa
ring
who
le n
umbe
rs
Rev
ise
the
follo
win
g do
ne in
Gra
de 6
:
Ord
er, c
ompa
re a
nd re
pres
ent n
umbe
rs
to a
t lea
st
9-di
git n
umbe
rs
R
ecog
nise
and
repr
esen
t prim
e nu
mbe
rs
to a
t lea
st 1
00
R
ound
ing
off n
umbe
rs to
the
near
est 5
, 10
, 100
or
1
000
Pr
oper
ties
of w
hole
num
bers
R
evis
e th
e fo
llow
ing
done
in G
rade
6:
R
ecog
nise
and
use
the
com
mut
ativ
e;
asso
ciat
ive;
dis
tribu
tive
prop
ertie
s w
ith
who
le n
umbe
rs
R
ecog
nise
and
use
0 in
term
s of
its
addi
tive
prop
erty
(ide
ntity
ele
men
t for
ad
ditio
n)
R
ecog
nise
and
use
1 in
term
s of
its
mul
tiplic
ativ
e pr
oper
ty (i
dent
ity e
lem
ent
for m
ultip
licat
ion)
C
alcu
latio
ns w
ith w
hole
num
bers
R
evis
e th
e fo
llow
ing
done
in G
rade
6,
with
out u
se o
f cal
cula
tors
:
Addi
tion
and
subt
ract
ion
of w
hole
nu
mbe
rs to
at l
east
6 –
digi
t num
bers
Ord
erin
g an
d co
mpa
ring
who
le
num
bers
R
evis
e Pr
ime
num
bers
to a
t lea
st 1
00
Prop
ertie
s of
who
le n
umbe
rs
Rev
ise:
The
com
mut
ativ
e; a
ssoc
iativ
e;
dist
ribut
ive
prop
ertie
s of
who
le
num
bers
0 in
term
s of
its
addi
tive
prop
erty
(id
entit
y el
emen
t for
add
ition
)
1 in
term
s of
its
mul
tiplic
ativ
e pr
oper
ty
(iden
tify
elem
ent f
or m
ultip
licat
ion)
Rec
ogni
se th
e di
visi
on p
rope
rty o
f 0,
whe
reby
any
num
ber d
ivid
ed b
y 0
is
unde
fined
C
alcu
latio
ns u
sing
who
le n
umbe
rs
Rev
ise:
C
alcu
latio
ns u
sing
all
four
ope
ratio
ns o
n w
hole
num
bers
, est
imat
ing
and
usin
g ca
lcul
ator
s w
here
app
ropr
iate
Prop
ertie
s of
num
bers
D
escr
ibe
the
real
num
ber s
yste
m b
y re
cogn
isin
g, d
efin
ing
and
dist
ingu
ishi
ng
prop
ertie
s of
:
natu
ral n
umbe
rs
w
hole
num
bers
inte
gers
ratio
nal n
umbe
rs
ir
ratio
nal n
umbe
rs
Cal
cula
tions
usi
ng w
hole
num
bers
R
evis
e:
Cal
cula
tions
usi
ng a
ll fo
ur o
pera
tions
on
who
le n
umbe
rs, e
stim
atin
g an
d us
ing
calc
ulat
ors
whe
re a
ppro
pria
te
12 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
M
ultip
licat
ion
of a
t lea
st w
hole
4-d
igit
by
2-di
git n
umbe
rs
D
ivis
ion
of a
t lea
st w
hole
4-d
igit
by 2
-di
git n
umbe
rs
Perfo
rm c
alcu
latio
ns u
sing
all
four
op
erat
ions
on
who
le n
umbe
rs, e
stim
atin
g an
d us
ing
calc
ulat
ors
whe
re a
ppro
pria
te
Cal
cula
tion
tech
niqu
es
Use
a ra
nge
of te
chni
ques
to p
erfo
rm a
nd
chec
k w
ritte
n an
d m
enta
l cal
cula
tions
of
who
le n
umbe
rs in
clud
ing:
Estim
atio
n
addi
ng, s
ubtra
ctin
g an
d m
ultip
lyin
g in
co
lum
ns
lo
ng d
ivis
ion
ro
undi
ng o
ff an
d co
mpe
nsat
ing
u
sing
a c
alcu
lato
r M
ultip
les
and
fact
ors
Rev
ise
the
follo
win
g do
ne in
Gra
de 6
:
Mul
tiple
s of
2-d
igit
and
3-di
git w
hole
nu
mbe
rs
F
acto
rs o
f 2-d
igit
and
3-di
git w
hole
nu
mbe
rs
P
rime
fact
ors
of n
umbe
rs to
at l
east
100
Li
st p
rime
fact
ors
of n
umbe
rs to
at l
east
3-
digi
t who
le n
umbe
rs
Find
the
LCM
and
HC
F of
num
bers
to a
t le
ast 3
-dig
it w
hole
num
bers
, by
insp
ectio
n or
fa
ctor
isat
ion
Cal
cula
tion
tech
niqu
es
Use
a ra
nge
of s
trate
gies
to p
erfo
rm a
nd
chec
k w
ritte
n an
d m
enta
l cal
cula
tions
w
ith w
hole
num
bers
in
clud
ing:
Estim
atio
n
add
ing,
sub
tract
ing
and
mul
tiply
ing
in
colu
mns
long
div
isio
n
ro
undi
ng o
ff an
d co
mpe
nsat
ing
us
ing
a ca
lcul
ator
M
ultip
les
and
fact
ors
Rev
ise:
Prim
e fa
ctor
s of
num
bers
to a
t lea
st 3
-di
git w
hole
num
bers
LCM
and
HC
F of
num
bers
to a
t lea
st
3-di
git w
hole
num
bers
, by
insp
ectio
n or
fact
oris
atio
n
Cal
cula
tion
tech
niqu
es
Use
a ra
nge
of s
trate
gies
to p
erfo
rm a
nd
chec
k w
ritte
n an
d m
enta
l cal
cula
tions
of
who
le n
umbe
rs
incl
udin
g:
Es
timat
ion
a
ddin
g, s
ubtra
ctin
g an
d m
ultip
lyin
g in
co
lum
ns
lo
ng d
ivis
ion
ro
undi
ng o
ff an
d co
mpe
nsat
ing
u
sing
a c
alcu
lato
r M
ultip
les
and
fact
ors
Use
prim
e fa
ctor
isat
ion
of n
umbe
rs to
find
LC
M a
nd H
CF
13MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Solv
ing
prob
lem
s So
lve
prob
lem
s in
volv
ing
who
le n
umbe
rs,
incl
udin
g:
C
ompa
ring
two
or m
ore
quan
titie
s of
the
sam
e ki
nd (r
atio
)
Com
parin
g tw
o qu
antit
ies
of d
iffer
ent
kind
s (ra
te)
Sh
arin
g in
a g
iven
ratio
whe
re th
e w
hole
is
giv
en
Solv
e pr
oble
ms
that
invo
lve
who
le n
umbe
rs,
perc
enta
ges
and
deci
mal
frac
tions
in
finan
cial
con
text
s su
ch a
s:
Pr
ofit,
loss
and
dis
coun
t
Budg
ets
Ac
coun
ts
Lo
ans
Si
mpl
e in
tere
st
Solv
ing
prob
lem
s So
lve
prob
lem
s in
volv
ing
who
le n
umbe
rs,
incl
udin
g:
C
ompa
ring
two
or m
ore
quan
titie
s of
th
e sa
me
kind
(rat
io)
C
ompa
ring
two
quan
titie
s of
diff
eren
t ki
nds
(rate
)
Shar
ing
in a
giv
en ra
tio w
here
the
who
le is
giv
en
In
crea
sing
or d
ecre
asin
g of
a n
umbe
r in
a g
iven
ratio
So
lve
prob
lem
s th
at in
volv
e w
hole
nu
mbe
rs, p
erce
ntag
es a
nd d
ecim
al
fract
ions
in fi
nanc
ial c
onte
xts
such
as:
Prof
it, lo
ss, d
isco
unt a
nd V
AT
Bu
dget
s
Acco
unts
Loan
s
Sim
ple
inte
rest
Hire
Pur
chas
e
Exch
ange
rate
s
Solv
ing
prob
lem
s So
lve
prob
lem
s in
con
text
s in
volv
ing
ratio
and
rate
dire
ct a
nd in
dire
ct p
ropo
rtion
So
lve
prob
lem
s th
at in
volv
e w
hole
nu
mbe
rs, p
erce
ntag
es a
nd d
ecim
al
fract
ions
in fi
nanc
ial c
onte
xts
such
as:
prof
it, lo
ss, d
isco
unt a
nd V
AT
bu
dget
s
acco
unts
and
loan
s
sim
ple
inte
rest
and
hire
pur
chas
e
exch
ange
rate
s an
d co
mm
issi
on
R
enta
ls
co
mpo
und
inte
rest
Sugg
este
d m
etho
dolo
gy
Ord
erin
g an
d co
mpa
ring
num
bers
Le
arne
rs s
houl
d be
giv
en a
rang
e of
exe
rcis
es s
uch
as:
Ar
rang
e gi
ven
num
bers
from
the
smal
lest
to th
e bi
gges
t or
big
gest
to s
mal
lest
Fill
in m
issi
ng n
umbe
rs in
a s
eque
nce;
on
a nu
mbe
r grid
; on
a nu
mbe
r lin
e e.
g. w
hich
who
le n
umbe
r is
halfw
ay b
etw
een
471
340
and
471
350.
Fi
ll in
<, =
or >
Ex
ampl
es:
a)
247
889
* 247
898
b)
78
4 10
9 * 7
85 1
90
Prop
ertie
s of
who
le n
umbe
rs
R
evis
ing
the
prop
ertie
s of
who
le n
umbe
rs s
houl
d be
the
star
ting
poin
t for
wor
k w
ith w
hole
num
bers
. The
pro
perti
es o
f num
bers
sh
ould
pro
vide
the
mot
ivat
ion
for w
hy a
nd h
ow o
pera
tions
with
num
bers
wor
k.
14 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
W
hen
lear
ners
are
intro
duce
d to
new
num
bers
, suc
h as
inte
gers
for e
xam
ple,
they
can
aga
in e
xplo
re h
ow th
e pr
oper
ties
of
num
bers
wor
k fo
r the
new
set
of n
umbe
rs.
Le
arne
rs a
lso
have
to a
pply
the
prop
ertie
s of
num
bers
in a
lgeb
ra, w
hen
they
wor
k w
ith v
aria
bles
in p
lace
of n
umbe
rs.
Le
arne
rs s
houl
d kn
ow a
nd b
e ab
le to
use
the
follo
win
g pr
oper
ties:
o
The
com
mut
ativ
e pr
oper
ty o
f add
ition
and
mul
tiplic
atio
n:
♦♦
a +
b =
b +
a
♦♦
a x
b =
b x
a
o Th
e as
soci
ativ
e (g
roup
ing)
pro
perty
of a
dditi
on a
nd m
ultip
licat
ion:
♦♦
(a +
b) +
c =
a +
(b +
a)
♦♦
(a x
b) x
c =
a x
(b x
c)
o Th
e di
strib
utiv
e pr
oper
ty o
f mul
tiplic
atio
n ov
er a
dditi
on a
nd s
ubtra
ctio
n:
♦♦
a(b
+ c)
= (a
x b
) + (a
x c
)
♦
♦ a(
b –
c) =
(a x
b) –
(a x
c)
o Ad
ditio
n an
d su
btra
ctio
n as
inve
rse
oper
atio
ns
o M
ultip
licat
ion
and
divi
sion
as
inve
rse
oper
atio
ns
o 0
is th
e id
entit
y el
emen
t for
add
ition
: t +
0 =
t o
1 is
the
iden
tity
elem
ent f
or m
ultip
licat
ion:
t x
1 =
t Ill
ustr
atin
g th
e pr
oper
ties
with
who
le n
umbe
rs
Exam
ples
a)
33
+ 9
9 =
99 +
33
= 13
2 b)
51
+ (1
9 +
46) =
(51
+ 19
) + 4
6 =
116
c)
4(12
+ 9
) = (4
x 1
2) +
(4 x
9) =
48
+ 36
= 8
4 d)
(9
x 6
4) +
(9 x
36)
= 9
x (6
4 +
36) =
9 x
100
= 9
00
e)
If 33
+ 9
9 =
132,
then
132
– 9
9 =
33 a
nd 1
32 –
33
= 99
f)
If 20
x 5
= 1
10, t
hen
110
÷ 20
= 5
and
110
÷ 5
= 2
0 C
alcu
latio
ns w
ith w
hole
num
bers
Lear
ners
sho
uld
do c
onte
xt fr
ee c
alcu
latio
ns a
nd s
olve
pro
blem
s in
con
text
s
Lear
ners
sho
uld
beco
me
mor
e co
nfid
ent i
n an
d m
ore
inde
pend
ent a
t m
athe
mat
ics,
if th
ey h
ave
tech
niqu
es
o to
che
ck th
eir s
olut
ions
them
selv
es, e
.g. u
sing
inve
rse
oper
atio
ns; u
sing
cal
cula
tors
o
to ju
dge
the
reas
onab
lene
ss o
f the
ir so
lutio
ns e
.g. e
stim
ate
by ro
undi
ng o
ff; e
stim
ate
by d
oubl
ing
or h
alvi
ng;
Ad
ding
, sub
tract
ing
and
mul
tiply
ing
in c
olum
ns, a
nd lo
ng d
ivis
ion,
sho
uld
only
be
used
to p
ract
ice
num
ber f
acts
and
cal
cula
tion
15MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
tech
niqu
es, a
nd h
ence
sho
uld
be d
one
with
fam
iliar a
nd s
mal
ler n
umbe
r ran
ges.
For
big
and
unw
ield
y ca
lcul
atio
ns, l
earn
ers
shou
ld
be e
ncou
rage
d to
use
a c
alcu
lato
r. M
ultip
les
and
fact
ors
Pr
actic
e w
ith fi
ndin
g m
ultip
les
and
fact
ors
of w
hole
num
bers
are
esp
ecia
lly im
porta
nt w
hen
lear
ners
do
calc
ulat
ions
with
frac
tions
. Th
ey u
se th
is k
now
ledg
e to
find
the
LCM
whe
n on
e de
nom
inat
or is
a m
ultip
le o
f ano
ther
, and
als
o w
hen
they
sim
plify
frac
tions
or
have
to fi
nd e
quiv
alen
t fra
ctio
ns.
Fa
ctor
isin
g w
hole
num
bers
lays
the
foun
datio
n fo
r fac
toris
atio
n of
alg
ebra
ic e
xpre
ssio
ns.
U
sing
the
defin
ition
of p
rime
num
bers
and
em
phas
ize
that
1 is
not
cla
ssifi
ed a
s a
prim
e nu
mbe
r
Exam
ples
a)
Th
e m
ultip
les
of 6
are
6 ,
12 ,
18 ,
24,..
. or M
6 =
{6; 1
2; 1
8; 2
4;...
} b)
LC
M o
f 6 a
nd 1
8 is
18
and
LCM
of 6
and
7 is
42
c)
The
fact
ors
of 2
4 ar
e 1,
2, 3
, 4, 6
, 12
and
24 b
y in
spec
tion
and,
the
prim
e fa
ctor
s of
24
are
2 an
d 3
d)
The
fact
ors
of 1
40 a
re 1
, 2, 5
, 7, 1
0, 1
4, 2
8, 3
5, 7
0 an
d 14
0 e)
D
eter
min
e th
e H
CF
of 1
20; 3
00 a
nd 9
00
Lear
ners
do
this
by
findi
ng th
e pr
ime
fact
ors
of th
e nu
mbe
rs fi
rst.
120
= 5
x 3
x 23 .
Initi
ally
lear
ners
may
writ
e th
is a
s: 5
x 3
x 2
x 2
x 2
30
0 =
52 x 3
x 2
2
900
= 52
x 3
2 x 2
2
HC
F =
5 x
3 x
22 =
60
(Mul
tiply
the
com
mon
prim
e fa
ctor
s of
the
thre
e nu
mbe
rs)
Solv
ing
prob
lem
s So
lvin
g pr
oble
ms
in c
onte
xts
shou
ld ta
ke a
ccou
nt o
f the
num
ber r
ange
s le
arne
rs a
re fa
milia
r with
. o
Con
text
s in
volv
ing
ratio
and
rate
sho
uld
incl
ude
spee
d, d
ista
nce
and
time
prob
lem
s.
o In
fina
ncia
l con
text
s, le
arne
rs a
re n
ot e
xpec
ted
to u
se fo
rmul
ae fo
r cal
cula
ting
sim
ple
inte
rest
. In
Gra
de 8
, rev
isin
g th
e pr
oper
ties
of w
hole
num
bers
sho
uld
be th
e st
artin
g po
int f
or w
ork
with
who
le n
umbe
rs. T
he p
rope
rties
of
num
bers
sho
uld
prov
ide
the
mot
ivat
ion
for w
hy a
nd h
ow o
pera
tions
with
num
bers
wor
k. W
hen
lear
ners
are
intro
duce
d to
new
nu
mbe
rs, s
uch
as in
tege
rs fo
r exa
mpl
e, th
ey c
an a
gain
exp
lore
how
the
prop
ertie
s of
num
bers
wor
k fo
r the
new
set
of n
umbe
rs.
Lear
ners
als
o ha
ve to
app
ly th
e pr
oper
ties
of n
umbe
rs in
alg
ebra
, whe
n th
ey w
ork
with
var
iabl
es in
pla
ce o
f num
bers
. In
Gra
de 9
lear
ners
con
solid
ate
num
ber k
now
ledg
e an
d ca
lcul
atio
n te
chni
ques
for w
hole
num
bers
, dev
elop
ed in
Gra
de 8
. The
focu
s in
G
rade
9 s
houl
d be
on
deve
lopi
ng a
n un
ders
tand
ing
of d
iffer
ent n
umbe
r sys
tem
s an
d th
e pr
oper
ties
of o
pera
tions
that
app
ly fo
r di
ffere
nt n
umbe
r sys
tem
s. T
he c
onte
xts
for s
olvi
ng p
robl
ems
shou
ld b
e m
ore
com
plex
and
var
ied,
invo
lvin
g w
hole
num
bers
, int
eger
s an
d ra
tiona
l num
bers
. Fin
anci
al c
onte
xts
are
espe
cial
ly ri
ch in
this
rega
rd. L
earn
ers
shou
ld b
e gi
ven
a cl
ear i
ndic
atio
n of
whe
n th
e us
e of
cal
cula
tors
is p
erm
issi
ble
or n
ot. C
alcu
lato
rs s
houl
d be
use
d ro
utin
ely
for c
alcu
latio
ns w
ith b
ig n
umbe
rs a
nd w
here
kno
wle
dge
of
16 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
num
ber f
acts
or c
once
pts
is n
ot e
xplic
itly
asse
ssed
. How
ever
, gua
rd a
gain
st le
arne
rs b
ecom
ing
depe
nden
t on
calc
ulat
ors
for a
ll ca
lcul
atio
ns. C
alcu
lato
rs re
mai
n a
usef
ul to
ol fo
r che
ckin
g so
lutio
ns. C
ompe
tenc
y in
find
ing
mul
tiple
s an
d fa
ctor
s, a
nd p
rime
fact
oris
atio
n of
who
le n
umbe
rs, r
emai
ns im
porta
nt fo
r dev
elop
ing
com
pete
ncy
in fa
ctor
isin
g al
gebr
aic
expr
essi
ons
and
solv
ing
alge
brai
c eq
uatio
ns.
By d
istin
guis
hing
the
prop
ertie
s of
diff
eren
t num
ber s
yste
ms,
lear
ners
sho
uld
reco
gniz
e th
at n
atur
al n
umbe
rs is
a s
ubse
t of w
hole
nu
mbe
rs, w
hich
in tu
rn is
a s
ubse
t of i
nteg
ers,
whi
ch in
turn
is a
sub
set o
f rat
iona
l num
bers
. All
of th
ese
num
bers
form
par
t of t
he re
al
num
ber s
yste
m.
Not
e th
at 0
may
som
etim
es b
e in
clud
ed in
the
set o
f nat
ural
num
bers
. Le
arne
rs s
houl
d re
cogn
ize
the
follo
win
g di
stin
guis
hing
feat
ures
of t
he n
umbe
r sys
tem
s
in
tege
rs e
xten
d th
e na
tura
l and
who
le n
umbe
r sys
tem
s by
incl
udin
g th
e op
erat
ion
a –
b, w
here
a <
b.
ra
tiona
l num
bers
ext
end
the
set o
f int
eger
s by
incl
udin
g th
e op
erat
ion
𝑎𝑎 𝑏𝑏 whe
re a
< b
ratio
nal n
umbe
rs a
re d
efin
ed a
s nu
mbe
rs th
at c
an b
e w
ritte
n in
the
form
𝑎𝑎 𝑏𝑏 whe
re a
and
b a
re in
tege
rs a
nd b
≠ 0
sinc
e in
tege
rs a
re a
sub
set o
f rat
iona
l num
bers
, eve
ry in
tege
r, ca
n be
exp
ress
ed a
s a
ratio
nal n
umbe
r a
irr
atio
nal n
umbe
rs a
re n
umbe
rs th
at c
anno
t be
expr
esse
d as
ratio
nal n
umbe
rs
Pi
(π) i
s an
irra
tiona
l num
ber,
even
thou
gh w
e us
e 22 7
or 3
,14
as ra
tiona
l num
ber a
ppro
xim
atio
ns fo
r π in
cal
cula
tions
R
atio
and
rate
pro
blem
s in
clud
e pr
oble
ms
invo
lvin
g sp
eed,
dis
tanc
e an
d tim
e. L
earn
ers
shou
ld b
e fa
milia
r with
the
follo
win
g fo
rmul
ae
for t
hese
cal
cula
tions
:
a)
spee
d =
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑎𝑎𝑑𝑑
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑡𝑡𝑑𝑑
b)
di
stan
ce =
spe
ed x
tim
e c)
tim
e =
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑎𝑎𝑑𝑑
𝑑𝑑𝑑𝑑𝑑𝑑𝑠𝑠𝑑𝑑𝑑𝑑𝑑𝑑
Sp
eed
is u
sual
ly g
iven
as
cons
tant
spe
ed o
r ave
rage
spe
ed. M
ake
sure
lear
ners
reco
gniz
e an
d ar
e ab
le to
con
vert
corre
ctly
bet
wee
n un
its fo
r tim
e an
d di
stan
ce.
Exam
ples
a)
A ca
r tra
vellin
g at
a c
onst
ant s
peed
trav
els
60 k
m in
18
min
utes
. How
far,
trave
lling
at th
e sa
me
cons
tant
spe
ed, w
ill th
e ca
r tra
vel i
n 1
hour
12
min
utes
? b)
A
car t
rave
lling
at a
n av
erag
e sp
eed
of 1
00 k
m/h
cov
ers
a ce
rtain
dis
tanc
e in
3 h
ours
20
min
utes
. At w
hat c
onst
ant s
peed
m
ust t
he c
ar tr
avel
to c
over
the
sam
e di
stan
ce in
2 h
ours
40
min
utes
?
17MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Dire
ct a
nd In
dire
ct p
ropo
rtio
n G
rade
9 le
arne
rs s
houl
d be
fam
iliar w
ith th
e fo
llow
ing
rela
tions
hips
:
x is
dire
ctly
pro
porti
onal
to y
if 𝑥𝑥 𝑦𝑦 =
con
stan
t
x an
d y
are
dire
ctly
pro
porti
onal
if, a
s th
e va
lue
of x
incr
ease
s th
e va
lue
of y
incr
ease
s in
the
sam
e pr
opor
tion,
and
as
the
valu
e of
x d
ecre
ases
the
valu
e of
y d
ecre
ases
in th
e sa
me
prop
ortio
n
The
dire
ct p
ropo
rtion
al re
latio
nshi
p is
repr
esen
ted
by a
stra
ight
line
gra
ph
x
is in
dire
ctly
or i
nver
sely
pro
porti
onal
to y
if x
x y
= a
con
stan
t. In
oth
er w
ords
y =
𝑐𝑐 𝑥𝑥
x an
d y
are
indi
rect
ly p
ropo
rtion
al if
, as
the
valu
e of
x in
crea
ses
the
valu
e of
y d
ecre
ases
and
as
the
valu
e of
x d
ecre
ases
the
valu
e of
y in
crea
ses
an
indi
rect
pro
porti
onal
rela
tions
hip
is re
pres
ente
d by
a n
on-li
near
cur
ve
Fina
ncia
l con
text
s O
nce
Gra
de 9
lear
ners
hav
e do
ne s
uffic
ient
cal
cula
tions
for s
impl
e an
d co
mpo
und
inte
rest
thro
ugh
repe
ated
cal
cula
tions
, the
y co
uld
use
give
n fo
rmul
ae fo
r the
se c
alcu
latio
ns.
Exam
ples
a)
Cal
cula
te th
e si
mpl
e in
tere
st o
n R
600
at 7
% p
.a fo
r 3 y
ears
usi
ng th
e fo
rmul
a S
I = 𝑃𝑃𝑃𝑃
𝑃𝑃10
0 or s
i = P
.n.i
for i
= 𝑖𝑖 100 .
b)
R80
0 in
vest
ed a
t r%
per
ann
um s
impl
e in
tere
st fo
r a p
erio
d of
3 y
ears
yie
lds
R16
8. C
alcu
late
the
valu
e of
r c)
H
ow lo
ng w
ill it
take
for R
3 00
0 in
vest
ed a
t 6%
per
ann
um s
impl
e in
tere
st to
gro
w to
R4
260?
d)
Te
mos
o bo
rrow
ed R
500
from
the
bank
for 3
yea
rs a
t 8%
p.a
. com
poun
d in
tere
st. W
ithou
t usi
ng a
form
ula,
cal
cula
te h
ow m
uch
Tem
oso
owes
the
bank
at t
he e
nd o
f thr
ee y
ears
. e)
U
se th
e fo
rmul
a A
= P(
1 +
𝑃𝑃 100 )n
to c
alcu
late
the
com
poun
d in
tere
st o
n a
loan
of R
3 45
0 at
6,5
% p
er a
nnum
for 5
yea
rs.
LT
SM
Res
ourc
es
Cal
cula
tors
, num
ber l
ine.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
G
rade
7 W
B 1
Activ
ities
(R
2a&b
; R3;
R4;
R5a
&b; R
6; 1
7-23
R13
G
rade
8 W
B 1
Activ
ities
(1
; 2a&
b; 3
-5; 6
-10)
G
rade
9 W
B 1
Activ
ities
(1
a&b;
3a&
b - 9
)
18 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Te
xtbo
ok
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t
19MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
2
TER
M 1
Dat
e : f
rom
....
to ..
....
Tim
e:
9 h
ours
G
rade
7
8 9
Topi
c IN
TEG
ERS
CAP
S pa
ges
67 –
68
78 -
80
121
Skills
and
Kno
wle
dge
Cou
ntin
g, o
rder
ing
and
com
parin
g in
tege
rs
C
ount
forw
ards
and
bac
kwar
ds in
in
tege
rs fo
r any
inte
rval
Rec
ogni
ze, o
rder
and
com
pare
in
tege
rs
Cal
cula
tions
with
inte
gers
• A
dd a
nd s
ubtra
ct w
ith in
tege
rs
Prop
ertie
s of
inte
gers
• R
ecog
nize
and
use
com
mut
ativ
e an
d as
soci
ativ
e pr
oper
ties
of a
dditi
on a
nd
mul
tiplic
atio
n fo
r int
eger
s
Cou
ntin
g, o
rder
ing
and
com
parin
g in
tege
rs
Rev
ise:
coun
ting
forw
ards
and
bac
kwar
ds in
in
tege
rs fo
r any
inte
rval
reco
gniz
ing,
ord
erin
g an
d co
mpa
ring
inte
gers
C
alcu
latio
ns w
ith in
tege
rs
R
evis
e ad
ditio
n an
d su
btra
ctio
n w
ith
inte
gers
Mul
tiply
and
div
ide
with
inte
gers
Perfo
rm c
alcu
latio
ns in
volv
ing
all f
our
oper
atio
ns w
ith in
tege
rs
Pe
rform
cal
cula
tions
invo
lvin
g al
l fou
r op
erat
ions
with
num
bers
that
invo
lve
the
squa
res,
cub
es, s
quar
e ro
ots
and
cube
root
s of
inte
gers
Prop
ertie
s of
inte
gers
Rec
ogni
ze a
nd u
se c
omm
utat
ive,
as
soci
ativ
e an
d di
strib
utiv
e pr
oper
ties
of a
dditi
on a
nd m
ultip
licat
ion
for
inte
gers
Rec
ogni
ze a
nd u
se a
dditi
ve a
nd
mul
tiplic
ativ
e in
vers
es fo
r int
eger
s
Cou
ntin
g, o
rder
ing
and
com
parin
g in
tege
rs -
Rev
isio
n R
evis
e:
co
untin
g fo
rwar
ds a
nd b
ackw
ards
in
inte
gers
for a
ny in
terv
al
re
cogn
izin
g, o
rder
ing
and
com
parin
g in
tege
rs
Cal
cula
tions
with
inte
gers
R
evis
e:
pe
rform
cal
cula
tions
invo
lvin
g al
l fou
r op
erat
ions
with
inte
gers
perfo
rm c
alcu
latio
ns in
volv
ing
all f
our
oper
atio
ns w
ith n
umbe
rs th
at in
volv
e th
e sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be
root
s of
inte
gers
Pr
oper
ties
of in
tege
rs
Rev
ise:
com
mut
ativ
e, a
ssoc
iativ
e an
d di
strib
utiv
e pr
oper
ties
of a
dditi
on a
nd
mul
tiplic
atio
n fo
r int
eger
s
ad
ditiv
e an
d m
ultip
licat
ive
inve
rses
for
inte
gers
20 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Solv
ing
prob
lem
s • S
olve
pro
blem
s in
con
text
s in
volv
ing
ad
ditio
n an
d su
btra
ctio
n of
inte
gers
Solv
ing
prob
lem
s So
lve
prob
lem
s in
con
text
s in
volv
ing
mul
tiple
ope
ratio
ns w
ith in
tege
rs
Solv
ing
prob
lem
s So
lve
prob
lem
s in
con
text
s in
volv
ing
mul
tiple
ope
ratio
ns w
ith in
tege
rs
Su
gges
ted
met
hodo
logy
In
tege
rs a
re n
ew n
umbe
rs in
trodu
ced
in G
rade
7.
Cou
ntin
g, o
rder
ing
and
com
parin
g in
tege
rs
Cou
ntin
g sh
ould
not
onl
y be
thou
ght o
f as
verb
al c
ount
ing.
Lea
rner
s ca
n co
unt u
sing
:
stru
ctur
ed, s
emi-s
truct
ured
or e
mpt
y nu
mbe
r lin
es
ch
ain
diag
ram
s fo
r cou
ntin
g Le
arne
rs s
houl
d be
giv
en a
rang
e of
exe
rcis
es s
uch
as:
ar
rang
e gi
ven
num
bers
from
the
smal
lest
to th
e bi
gges
t: or
big
gest
to s
mal
lest
fill i
n m
issi
ng n
umbe
rs in
a s
eque
nce;
on
a nu
mbe
r grid
; or o
n a
num
ber l
ine
fil
l in
<, =
or >
Ex
ampl
e:
– 4
25 *
– 45
0 C
alcu
latio
ns u
sing
inte
gers
• St
art c
alcu
latio
ns u
sing
inte
gers
in s
mal
l num
ber r
ange
s.
• D
evel
op a
n un
ders
tand
ing
that
sub
tract
ing
an in
tege
r is
the
sam
e as
add
ing
its a
dditi
ve in
vers
e.
Exam
ple:
7
– 4
= 7
+ (–
4) =
3 O
R –
7 –
4 =
–7 +
(– 4
) = –
11
So to
o, 7
– (–
4) =
7 +
(+4)
= 1
1 O
R –
7 –
(– 4
) = –
7 +
(+4)
= –
3 H
ere
the
use
of b
rack
ets
arou
nd th
e in
tege
rs is
use
ful.
Prop
ertie
s of
inte
gers
•
Lear
ners
sho
uld
inve
stig
ate
the
prop
ertie
s fo
r ope
ratio
ns u
sing
who
le n
umbe
rs o
n th
e se
t of i
nteg
ers.
•
Thes
e pr
oper
ties
shou
ld s
erve
as
mot
ivat
ion
for t
he o
pera
tions
they
can
per
form
usi
ng in
tege
rs.
• Le
arne
rs s
houl
d se
e th
at th
e co
mm
utat
ive
prop
erty
for a
dditi
on h
olds
for i
nteg
ers
e.g.
8 +
(–3)
= (–
3) +
8 =
5
• Le
arne
rs s
houl
d se
e th
at th
ey c
an s
till u
se s
ubtra
ctio
n to
che
ck a
dditi
on o
r vic
e ve
rsa.
Exa
mpl
e: If
8 +
(–3)
= 5
, the
n 5
– 8
= –
3 an
d 5
–(–3
) = 8
•
Lear
ners
sho
uld
see
that
the
asso
ciat
ive
prop
erty
for a
dditi
on h
olds
for i
nteg
ers.
Exa
mpl
e: [(
–6) +
4] +
(–1)
= (–
6) +
[4 +
(–1)
] = –
3 •
Lear
ners
sho
uld
only
exp
lore
the
dist
ribut
ive
prop
erty
onc
e th
ey c
an m
ultip
ly w
ith in
tege
rs
21MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
In G
rade
8 le
arne
rs:
• C
onso
lidat
e nu
mbe
r kno
wle
dge
and
calc
ulat
ion
tech
niqu
es fo
r int
eger
s, d
evel
oped
in G
rade
7.
• M
ultip
ly a
nd d
ivid
e w
ith in
tege
rs
• Pe
rform
all
four
ope
ratio
ns w
ith in
tege
rs
• Pe
rform
all
four
ope
ratio
ns w
ith s
quar
es, c
ubes
, squ
are
and
cube
root
s of
inte
gers
C
ount
ing,
ord
erin
g an
d co
mpa
ring
inte
gers
Lear
ners
sho
uld
cont
inue
pra
ctis
ing
coun
ting,
ord
erin
g an
d co
mpa
ring
inte
gers
. Cou
ntin
g sh
ould
not
onl
y be
thou
ght o
f as
verb
al
coun
ting.
Lea
rner
s ca
n co
unt u
sing
: o
stru
ctur
ed, s
emi-s
truct
ured
or e
mpt
y nu
mbe
r lin
es
o ch
ain
diag
ram
s fo
r cou
ntin
g
Lear
ners
sho
uld
be g
iven
a ra
nge
of e
xerc
ises
Th
ey s
houl
d
o Ar
rang
e gi
ven
num
bers
from
the
smal
lest
to th
e bi
gges
t: or
big
gest
to s
mal
lest
o
Fill
in m
issi
ng n
umbe
rs in
a s
eque
nce;
on
a nu
mbe
r grid
; or o
n a
num
ber l
ine
o Fi
ll in
<, =
or >
e.g
. – 4
25 *
– 45
0;
Cal
cula
tions
usi
ng in
tege
rs
See
note
s fo
r Gra
de 7
abo
ve.
A us
eful
stra
tegy
is to
use
repe
ated
add
ition
and
num
ber p
atte
rns
to s
how
lear
ners
the
reas
onab
lene
ss o
f rul
es fo
r the
resu
ltant
sig
n fo
r mul
tiplic
atio
n w
ith in
tege
rs.
Exam
ple:
a)
R
epea
ted
addi
tion
of (–
3): (
–3) +
(–3)
+ (–
3) =
–9
= 3
x (–
3)
b)
Rep
eate
d ad
ditio
n of
(–2)
: (–2
) + (–
2) +
(–2)
+ (–
2) =
–8
= 4
x (–
2)
c)
Cou
ntin
g do
wn
in in
terv
als
of 3
from
9:
3 x
3 =
9 3
x 2
= 6
3 x
1 =
3 3
x 0
= 0
3 x
–1 =
–3
3 x
–2 =
?
3 x
–3 =
?
Hen
ce th
e ru
le: a
pos
itive
inte
ger x
a n
egat
ive
inte
ger =
a n
egat
ive
inte
ger
22 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
d)
U
sing
the
rule
that
a p
ositi
ve in
tege
r x a
neg
ativ
e in
tege
r = a
neg
ativ
e in
tege
r, es
tabl
ishe
d fro
m e
xam
ples
abo
ve, t
he
follo
win
g pa
ttern
can
be
used
: –1
x 3
= –
3 –1
x 2
= –
2 –1
x 1
= –
1 –1
x 0
= 0
–1
x –
1 =
1 –1
x –
2 =
? –1
x –
3 =
? H
ence
the
rule
: a n
egat
ive
inte
ger x
a n
egat
ive
inte
ger =
a p
ositi
ve in
tege
r U
se th
e in
vers
e op
erat
ion
for m
ultip
licat
ion
and
divi
sion
to d
evel
op a
rule
for t
he re
sulta
nt s
ign
for d
ivis
ion
with
inte
gers
. Ex
ampl
e:
a)
If 4
x (–
2) =
–8,
then
–8
÷ 4
= –2
and
–8
÷ (–
2) =
4
b)
If (–
1) x
(–3)
= 3
, the
n 3
÷ (–
1) =
–3
and
3 ÷
(–3)
= –
1 H
ence
the
rule
s: d
ivis
ion
of a
pos
itive
and
neg
ativ
e in
tege
r equ
als
a ne
gativ
e in
tege
r and
div
isio
n of
two
nega
tive
inte
gers
equ
al a
po
sitiv
e in
tege
r. Fi
ndin
g th
e sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of in
tege
rs a
re a
lso
oppo
rtuni
ties
to c
heck
that
lear
ners
kno
w th
e ru
les
for
resu
ltant
sig
ns w
hen
mul
tiply
ing
inte
gers
. The
refo
re, m
ake
sure
that
lear
ners
und
erst
and
why
you
can
not f
ind
the
squa
re ro
ot o
f a
nega
tive
inte
ger,
and
that
the
squa
re o
f a n
egat
ive
inte
ger i
s al
way
s po
sitiv
e.
Exam
ple:
a)
(−
5)2
= (–
5) x
(–5)
= 2
5 b)
)
(− 4)
3 = (–
4) x
(–4)
x (–
4) =
– 6
4 c)
√−
273
= –
3 be
caus
e –3
x –
3 x–
3 =
–27
Prop
ertie
s of
inte
gers
Le
arne
rs s
houl
d in
vest
igat
e th
e pr
oper
ties
for o
pera
tions
with
who
le n
umbe
rs o
n th
e se
t of i
nteg
ers.
The
se p
rope
rties
sho
uld
serv
e as
mot
ivat
ion
for t
he o
pera
tions
they
can
per
form
with
inte
gers
. Lea
rner
s sh
ould
: •
see
that
the
com
mut
ativ
e pr
oper
ty fo
r add
ition
and
mul
tiplic
atio
n ho
lds
for i
nteg
ers,
e.g
. 8 +
(–3)
= (–
3) +
8 =
5; 8
x (–
3) =
(–3)
x 8
= –
24
• se
e th
at th
ey c
an s
till u
se s
ubtra
ctio
n to
che
ck a
dditi
on o
r vic
e ve
rsa,
e.g
. if 8
+ (–
3) =
5, t
hen
5 –
8 =
–3 a
nd 5
–(–
3) =
8
• se
e th
at th
e as
soci
ativ
e pr
oper
ty fo
r add
ition
hol
ds fo
r int
eger
s, e
.g. [
(–6)
+ 4
] + (–
1) =
(–6)
+ [4
+ (–
1)] =
–3
23MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
• se
e th
at th
e in
vers
e op
erat
ion
for m
ultip
licat
ion
and
divi
sion
hol
ds fo
r int
eger
s, e
.g. i
f 5 x
(–6)
= –
30, t
hen
–30
÷ 5
= –6
and
–30
÷ (–
6) =
5
• de
velo
p th
e ru
les,
thro
ugh
patte
rnin
g, fo
r res
ulta
nt s
igns
whe
n m
ultip
lyin
g an
d di
vidi
ng in
tege
rs:
Exam
ples
(+
5) x
(+5)
= (+
25);
(–5)
x (–
5) =
(+25
); (–
5) x
(+5)
= (–
25);
(+25
) ÷ (+
5) =
(+5)
; (–
25) ÷
(–5)
= (+
5);
(–25
) ÷ (+
5) =
(–5)
; In
Gra
de 9
lear
ners
: •
cons
olid
ate
num
ber k
now
ledg
e an
d ca
lcul
atio
n te
chni
ques
for i
nteg
ers,
dev
elop
ed in
Gra
de 8
. •
wor
k w
ith in
tege
rs m
ostly
as
coef
ficie
nts
in a
lgeb
raic
exp
ress
ions
and
equ
atio
ns.
The
y ar
e ex
pect
ed to
be
com
pete
nt in
per
form
ing
all f
our o
pera
tions
with
inte
gers
and
usi
ng th
e pr
oper
ties
of in
tege
rs a
ppro
pria
tely
w
here
nec
essa
ry.
LTSM
Res
ourc
es
Num
ber l
ine
with
neg
ativ
e an
d po
sitiv
e nu
mbe
rs, t
herm
omet
er, m
easu
ring
jugs
, ice
, cha
rts.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
G
rade
7 W
B 2
Activ
ities
(1
05 -
113)
Gra
de 8
WB
1 Ac
tiviti
es
(1; 1
1; 1
2; 1
3)
Gra
de 9
WB
1 Ac
tiviti
es
(R4)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – te
st 1
5 m
arks
max
24 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
25MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTIG
RAD
E LE
SSO
N P
LAN
3
TER
M 1
Dat
e: fr
om...
. to.
.....
Tim
e:
9 ho
urs
Gra
de
7 8
9 To
pic
EXPO
NEN
TS
CAP
S pa
ges
43 –
44
81 -
84
124
- 126
Sk
ills a
nd
Know
ledg
e C
ompa
ring
and
repr
esen
ting
num
bers
in
exp
onen
tial f
orm
C
ompa
re a
nd re
pres
ent w
hole
num
bers
in
exp
onen
tial f
orm
:
b a=
a x
a x
a x.
.. fo
r b n
umbe
r of
fact
ors
Cal
cula
tions
usi
ng n
umbe
rs in
ex
pone
ntia
l for
m
R
ecog
nize
and
use
the
appr
opria
te
law
s of
ope
ratio
ns w
ith n
umbe
rs
invo
lvin
g ex
pone
nts
and
squa
re a
nd
cube
root
s
Com
parin
g an
d re
pres
entin
g nu
mbe
rs in
exp
onen
tial f
orm
Rev
ise
com
pare
and
repr
esen
t who
le
num
bers
in e
xpon
entia
l for
m
C
ompa
re a
nd re
pres
ent i
nteg
ers
in
expo
nent
ial f
orm
Com
pare
and
repr
esen
t num
bers
in
scie
ntifi
c no
tatio
n, li
mite
d to
pos
itive
ex
pone
nts
Cal
cula
tions
usi
ng n
umbe
rs in
ex
pone
ntia
l for
m
Es
tabl
ish
gene
ral l
aws
of e
xpon
ents
, lim
ited
to n
atur
al n
umbe
r exp
onen
ts:
o n
mn
ma
aa
o n
mn
ma
aa
o
nm
nm
aa
)(
o n
nn
ta
ta
)
(
o 1
)(
0
a
R
ecog
nize
and
use
the
appr
opria
te
law
s of
ope
ratio
ns u
sing
num
bers
Com
parin
g an
d re
pres
entin
g nu
mbe
rs in
ex
pone
ntia
l for
m
R
evis
e:
o co
mpa
re a
nd re
pres
ent i
nteg
ers
in
expo
nent
ial f
orm
o
com
pare
and
repr
esen
t num
bers
in
scie
ntifi
c no
tatio
n
Exte
nd s
cien
tific
not
atio
n to
incl
ude
nega
tive
expo
nent
s C
alcu
latio
ns u
sing
num
bers
in
expo
nent
ial f
orm
Rev
ise
the
follo
win
g ge
nera
l law
s of
ex
pone
nts:
o
am x
an =
am
+n
o am
÷ a
n = a
m-n
, if m
>n
o (a
m)n =
am
x n
o
(a x
t)n =
an x
t n
o a0 =
1
Ex
tend
the
gene
ral l
aws
of e
xpon
ents
to
incl
ude
inte
ger e
xpon
ents
𝑎𝑎−𝑛𝑛
= 1 𝑎𝑎𝑛𝑛
26 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Perfo
rm c
alcu
latio
ns in
volv
ing
all f
our
oper
atio
ns u
sing
num
bers
in
expo
nent
ial f
orm
, lim
ited
to
expo
nent
s up
to 5
, and
squ
are
and
cube
root
s
So
lvin
g pr
oble
ms
Solv
e pr
oble
ms
in c
onte
xts
invo
lvin
g nu
mbe
rs in
exp
onen
tial f
orm
invo
lvin
g ex
pone
nts
and
squa
re a
nd
cube
root
s
Pe
rform
cal
cula
tions
invo
lvin
g al
l fou
r op
erat
ions
with
num
bers
that
invo
lve
squa
res,
cub
es, s
quar
e ro
ots
and
cube
ro
ots
of in
tege
rs
C
alcu
late
the
squa
res,
cub
es, s
quar
e ro
ots
and
cube
root
s of
ratio
nal
num
bers
So
lvin
g pr
oble
ms
Solv
e pr
oble
ms
in c
onte
xts
invo
lvin
g nu
mbe
rs in
exp
onen
tial f
orm
.
Pe
rform
cal
cula
tions
invo
lvin
g al
l fou
r op
erat
ions
usi
ng n
umbe
rs in
exp
onen
tial
form
So
lvin
g pr
oble
ms
Solv
e pr
oble
ms
in c
onte
xts
invo
lvin
g nu
mbe
rs in
exp
onen
tial f
orm
, inc
ludi
ng
scie
ntifi
c no
tatio
n
Sugg
este
d m
etho
dolo
gy
Alth
ough
Gra
de 7
lear
ners
may
hav
e en
coun
tere
d sq
uare
num
bers
in G
rade
6, t
hey
wer
e no
t exp
ecte
d to
writ
e th
ese
num
bers
in
expo
nent
ial f
orm
. C
ompa
ring
and
repr
esen
ting
num
bers
in e
xpon
entia
l for
m
Lear
ners
nee
d to
und
erst
and
that
in th
e ex
pone
ntia
l for
m a
b , th
e nu
mbe
r is
read
as
‘a to
the
pow
er b
’, w
here
a is
cal
led
the
base
an
d b
is c
alle
d th
e ex
pone
nt o
r ind
ex.(b
) ind
icat
es th
e nu
mbe
r of f
acto
rs th
at a
re m
ultip
lied.
Ex
ampl
e:
a) a
3 = a
x a
x a
; b)
a5 =
a x
a x
a x
a x
a
Lear
ners
can
repr
esen
t any
num
ber i
n ex
pone
ntia
l for
m, w
ithou
t nee
ding
to c
ompu
te th
e va
lue.
27MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Exam
ple:
50
x 5
0 x
50 x
50
x 50
x 5
0 x
50 =
(50)
7
Mak
e su
re le
arne
rs u
nder
stan
d th
at s
quar
e ro
ots
and
cube
root
s ar
e th
e in
vers
e op
erat
ions
of s
quar
ing
and
cubi
ng n
umbe
rs.
Exam
ples
: 32 =
9 th
eref
ore √92
= 3
M
ake
sure
lear
ners
und
erst
and
that
any
num
ber r
aise
d to
the
pow
er 1
is e
qual
to th
e nu
mbe
r. Ex
ampl
e:
m1 =
m
At th
is p
oint
, lea
rner
s do
not
nee
d to
kno
w th
e ru
le fo
r rai
sing
a n
umbe
r to
the
pow
er 0
. Thi
s w
ill on
ly b
e in
trodu
ced
in G
rade
8 w
hen
they
use
oth
er la
ws
of e
xpon
ents
in c
alcu
latio
ns. T
o av
oid
com
mon
mis
conc
eptio
ns, e
mph
asiz
e th
e fo
llow
ing
with
exa
mpl
es:
(1
2)2 =
12
x 12
and
not
12
x 2
13 m
eans
1 x
1 x
1 a
nd n
ot 1
x 3
1001 =
100
√81
2 =
9 b
ecau
se 9
2 = 8
1
√27
3 =
3 b
ecau
se 3
3 = 2
7
The
squa
re o
f 9
= 81
, whe
reas
the
squa
re ro
ot o
f 9 =
3
Le
arne
rs s
houl
d us
e th
eir k
now
ledg
e of
repr
esen
ting
num
bers
in e
xpon
entia
l l fo
rm w
hen
sim
plify
ing
and
expa
ndin
g al
gebr
aic
expr
essi
ons
and
solv
ing
alge
brai
c e
quat
ions
C
alcu
latio
ns u
sing
num
bers
in e
xpon
entia
l for
m
Know
ing
the
rule
s of
ope
ratio
ns fo
r cal
cula
tions
invo
lvin
g ex
pone
nts
is im
porta
nt.
Exam
ples
: a)
(7 –
4)3 =
33 a
nd N
OT
73 – 4
3
b) √
16+
9 =
√25
, and
NO
T √1
6 + √9
In
Gra
de 8
:
• In
tege
rs a
nd ra
tiona
l num
bers
in e
xpon
entia
l for
m
28 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
• Sc
ient
ific
nota
tion
of n
umbe
rs
Law
s of
exp
onen
ts
The
law
s of
exp
onen
ts s
houl
d be
intro
duce
d th
roug
h a
rang
e of
num
eric
exa
mpl
es fi
rst,
then
var
iabl
es c
an b
e us
ed. I
n ot
her w
ords
, th
e nu
mbe
rs a
re re
plac
ed w
ith le
tters
, but
the
rule
s w
ork
the
sam
e. T
he fo
llow
ing
law
s of
exp
onen
ts s
houl
d be
intro
duce
d, w
here
m
and
n ar
e na
tura
l num
bers
and
a a
nd t
are
not e
qual
to 0
:
nm
nm
aa
a
if
m >
n Ex
ampl
es
a)
74
34
32
22
2
b)
74
34
3x
xx
x
n
mn
ma
aa
if
m >
n
Exam
ples
a)
273
33
32
5
b)
23
53
5x
xx
x
nm
nm
aa
)(
Exam
ples
: a)
62
32
32
2)
2(
b)
62
32
3)
(x
xx
29MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
nn
nt
at
a
)(
Ex
ampl
es:
a)
63
23
32
27)
(3
)3(
xx
x
b)
4000
625
6425
8)
5()
2()
52(
22
22
23
22
3
1
)(
0
a
Exam
ples
: a)
1)
37(0
b)
1)
4(0
2
x
Mak
e su
re th
at le
arne
rs u
nder
stan
d th
e la
ws
of e
xpon
ents
read
ing
from
bot
h si
des
of th
e eq
ual s
ign
i.e. i
f the
LH
S =
RH
S, th
en th
e R
HS
= LH
S.
The
law
1
)(
0
aca
n be
der
ived
by
usin
g th
e la
w o
f exp
onen
ts fo
r div
isio
n in
a fe
w e
xam
ples
.
1
44
a
aa
aa
aa
aa
a
ther
efor
e 1
04
4
aa
Lear
ners
sho
uld
be a
ble
to u
se th
e la
ws
of e
xpon
ents
in c
alcu
latio
ns a
nd fo
r sol
ving
sim
ple
expo
nent
ial e
quat
ions
as
wel
l as
expa
ndin
g or
sim
plify
ing
alge
brai
c ex
pres
sion
s.
Look
out
for t
he fo
llow
ing
com
mon
mis
conc
eptio
ns w
here
: •
Lear
ners
mul
tiply
unl
ike
base
s an
d ad
d th
e ex
pone
nt.
Ex
ampl
e:
a)
nm
nm
xyy
x
)(
•
Lear
ners
mul
tiply
like
bas
es a
nd a
dd th
e ex
pone
nts
30 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Exam
ple:
127
57
54
22
2
In
stea
d of
the
corre
ct a
nsw
er 2
12.
• Le
arne
rs fo
rget
, for
exa
mpl
e, th
at w
hen
squa
ring
a bi
nom
ial t
here
is a
mid
dle
term
. Ex
ampl
e:
mm
my
xy
x
)(
Lear
ners
con
fuse
add
ing
the
expo
nent
s an
d ad
ding
the
term
s
Exam
ple:
n
mn
mx
xx
o
r m
nx
C
alcu
latio
ns u
sing
num
bers
in e
xpon
entia
l for
m
Know
ing
the
rule
s of
ope
ratio
ns fo
r cal
cula
tions
invo
lvin
g ex
pone
nts
are
impo
rtant
, e.g
. a)
(7
– 4
)3 = 3
3 and
NO
T 73 –
43
b) √1
6+
9 =
√25
, and
NO
T √1
6 + √9
Le
arne
rs c
an a
lso
do s
impl
e ca
lcul
atio
ns w
here
the
num
erat
or a
nd d
enom
inat
or o
f a fr
actio
n ar
e w
ritte
n in
exp
onen
tial f
orm
, e.g
., 23 22
= 2
× 2
× 2
2 ×2
= 8 4
=
2
Lear
ners
can
als
o fin
d sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of d
ecim
al a
nd c
omm
on fr
actio
ns b
y in
spec
tion.
31MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Exam
ples
a)
(0,7
)2 = 0
,49
b)
(0,1
)3 = 0
,001
c)
09,0
=0,3
d)
16943
2222
23
e)
259=
259=
53
Scie
ntifi
c no
tatio
n W
hen
writ
ing
num
bers
in s
cien
tific
not
atio
n, le
arne
rs h
ave
to u
nder
stan
d th
e re
latio
nshi
p be
twee
n th
e nu
mbe
r of d
ecim
al p
lace
s an
d th
e in
dex
of 1
0 Ex
ampl
es:
a)
25
= 2
,5 x
101
b)
250
= 2,
5 x
102
c)
2 50
0 =
2,5
x 10
3
Scie
ntifi
c no
tatio
n lim
ited
to p
ositi
ve e
xpon
ents
, inc
lude
s w
ritin
g ve
ry la
rge
num
bers
in s
cien
tific
not
atio
n.
Exam
ple:
25
milli
on =
2,5
x 1
07
Lear
ners
pra
ctic
e w
ritin
g la
rge
num
bers
in s
cien
tific
not
atio
n, th
ey w
ill re
aliz
e th
ey h
ave
enco
unte
red
thes
e in
Nat
ural
Sci
ence
. It i
s us
eful
to re
fer t
o th
ese
cont
exts
whe
n ta
lkin
g ab
out s
cien
tific
not
atio
n.
In G
rade
9 le
arne
rs
C
onso
lidat
e nu
mbe
r kno
wle
dge
and
calc
ulat
ion
tech
niqu
es fo
r exp
onen
ts, d
evel
oped
in G
rade
8.
D
o ad
ditio
nal l
aws
of e
xpon
ents
invo
lvin
g in
tege
r exp
onen
ts
32 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
D
o sc
ient
ific
nota
tion
for n
umbe
rs, i
nclu
ding
neg
ativ
e ex
pone
nts
Law
s of
exp
onen
ts in
volv
ing
inte
ger e
xpon
ents
mm
aa
1
Exam
ples
:
a)
1251
515
33
b)
49171
77
77
22
53
53
Lo
ok o
ut fo
r the
follo
win
g co
mm
on m
isco
ncep
tions
whe
re:
Lear
ners
con
fuse
the
expo
nent
of t
he v
aria
ble
and
the
coef
ficie
nt, e
.g.,
33
212
xx
inst
ead
of
312
x
C
alcu
latio
ns a
nd s
impl
e eq
uatio
ns u
sing
num
bers
in e
xpon
entia
l for
m
The
calc
ulat
ions
and
equ
atio
ns s
houl
d pr
ovid
e op
portu
nitie
s to
app
ly th
e la
ws
of e
xpon
ents
and
sho
uld
not b
e un
duly
com
plex
. Ex
ampl
es
a)
Cal
cula
te:
23
13
62
b)
Si
mpl
ify:
22
)2
)(2
(
xx
c)
So
lve
x:
93
x
d)
Solv
e x:
(i)
412
x
33MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
(ii)
1
51
x
Sc
ient
ific
nota
tion
W
hen
writ
ing
num
bers
in s
cien
tific
not
atio
n, le
arne
rs h
ave
to u
nder
stan
d th
e re
latio
nshi
p be
twee
n th
e nu
mbe
r of d
ecim
al p
lace
s an
d th
e in
dex
of 1
0.
Exam
ple:
1
105.2
25
a
nd
210
5.225
0
Sc
ient
ific
nota
tion
that
ext
ends
to n
egat
ive
expo
nent
s in
clud
es w
ritin
g ve
ry s
mal
l num
bers
in s
cien
tific
not
atio
n Ex
ampl
e:
510
5.225
m
illio
nth
Le
arne
rs p
ract
ice
writ
ing
smal
l and
larg
e nu
mbe
rs in
sci
entif
ic n
otat
ion,
whi
ch th
ey m
ight
alre
ady
have
enc
ount
ered
in N
atur
al
Scie
nce.
It is
use
ful t
o re
fer t
o th
ese
cont
exts
whe
n di
scus
sing
sci
entif
ic n
otat
ion.
C
alcu
latio
ns c
an b
e do
ne w
ith o
r with
out a
cal
cula
tor.
Exam
ples
a)
C
alcu
late
: 7
510
910
6.2
with
out u
sing
a c
alcu
lato
r and
giv
e an
swer
in s
cien
tific
not
atio
n.
b)
Writ
e in
sci
entif
ic n
otat
ion:
0,0
0053
c)
C
alcu
late
: 5
410
3.210
8.5
with
out u
sing
a c
alcu
lato
r.
LTSM
Res
ourc
es
Cha
rts.
G
rade
7
Gra
de 8
G
rade
8
Wor
kboo
k re
fere
nce
Gra
de 7
WB
1 Ac
tiviti
es
(14;
15;
16;
17;
18;
19
Gra
de 8
WB
1 Ac
tiviti
es
(R3a
&b; 1
4 –
24; 2
6)
Gra
de 9
WB
1 Ac
tiviti
es
(R3a
&b; 2
2-26
a&b)
34 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – te
st
15 m
arks
max
35MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
4
TER
M 1
Dat
e : f
rom
....
to ..
....
T
ime:
6 ho
urs
G
rade
7
8 9
Topi
c N
UM
ERIC
AN
D G
EOM
ETR
IC P
ATTE
RN
S C
APS
page
s 58
– 6
1; 6
8 85
– 8
8 12
6 - 1
29
Skills
and
Kn
owle
dge
Inve
stig
ate
and
exte
nd p
atte
rns
In
vest
igat
e an
d ex
tend
num
eric
and
ge
omet
ric p
atte
rns
look
ing
for
rela
tions
hips
bet
wee
n nu
mbe
rs, i
nclu
ding
pa
ttern
s:
o re
pres
ente
d in
phy
sica
l or d
iagr
am
form
o
not l
imite
d to
seq
uenc
es in
volv
ing
a co
nsta
nt d
iffer
ence
or r
atio
o
of le
arne
r’s o
wn
crea
tion
o re
pres
ente
d in
tabl
es
D
escr
ibe
and
just
ify th
e ge
nera
l rul
es fo
r ob
serv
ed re
latio
nshi
ps b
etw
een
num
bers
in
ow
n w
ords
Inve
stig
ate
and
exte
nd p
atte
rns
In
vest
igat
e an
d ex
tend
num
eric
and
ge
omet
ric p
atte
rns
look
ing
for
rela
tions
hips
bet
wee
n nu
mbe
rs,
incl
udin
g pa
ttern
s:
o re
pres
ente
d in
phy
sica
l or d
iagr
am
form
o
not l
imite
d to
seq
uenc
es in
volv
ing
a co
nsta
nt d
iffer
ence
or r
atio
o
of le
arne
r’s o
wn
crea
tion
o re
pres
ente
d in
tabl
es
o re
pres
ente
d al
gebr
aica
lly
Des
crib
e an
d ju
stify
the
gene
ral r
ules
fo
r obs
erve
d re
latio
nshi
ps b
etw
een
num
bers
in o
wn
wor
ds o
r in
alge
brai
c la
ngua
ge
Inve
stig
ate
and
exte
nd p
atte
rns
In
vest
igat
e an
d ex
tend
num
eric
and
ge
omet
ric p
atte
rns
look
ing
for
rela
tions
hips
bet
wee
n nu
mbe
rs
incl
udin
g pa
ttern
s:
o re
pres
ente
d in
phy
sica
l or d
iagr
am
form
o
not l
imite
d to
seq
uenc
es in
volv
ing
a co
nsta
nt d
iffer
ence
or r
atio
o
of le
arne
r’s o
wn
crea
tion
o re
pres
ente
d in
tabl
es
o re
pres
ente
d al
gebr
aica
lly
Des
crib
e an
d ju
stify
the
gene
ral r
ules
for
obse
rved
rela
tions
hips
bet
wee
n nu
mbe
rs in
ow
n w
ords
or i
n al
gebr
aic
lang
uage
Sugg
este
d m
etho
dolo
gy
In th
e Se
nior
Pha
se th
e em
phas
is is
less
on
mer
ely
exte
ndin
g a
patte
rn, a
nd m
ore
on d
escr
ibin
g a
gene
ral r
ule
for t
he p
atte
rn o
r seq
uenc
e an
d be
ing
able
to p
redi
ct u
nkno
wn
term
s in
a s
eque
nce
usin
g a
gene
ral r
ule.
Inve
stig
atin
g nu
mbe
r pat
tern
s is
an
oppo
rtuni
ty to
gen
eral
ize
– to
giv
e ge
nera
l alg
ebra
ic d
escr
iptio
ns o
f the
rela
tions
hip
betw
een
term
s an
d its
pos
ition
in a
seq
uenc
e an
d to
just
ify s
olut
ions
. The
rang
e of
num
ber p
atte
rns
are
exte
nded
to in
clud
e pa
ttern
s in
tege
rs, s
quar
e nu
mbe
rs a
nd c
ubic
num
bers
As
lear
ners
bec
ome
used
to d
escr
ibin
g pa
ttern
s in
thei
r ow
n w
ords
, the
ir de
scrip
tions
sho
uld
beco
me
mor
e pr
ecis
e an
d ef
ficie
nt w
ith th
e us
e of
alg
ebra
ic la
ngua
ge to
des
crib
e ge
nera
l rul
es o
f pat
tern
s. It
is u
sefu
l als
o to
intro
duce
the
lang
uage
of ‘
term
in a
seq
uenc
e’ in
ord
er
to d
istin
guis
h th
e te
rm fr
om th
e po
sitio
n of
a te
rm in
a s
eque
nce
36 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Kin
ds o
f num
eric
pat
tern
s Pr
ovid
e a
sequ
ence
of n
umbe
rs; l
earn
ers
have
to id
entif
y a
patte
rn o
r rel
atio
nshi
p be
twee
n co
nsec
utiv
e te
rms
in o
rder
to e
xten
d th
e pa
ttern
. Ex
ampl
es
Giv
e a
rule
to d
escr
ibe
the
rela
tions
hip
betw
een
the
num
bers
in th
e se
quen
ces
belo
w. U
se th
is ru
le to
giv
e th
e ne
xt th
ree
num
bers
in th
e se
quen
ce:
a)
3
; 7 ;
11 ;
15 ;
... ;
...
b)
120
; 115
; 11
0 ; 1
05 ;
... ;
...
c)
2 ; 4
; 8
; 16
; ...
; ...
d)
1 ; 2
; 4
; 7 ;
11 ;
16 ;
... ;
...
Her
e le
arne
rs c
ould
iden
tify
the
cons
tant
diff
eren
ce b
etw
een
cons
ecut
ive
term
s in
ord
er to
ext
end
the
patte
rn. T
hese
pat
tern
s ca
n be
de
scrib
ed in
lear
ners
’ ow
n w
ords
as
a)
‘add
ing
4’ o
r ‘co
untin
g in
4s’
or ‘
add
4 to
the
prev
ious
num
ber i
n th
e pa
ttern
’ b)
‘s
ubtra
ctin
g 5’
or ‘
coun
ting
dow
n in
5s’
, or ‘
subt
ract
5 fr
om th
e pr
evio
us n
umbe
r in
the
patte
rn’.
c)
Her
e le
arne
rs c
ould
iden
tify
the
cons
tant
ratio
bet
wee
n co
nsec
utiv
e te
rms.
Thi
s pa
ttern
can
be
desc
ribed
in le
arne
rs’ o
wn
wor
ds
as ‘m
ultip
ly th
e pr
evio
us n
umbe
r by
2’.
d)
This
pat
tern
has
nei
ther
a c
onst
ant d
iffer
ence
nor
con
stan
t rat
io. T
his
patte
rn c
an b
e de
scrib
ed in
lear
ners
’ ow
n w
ords
as
‘incr
ease
the
diffe
renc
e be
twee
n co
nsec
utiv
e te
rms
by 1
eac
h tim
e’ o
r ‘ad
d 1
mor
e th
an w
as a
dded
to g
et th
e pr
evio
us te
rm’.
Usi
ng th
is ru
le, t
he n
ext 3
term
s w
ill be
22
, 29
, 37.
Pr
ovid
e a
sequ
ence
of n
umbe
rs; l
earn
ers
have
to id
entif
y a
patte
rn o
r rel
atio
nshi
p be
twee
n th
e te
rm a
nd it
s po
sitio
n in
the
sequ
ence
. Th
is e
nabl
es le
arne
rs to
pre
dict
a te
rm in
a s
eque
nce
base
d on
the
posi
tion
of th
at te
rm in
the
sequ
ence
. It i
s us
eful
for l
earn
ers
to
repr
esen
t the
se s
eque
nces
in ta
bles
so
that
they
can
con
side
r the
pos
ition
of t
he te
rm.
Exam
ples
:
a)
Prov
ide
a ru
le to
des
crib
e th
e re
latio
nshi
p be
twee
n th
e nu
mbe
rs in
this
seq
uenc
e:1;
4; 9
; 16;
... ..
. Use
the
rule
to fi
nd th
e 10
th te
rm
in th
is s
eque
nce.
F
irstly
, lea
rner
s ha
ve to
und
erst
and
that
the
‘10t
h te
rm’ r
efer
s to
pos
ition
10
in th
e nu
mbe
r seq
uenc
e. T
hey
have
to fi
nd a
rule
in o
rder
to
dete
rmin
e th
e 10
th te
rm,
rath
er th
an c
ontin
uing
the
sequ
ence
to th
e te
nth
term
. Thi
s se
quen
ce c
an b
e re
pres
ente
d in
the
follo
win
g ta
ble:
37MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Lear
ners
sho
uld
reco
gniz
e th
at e
ach
term
in th
e bo
ttom
row
is o
btai
ned
by s
quar
ing
the
posi
tion
num
ber i
n th
e to
p ro
w. T
hus
the
10th
term
w
ill be
'10
squa
red'
or 1
02 w
hich
is 1
00. U
sing
the
sam
e ru
le, l
earn
ers
can
also
be
aske
d w
hat t
erm
num
ber o
r pos
ition
will
625
be?
If th
e te
rm is
obt
aine
d by
squ
arin
g th
e po
sitio
n nu
mbe
r of t
he te
rm, t
hen
the
posi
tion
num
ber c
an b
e ob
tain
ed b
y fin
ding
the
squa
re ro
ot o
f the
te
rm. H
ence
, 625
will
be th
e 25
th te
rm in
the
sequ
ence
sin
ce √
625
= 2
5.
b)
Pr
ovid
e a
rule
to d
escr
ibe
the
rela
tions
hip
betw
een
the
num
bers
in th
is s
eque
nce:
4 ;
7 ; 1
0 ; 1
3 ; …
... U
se th
e ru
le to
find
the
20th
te
rm in
the
sequ
ence
. If
lear
ners
con
side
r onl
y th
e re
latio
nshi
p be
twee
n co
nsec
utiv
e te
rms,
then
they
can
con
tinue
the
patte
rn (‘
add
3 to
pre
viou
s nu
mbe
r’)
to th
e 20
th te
rm to
find
the
answ
er. H
owev
er, i
f the
y lo
ok fo
r a re
latio
nshi
p or
rule
bet
wee
n th
e te
rm a
nd th
e po
sitio
n of
the
term
, the
y ca
n pr
edic
t the
ans
wer
with
out c
ontin
uing
the
patte
rn. U
sing
num
ber s
ente
nces
can
be
usef
ul to
find
the
rule
: 1s
t ter
m: 4
= 3
(1) +
1
2nd
term
: 7 =
3 (2
) + 1
3r
d te
rm: 1
0 =
3 (3
) + 1
4t
h te
rm: 1
3 =
3 (4
) + 1
Th
e nu
mbe
r in
the
brac
kets
cor
resp
onds
to th
e po
sitio
n of
the
term
. Hen
ce, t
he 2
0th
term
will
be: 3
(20)
+ 1
= 6
1 Th
e ru
le in
lear
ners
’ ow
n w
ords
can
be
writ
ten
as ‘3
x th
e po
sitio
n of
the
term
+ 1
. Th
ese
type
s of
num
eric
pat
tern
s de
velo
p an
und
erst
andi
ng o
f fun
ctio
nal
rela
tions
hips
, in
whi
ch y
ou h
ave
a de
pend
ent v
aria
ble
(pos
ition
of t
he te
rm) a
nd in
depe
nden
t var
iabl
e (th
e te
rm it
self)
, and
whe
re y
ou
have
a u
niqu
e ou
tput
for a
ny g
iven
inpu
t val
ue.
K
inds
of g
eom
etric
pat
tern
s
Geo
met
ric p
atte
rns
are
num
ber p
atte
rns
repr
esen
ted
diag
ram
mat
ical
ly. T
he d
iagr
amm
atic
repr
esen
tatio
n re
veal
s th
e st
ruct
ure
of th
e nu
mbe
r pat
tern
.
H
ence
, rep
rese
ntin
g th
e nu
mbe
r pat
tern
s in
tabl
es, m
akes
it e
asie
r for
lear
ners
to d
escr
ibe
the
gene
ral r
ule
for t
he p
atte
rn.
Exam
ple
Con
side
r thi
s pa
ttern
for b
uild
ing
hexa
gons
with
mat
chst
icks
. How
man
y m
atch
stic
ks w
ill be
use
d to
bui
ld th
e 10
th h
exag
on?
38 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Th
e ru
le fo
r the
pat
tern
is c
onta
ined
in th
e st
ruct
ure
(con
stru
ctio
n) o
f the
suc
cess
ive
hexa
gona
l sha
pes:
(1
) ad
d 1
on m
atch
stic
k pe
r sid
e (2
) th
ere
are
6 si
des,
so
(3)
add
on 6
mat
chst
icks
per
hex
agon
as
you
proc
eed
from
a g
iven
hex
agon
to th
e ne
xt o
ne.
For t
he 2
nd h
exag
on, y
ou h
ave
2 x
6 m
atch
es; f
or th
e 3r
d he
xago
n yo
u ha
ve 3
x 6
mat
ches
. Usi
ng th
is p
atte
rn fo
r bui
ldin
g he
xago
ns, t
he
10th
hex
agon
will
have
10
x 6
mat
ches
. Lea
rner
s ca
n al
so u
se a
tabl
e to
reco
rd th
e nu
mbe
r of m
atch
es u
sed
for e
ach
hexa
gon.
Thi
s w
ay th
ey c
an lo
ok a
t the
nu
mbe
r pat
tern
rela
ted
to th
e nu
mbe
r of m
atch
es u
sed
for e
ach
new
hex
agon
.
D
escr
ibin
g pa
ttern
s It
does
not
mat
ter i
f lea
rner
s ar
e al
read
y fa
milia
r with
a p
artic
ular
pat
tern
. The
ir de
scrip
tions
of t
he s
ame
patte
rn c
an b
e di
ffere
nt w
hen
they
en
coun
ter i
t at d
iffer
ent s
tage
s of
thei
r mat
hem
atic
al d
evel
opm
ent.
Exam
ple
The
rule
for t
he s
eque
nce:
4 ;
7 ; 1
0 ; 1
3 ca
n be
des
crib
ed in
the
follo
win
g w
ays:
a)
ad
d th
ree
to th
e pr
evio
us te
rm
b)
3 tim
es th
e po
sitio
n of
the
term
+ 1
or 3
x th
e po
sitio
n of
the
term
+ 1
c)
3(
n) +
1, w
here
n is
the
posi
tion
of th
e te
rm
d)
3(n)
+ 1
, whe
re n
is a
Nat
ural
num
ber.
In G
rade
8:
Th
e ra
nge
of n
umbe
r pat
tern
s ar
e ex
tend
ed to
incl
ude
patte
rns
with
mul
tiplic
atio
n an
d di
visi
on w
ith in
tege
rs, n
umbe
rs in
ex
pone
ntia
l for
m
A
s le
arne
rs b
ecom
e us
ed to
des
crib
ing
patte
rns
in th
eir o
wn
wor
ds, t
heir
desc
riptio
ns s
houl
d be
com
e m
ore
prec
ise
and
effic
ient
w
ith th
e us
e of
alg
ebra
ic la
ngua
ge to
des
crib
e ge
nera
l rul
es o
f pat
tern
s
39MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
It
is u
sefu
l als
o to
intro
duce
the
lang
uage
of ‘
term
in a
seq
uenc
e’ in
ord
er to
dis
tingu
ish
the
term
from
the
posi
tion
of a
term
in a
se
quen
ce
In
vest
igat
ing
num
ber p
atte
rns
is a
n op
portu
nity
to g
ener
aliz
e –
to g
ive
gene
ral a
lgeb
raic
des
crip
tions
of t
he re
latio
nshi
p be
twee
n te
rms
and
thei
r pos
ition
in a
seq
uenc
e an
d to
just
ify s
olut
ions
. K
inds
of n
umer
ic p
atte
rns
Giv
en a
seq
uenc
e of
num
bers
, lea
rner
s ha
ve to
iden
tify
a pa
ttern
or r
elat
ions
hip
betw
een
cons
ecut
ive
term
s in
ord
er to
ext
end
the
patte
rn.
Exam
ples
Pr
ovid
e a
rule
to d
escr
ibe
the
rela
tions
hip
betw
een
the
num
bers
in th
e se
quen
ces
belo
w. U
se th
is ru
le to
pro
vide
the
next
thre
e nu
mbe
rs
in th
e se
quen
ce:
a)
–3
; –7
; –1
1 ; –
15 ;
... ;
...
Her
e le
arne
rs s
houl
d id
entif
y th
e co
nsta
nt d
iffer
ence
bet
wee
n co
nsec
utiv
e te
rms
in o
rder
to e
xten
d th
e pa
ttern
. Thi
s pa
ttern
can
be
desc
ribed
in le
arne
rs’ o
wn
wor
ds a
s ‘a
ddin
g –4
’ or ‘
coun
ting
in –
4s’ o
r ‘ad
d –4
to th
e pr
evio
us n
umbe
r in
the
patte
rn’.
b)
2
; –4
; 8 ;
–16
; 32
; ...
; ...
Her
e le
arne
rs s
houl
d id
entif
y th
e co
nsta
nt ra
tio b
etw
een
cons
ecut
ive
term
s. T
his
patte
rn c
an b
e de
scrib
ed in
lear
ners
’ ow
n w
ords
as
‘mul
tiply
the
prev
ious
num
ber b
y
–2’.
c)
1
; 2 ;
4 ; 7
; 11
; 16
; ...
; ...
Th
is p
atte
rn h
as n
eith
er a
con
stan
t diff
eren
ce n
or c
onst
ant r
atio
. Thi
s pa
ttern
can
be
desc
ribed
in le
arne
rs’ o
wn
wor
ds a
s ‘in
crea
se th
e di
ffere
nce
betw
een
cons
ecut
ive
term
s by
1 e
ach
time’
or ‘
add
1 m
ore
than
was
add
ed to
get
the
prev
ious
term
’. U
sing
this
rule
, the
ne
xt 3
term
s w
ill be
22
; 29
; 37.
G
iven
a s
eque
nce
of n
umbe
rs, l
earn
ers
have
to id
entif
y a
patte
rn o
r rel
atio
nshi
p be
twee
n th
e te
rm a
nd it
s po
sitio
n in
the
sequ
ence
. Th
is e
nabl
es le
arne
rs to
pre
dict
a te
rm in
a s
eque
nce
base
d on
the
posi
tion
of th
at te
rm in
the
sequ
ence
. It i
s us
eful
for l
earn
ers
to
repr
esen
t the
se s
eque
nces
in ta
bles
so
that
they
can
con
side
r the
pos
ition
of t
he te
rm.
In G
rade
9:
Le
arne
rs c
onso
lidat
e w
ork
invo
lvin
g nu
mer
ic a
nd g
eom
etric
pat
tern
s do
ne in
Gra
de 8
.
Inve
stig
atin
g nu
mbe
r pat
tern
s is
an
oppo
rtuni
ty to
gen
eral
ize
– to
giv
e ge
nera
l alg
ebra
ic d
escr
iptio
ns o
f the
rela
tions
hip
betw
een
term
s an
d th
eir p
ositi
on in
a s
eque
nce
and
to ju
stify
sol
utio
ns.
40 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Kin
ds o
f num
eric
pat
tern
s G
iven
a s
eque
nce
of n
umbe
rs, l
earn
ers
have
to id
entif
y a
patte
rn o
r rel
atio
nshi
p be
twee
n co
nsec
utiv
e te
rms
in o
rder
to e
xten
d th
e pa
ttern
. Ex
ampl
es
Prov
ide
a ru
le to
des
crib
e th
e re
latio
nshi
p be
twee
n th
e nu
mbe
rs in
the
sequ
ence
s be
low
. Use
this
rule
to g
ive
the
next
thre
e nu
mbe
rs in
th
e se
quen
ce:
a) –
1; –
1,5;
–2;
–2,
5 ...
...
Her
e le
arne
rs s
houl
d id
entif
y th
e co
nsta
nt d
iffer
ence
bet
wee
n co
nsec
utiv
e te
rms
in o
rder
to e
xten
d th
e pa
ttern
. Thi
s pa
ttern
can
be
desc
ribed
in le
arne
rs’ o
wn
wor
ds a
s ‘a
ddin
g –0
,5’ o
r ‘co
untin
g in
–0,
5s ’
or ‘a
dd –
0,5
to th
e pr
evio
us n
umbe
r in
the
patte
rn’.
b). 2
; –1;
0,5
; –0,
25 ;
0,12
5 ...
...
Her
e le
arne
rs s
houl
d id
entif
y th
e co
nsta
nt ra
tio b
etw
een
cons
ecut
ive
term
s. T
his
patte
rn c
an b
e de
scrib
ed in
lear
ners
’ ow
n w
ords
as
‘mul
tiply
the
prev
ious
num
ber b
y –
0,5
’. c)
. 1; 0
; –2;
–5;
–9;
–14
... .
Th
is p
atte
rn h
as n
eith
er a
con
stan
t diff
eren
ce n
or c
onst
ant r
atio
. Thi
s pa
ttern
can
be
desc
ribed
in le
arne
rs’ o
wn
wor
ds a
s ‘s
ubtra
ct 1
mor
e th
an w
as s
ubtra
cted
to g
et th
e pr
evio
us te
rm’.
Usi
ng th
is ru
le, t
he n
ext 3
term
s w
ill be
–20
, –27
, –35
G
iven
a s
eque
nce
of n
umbe
rs, l
earn
ers
have
to id
entif
y a
patte
rn o
r rel
atio
nshi
p be
twee
n th
e te
rm a
nd it
s po
sitio
n in
the
sequ
ence
. Th
is e
nabl
es le
arne
rs to
pre
dict
a te
rm in
a s
eque
nce
base
d on
the
posi
tion
of th
at te
rm in
the
sequ
ence
. It i
s us
eful
for l
earn
ers
to
repr
esen
t the
se s
eque
nces
in ta
bles
so
that
they
can
con
side
r the
pos
ition
of t
he te
rm.
Exam
ples
a)
Ref
er to
exa
mpl
es g
iven
abo
ve.
b)
Prov
ide
a ru
le to
des
crib
e th
e re
latio
nshi
p be
twee
n th
e nu
mbe
rs in
this
seq
uenc
e: –
2 ; –
5 ; –
8 ;
–11
; ...
. Use
this
rule
to fi
nd th
e 20
th te
rm in
this
seq
uenc
e.
If le
arne
rs c
onsi
der o
nly
the
rela
tions
hip
betw
een
cons
ecut
ive
term
s, th
en th
ey c
an c
ontin
ue th
e pa
ttern
(‘ad
d -3
to p
revi
ous
num
ber’)
up
to th
e 20
th te
rm to
find
the
answ
er. H
owev
er, i
f the
y lo
ok fo
r a re
latio
nshi
p or
rule
bet
wee
n th
e te
rm a
nd th
e po
sitio
n of
the
term
, th
ey c
an p
redi
ct th
e an
swer
with
out c
ontin
uing
the
patte
rn. U
sing
num
ber s
ente
nces
can
be
usef
ul to
find
the
rule
: 1s
t ter
m: –
2 =
–3(1
) + 1
2n
d te
rm: –
5 =
–3(2
) + 1
3r
d te
rm: –
8 =
–3(3
) + 1
4t
h te
rm: –
11 =
–3(
4) +
1
Th
e nu
mbe
r in
the
brac
kets
cor
resp
onds
to th
e po
sitio
n of
the
term
. Hen
ce, t
he 2
0th
term
will
be: –
3(2
0) +
1 =
– 5
9. T
he ru
le in
lear
ners
’
41MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
own
wor
ds c
an b
e w
ritte
n as
–
3 x
the
posi
tion
of th
e te
rm +
1’ o
r –3n
+ 1
, whe
re n
is th
e po
sitio
n of
the
term
. The
se ty
pes
of n
umer
ic p
atte
rns
deve
lop
an
unde
rsta
ndin
g of
func
tiona
l re
latio
nshi
ps, i
n w
hich
ther
e is
a d
epen
dent
var
iabl
e (p
ositi
on o
f the
term
) and
an
inde
pend
ent v
aria
ble
(the
term
itse
lf), a
nd w
here
you
ha
ve a
uni
que
outp
ut fo
r any
giv
en in
put v
alue
. K
inds
of g
eom
etric
pat
tern
s G
eom
etric
pat
tern
s ar
e nu
mbe
r pat
tern
s re
pres
ente
d di
agra
mm
atic
ally
. The
dia
gram
mat
ic re
pres
enta
tion
reve
als
the
stru
ctur
e of
the
num
ber p
atte
rn. •
Hen
ce, r
epre
sent
ing
the
num
ber p
atte
rns
in ta
bles
mak
es it
eas
ier f
or le
arne
rs to
des
crib
e th
e ge
nera
l rul
e fo
r the
pa
ttern
. Ex
ampl
e C
onsi
der t
his
patte
rn fo
r bui
ldin
g he
xago
ns u
sing
mat
chst
icks
. How
man
y m
atch
stic
ks w
ill be
use
d to
bui
ld th
e 10
th h
exag
on?
Pr
ovid
e an
exp
ress
ion
to d
escr
ibe
the
gene
ral t
erm
for t
his
num
ber s
eque
nce.
The
rule
for t
he p
atte
rn is
con
tain
ed in
the
stru
ctur
e (c
onst
ruct
ion)
of t
he
succ
essi
ve h
exag
onal
sha
pes:
(1
) ad
d on
1 m
atch
stic
k pe
r sid
e (2
) th
ere
are
6 si
des,
so
(3)
add
on 6
mat
chst
icks
per
hex
agon
as
you
proc
eed
from
a g
iven
hex
agon
to th
e ne
xt o
ne.
So, f
or th
e 2n
d he
xago
n, y
ou h
ave
2 x
6 m
atch
es; f
or th
e 3r
d he
xago
n yo
u ha
ve 3
x 6
mat
ches
. Usi
ng th
is p
atte
rn fo
r bui
ldin
g he
xago
ns,
the
10th
hex
agon
will
have
10
x 6
mat
ches
. Lea
rner
s ca
n al
so u
se a
tabl
e to
reco
rd th
e nu
mbe
r of m
atch
es u
sed
for e
ach
hexa
gon.
Thi
s w
ay th
e nu
mbe
r pat
tern
is
rela
ted
to th
e nu
mbe
r of m
atch
es u
sed
for e
ach
new
hex
agon
.
Th
e nt
h te
rm fo
r thi
s se
quen
ce c
an b
e w
ritte
n as
6n
or 6
+ (n
– 1
)6
42 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LTSM
Res
ourc
es
Cha
rts w
ith p
ictu
res
or p
atte
rns
and
num
bers
, too
thpi
cks,
mat
ch s
ticks
, pat
tern
blo
cks.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce)
W
B 1
Activ
ities
(R9a
&b)
WB
2 Ac
tiviti
es (6
5 -7
1; 1
14 –
117
; 141
)
WB
1 Ac
tiviti
es (2
7)
W
B 1
Activ
ities
(2
7; 2
8)
TEXT
BO
OK
R
EFER
ENC
E
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – te
st 1
5 m
arks
max
43MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
5
TER
M 1
Dat
e : f
rom
....
to ..
....
T
ime:
4 ho
urs
G
rade
7
8 9
Topi
c FU
NC
TIO
NS
AND
REL
ATIO
NSH
IPS
CAP
S pa
ges
62
88 -
89
129
Skills
and
Kn
owle
dge
Inpu
t and
out
put v
alue
s D
eter
min
e in
put v
alue
s, o
utpu
t val
ues
or
rule
s fo
r pa
ttern
s an
d re
latio
nshi
ps u
sing
:
flow
dia
gram
s
tabl
es
fo
rmul
ae
Equi
vale
nt fo
rms
Det
erm
ine,
inte
rpre
t and
just
ify e
quiv
alen
ce
of d
iffer
ent d
escr
iptio
ns o
f the
sam
e re
latio
nshi
p or
rule
pre
sent
ed:
ve
rbal
ly
in
flow
dia
gram
s
in ta
bles
by fo
rmul
ae
by
num
ber s
ente
nces
Inpu
t and
out
put v
alue
s D
eter
min
e in
put v
alue
s, o
utpu
t val
ues
or
rule
s fo
r pat
tern
s an
d re
latio
nshi
ps u
sing
:
flow
dia
gram
s
tabl
es
fo
rmul
ae
eq
uatio
ns
Equi
vale
nt fo
rms
Det
erm
ine,
inte
rpre
t and
just
ify
equi
vale
nce
of d
iffer
ent d
escr
iptio
ns
of th
e sa
me
rela
tions
hip
or ru
le
pres
ente
d:
ve
rbal
ly
in
flow
dia
gram
s
in ta
bles
by fo
rmul
ae
--
by e
quat
ions
Inpu
t and
out
put v
alue
s D
eter
min
e in
put v
alue
s, o
utpu
t val
ues
or
rule
s fo
r pat
tern
s an
d re
latio
nshi
ps u
sing
:
flow
dia
gram
s
tabl
es
fo
rmul
ae
eq
uatio
ns
Equi
vale
nt fo
rms
Det
erm
ine,
inte
rpre
t and
just
ify e
quiv
alen
ce
of d
iffer
ent d
escr
iptio
ns o
f the
sam
e re
latio
nshi
p or
rule
pr
esen
ted:
verb
ally
in fl
ow d
iagr
ams
in
tabl
es
by
form
ulae
by e
quat
ions
by g
raph
s on
a C
arte
sian
pla
ne
Su
gges
ted
met
hodo
logy
In
this
pha
se, i
t is
usef
ul to
beg
in to
spe
cify
whe
ther
the
inpu
t val
ues
are
natu
ral n
umbe
rs, o
r int
eger
s or
ratio
nal n
umbe
rs. T
his
build
s le
arne
rs’ a
war
enes
s of
the
dom
ain
of in
put v
alue
s. H
ence
, to
find
outp
ut v
alue
s, le
arne
rs s
houl
d be
giv
en th
e ru
le/fo
rmul
a as
wel
l as
the
dom
ain
of th
e in
put v
alue
s.
In G
rade
7, t
he fo
cus
is o
n fin
ding
out
put v
alue
s fo
r giv
en fo
rmul
ae a
nd in
put v
alue
s. N
ote,
whe
n le
arne
rs fi
nd in
put o
r out
put v
alue
s fo
r gi
ven
rule
s or
form
ulae
, the
y ar
e ac
tual
ly fi
ndin
g th
e nu
mer
ical
val
ue o
f alg
ebra
ic e
xpre
ssio
ns u
sing
sub
stitu
tion.
44 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Exam
ples
U
se th
e fo
rmul
a fo
r the
are
a of
a re
ctan
gle:
A =
l x
b to
cal
cula
te th
e fo
llow
ing:
a)
Th
e ar
ea, i
f the
leng
th is
4 c
m a
nd th
e w
idth
is 2
cm
b)
Th
e le
ngth
, if t
he a
rea
is 2
0 cm
2 an
d th
e w
idth
is 4
cm
c)
Th
e w
idth
, if t
he a
rea
is 2
4 cm
2 an
d th
e le
ngth
is 8
cm
Le
arne
rs c
an w
rite
thes
e as
num
ber s
ente
nces
, and
sol
ve b
y in
spec
tion.
In
Gra
de 8
:
The
focu
s is
on
findi
ng in
put o
r out
put v
alue
s us
ing
give
n eq
uatio
ns
Th
e ru
les
and
num
ber p
atte
rns
for w
hich
lear
ners
hav
e to
find
inpu
t and
out
put v
alue
s ar
e ex
tend
ed to
incl
ude
patte
rns
with
m
ultip
licat
ion
and
divi
sion
of i
nteg
ers
and
num
bers
in e
xpon
entia
l for
m.
Flow
dia
gram
s ar
e re
pres
enta
tions
of f
unct
iona
l rel
atio
nshi
ps. H
ence
, whe
n us
ing
flow
dia
gram
s, th
e co
rresp
onde
nce
betw
een
inpu
t and
ou
tput
val
ues
shou
ld b
e cl
ear i
n its
repr
esen
tatio
nal f
orm
i.e.
the
first
inpu
t pro
duce
s th
e fir
st o
utpu
t; th
e se
cond
inpu
t pro
duce
s th
e se
cond
ou
tput
, etc
. Ex
ampl
es
a)
Use
the
give
n ru
le to
cal
cula
te th
e va
lues
of t
for e
ach
valu
e of
p, w
here
p is
a n
atur
al n
umbe
r. In
this
kin
d of
flow
dia
gram
, lea
rner
s ca
n al
so b
e as
ked
to d
eter
min
e th
e va
lue
of p
for a
giv
en t
valu
e.
b)
Det
erm
ine
the
rule
for c
alcu
latin
g th
e ou
tput
val
ue fo
r eve
ry g
iven
inpu
t val
ue in
the
flow
dia
gram
bel
ow.
45MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
In
flow
dia
gram
s su
ch a
s th
ese,
mor
e th
an o
ne ru
le m
ight
be
poss
ible
to d
escr
ibe
the
rela
tions
hip
betw
een
inpu
t and
out
put v
alue
s. T
he
rule
s ar
e ac
cept
able
if th
ey m
atch
the
give
n in
put v
alue
s to
the
corre
spon
ding
out
put v
alue
s.
c)
If
the
rule
for f
indi
ng y
in th
e ta
ble
belo
w is
y =
–3x
– 1
, cal
cula
te y
for t
he g
iven
x v
alue
s:
d)
Des
crib
e th
e re
latio
nshi
p be
twee
n th
e nu
mbe
rs in
the
top
row
and
bot
tom
row
in th
e ta
ble.
The
n w
rite
dow
n th
e va
lue
of m
and
n
In
tabl
es s
uch
as th
ese,
mor
e th
an o
ne ru
le m
ight
be
poss
ible
to d
escr
ibe
the
rela
tions
hip
betw
een
x an
d y
valu
es. T
he ru
les
are
acce
ptab
le if
they
mat
ch th
e gi
ven
inpu
t val
ues
to th
e co
rresp
ondi
ng o
utpu
t val
ues.
For
exa
mpl
e, th
e ru
le y
= x
–3
desc
ribes
the
rela
tions
hip
betw
een
the
x va
lues
and
giv
en y
val
ues.
To
find
m a
nd n
, lea
rner
s ha
ve to
sub
stitu
te th
e co
rresp
ondi
ng v
alue
s fo
r x o
r y in
th
e ru
le a
nd s
olve
the
equa
tion
by in
spec
tion.
In
Gra
de 9
, lea
rner
s co
nsol
idat
e w
ork
with
inpu
t and
out
put v
alue
s do
ne in
Gra
de 8
. The
y sh
ould
con
tinue
to fi
nd in
put o
r out
put v
alue
s in
flo
w d
iagr
ams,
tabl
es, f
orm
ulae
and
equ
atio
ns. L
earn
ers
shou
ld b
egin
to re
cogn
ize
equi
vale
nt re
pres
enta
tions
of t
he s
ame
rela
tions
hips
sh
own
as a
n eq
uatio
n, a
set
of o
rder
ed p
airs
in a
tabl
e or
on
a gr
aph.
46 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Exam
ples
a)
. If t
he ru
le fo
r fin
ding
in th
e ta
ble
belo
w is
y =
1 2 x +
1, d
eter
min
e th
e va
lues
of y
for t
he g
iven
x v
alue
s:
b)
. Des
crib
e th
e re
latio
nshi
p be
twee
n th
e nu
mbe
rs in
the
top
row
and
thos
e in
the
botto
m ro
w in
the
tabl
e. T
hen
writ
e do
wn
the
valu
e of
m
and
n
In
tabl
es s
uch
as th
ese,
mor
e th
an o
ne ru
le m
ight
be
poss
ible
to d
escr
ibe
the
rela
tions
hip
betw
een
x an
d y
valu
es. T
he ru
les
are
acce
ptab
le if
they
mat
ch th
e gi
ven
inpu
t val
ues
to th
e co
rresp
ondi
ng o
utpu
t val
ues.
For
exa
mpl
e, th
e ru
le y
= 2
x –
3 de
scrib
es th
e re
latio
nshi
p be
twee
n th
e gi
ven
valu
es fo
r x a
nd y
. To
find
m a
nd n
, lea
rner
s ha
ve to
sub
stitu
te th
e co
rresp
ondi
ng v
alue
s fo
r x o
r y in
to th
e ru
le a
nd s
olve
the
equa
tion
by in
spec
tion.
LTSM
Res
ourc
es
Cha
rts, s
quar
e pa
pers
.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
2 Ac
tiviti
es (6
5 - 7
0)
WB
1 Ac
tivity
(R7;
28)
WB
1 Ac
tiviti
es (R
7)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
47MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
E.g.
Info
rmal
as
sess
men
t – te
st 1
5 m
arks
max
48 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
6
TER
M 1
Dat
e : f
rom
....
to ..
....
T
ime:
4 ho
urs
G
rade
7
8 9
Topi
c G
RAP
HS
CAP
S pa
ges
65
114
145
Skills
and
Kno
wle
dge
Inte
rpre
ting
grap
hs
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
spe
cial
focu
s on
the
follo
win
g tre
nds
and
feat
ures
:
linea
r or n
on-li
near
cons
tant
, inc
reas
ing
or
decr
easi
ng
Dra
win
g gr
aphs
D
raw
glo
bal g
raph
s fro
m g
iven
de
scrip
tions
of a
pro
blem
situ
atio
n,
iden
tifyi
ng fe
atur
es li
sted
abo
ve
Inte
rpre
ting
grap
hs
Rev
ise
the
follo
win
g do
ne in
Gra
de 7
:
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
a s
peci
al fo
cus
on th
e fo
llow
ing
trend
s an
d fe
atur
es:
♦♦
linea
r or n
on-li
near
♦♦
cons
tant
, inc
reas
ing
or d
ecre
asin
g
Exte
nd th
e fo
cus
on fe
atur
es o
f gra
phs
to in
clud
e:
♦♦
max
imum
or m
inim
um
♦♦
dis
cret
e or
con
tinuo
us
Dra
win
g gr
aphs
Dra
w g
loba
l gra
phs
from
giv
en
desc
riptio
ns o
f a p
robl
em s
ituat
ion,
id
entif
ying
feat
ures
list
ed a
bove
Use
tabl
es o
f ord
ered
pai
rs to
plo
t po
ints
and
dra
w g
raph
s on
the
Car
tesi
an p
lane
Inte
rpre
ting
grap
hs
Rev
ise
the
follo
win
g do
ne in
Gra
de 8
:
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
a s
peci
al fo
cus
on th
e fo
llow
ing
trend
s an
d fe
atur
es:
♦♦
line
ar o
r non
-line
ar
♦♦
con
stan
t, in
crea
sing
or d
ecre
asin
g
♦♦
max
imum
or m
inim
um
♦♦
dis
cret
e or
con
tinuo
us
Ex
tend
the
abov
e w
ith s
peci
al fo
cus
on
the
follo
win
g fe
atur
es o
f lin
ear g
raph
s:
♦♦
x-in
terc
ept a
nd y
-inte
rcep
t
♦♦
grad
ient
D
raw
ing
grap
hs
Rev
ise
the
follo
win
g do
ne in
Gra
de 8
:
draw
glo
bal g
raph
s fro
m g
iven
de
scrip
tions
of a
pro
blem
situ
atio
n,
iden
tifyi
ng fe
atur
es li
sted
abo
ve.
u
se ta
bles
of o
rder
ed p
airs
to p
lot p
oint
s an
d dr
aw g
raph
s on
the
Car
tesi
an p
lane
Exte
nd th
e ab
ove
with
a s
peci
al fo
cus
on:
♦♦
dra
win
g lin
ear g
raph
s fro
m g
iven
eq
uatio
ns
♦♦
det
erm
ine
equa
tions
from
giv
en
linea
r gra
phs
49MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Sugg
este
d m
etho
dolo
gy
In th
e In
term
edia
te P
hase
lear
ners
enc
ount
ered
gra
phs
in th
e fo
rm o
f dat
a ba
r gra
phs
and
pie
char
ts. T
his
mea
ns th
ey d
o ha
ve
som
e ex
perie
nce
read
ing
and
inte
rpre
ting
grap
hs. H
owev
er, i
n th
e Se
nior
Pha
se, t
hey
are
intro
duce
d to
line
gra
phs
that
sho
w
func
tiona
l rel
atio
nshi
ps d
escr
ibed
in te
rms
of d
epen
dent
and
inde
pend
ent v
aria
bles
. In
Gra
de 7
, the
focu
s is
on
draw
ing,
ana
lysi
ng a
nd in
terp
retin
g gl
obal
gra
phs
only
. Tha
t is,
lear
ners
do
not h
ave
to p
lot p
oint
s to
dr
aw g
raph
s an
d th
ey fo
cus
on th
e fe
atur
es o
f the
glo
bal r
elat
ions
hip
show
n in
the
grap
h.
Exam
ples
of c
onte
xts
for g
loba
l gra
phs
incl
ude:
the
rela
tions
hip
betw
een
time
and
dist
ance
trav
elle
d
the
rela
tions
hip
betw
een
tem
pera
ture
and
tim
e ov
er w
hich
it is
mea
sure
d
the
rela
tions
hip
betw
een
rain
fall
and
time
over
whi
ch it
is m
easu
red,
etc
. In
Gra
de 8
:
New
feat
ures
of g
loba
l gra
phs,
nam
ely,
max
imum
and
min
imum
; dis
cret
e an
d co
ntin
uous
, are
intro
duce
d
Lear
ners
plo
t poi
nts
to d
raw
gra
phs
See
Gra
de 7
abo
ve fo
r exa
mpl
es o
f con
text
s fo
r glo
bal g
raph
s.
Exam
ples
of d
raw
ing
grap
hs b
y pl
ottin
g po
ints
a)
C
ompl
ete
the
tabl
e of
ord
ered
pai
rs b
elow
for t
he e
quat
ion
y =
x +
3
N
ow, p
lot t
he a
bove
co-
ordi
nate
poi
nts
on th
e C
arte
sian
pla
ne. J
oin
poin
ts to
form
a g
raph
.
b)
Com
plet
e th
e ta
ble
of o
rder
ed p
airs
bel
ow fo
r the
equ
atio
n y
= x
2 + 3
N
ow, p
lot t
he a
bove
co-
ordi
nate
poi
nts
on th
e C
arte
sian
pla
ne. J
oin
poin
ts to
form
a g
raph
. In
Gra
de 9
,
x-in
terc
ept,
y-in
terc
ept a
nd g
radi
ent o
f lin
ear g
raph
s ar
e in
trodu
ced
Le
arne
rs d
raw
line
ar g
raph
s fro
m g
iven
equ
atio
ns
Th
ey d
eter
min
e eq
uatio
ns o
f lin
ear g
raph
s Le
arne
rs s
houl
d co
ntin
ue to
ana
lyse
and
inte
rpre
t gra
phs
of p
robl
em s
ituat
ions
.
50 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Inve
stig
atin
g lin
ear g
raph
s
To
ske
tch
linea
r gra
phs
from
giv
en e
quat
ions
, lea
rner
s sh
ould
firs
t dra
w u
p a
tabl
e of
ord
ered
pai
rs, t
hat i
nclu
des
the
inte
rcep
t poi
nts
(x; 0
) and
(0; y
), an
d th
en p
lotti
ng th
e po
ints
.
Lear
ners
sho
uld
inve
stig
ate
grad
ient
s by
com
parin
g 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
𝑣𝑣ℎ𝑣𝑣𝑎𝑎
𝑎𝑎𝑣𝑣ℎ𝑜𝑜
𝑣𝑣𝑣𝑣𝑜𝑜𝑜𝑜𝑎𝑎𝑣𝑣𝑣𝑣𝑣𝑣
𝑣𝑣ℎ𝑣𝑣𝑎𝑎
𝑎𝑎𝑣𝑣 b
etw
een
any
two
poin
ts o
n a
stra
ight
line
gra
ph.
Le
arne
rs s
houl
d al
so in
vest
igat
e th
e re
latio
nshi
p be
twee
n th
e va
lue
of th
e gr
adie
nt a
nd th
e co
effic
ient
of x
in th
e eq
uatio
n of
a
stra
ight
line
gra
ph.
Le
arne
rs s
houl
d co
mpa
re y
-inte
rcep
ts o
f lin
ear g
raph
s to
the
valu
e of
the
cons
tant
in th
e eq
uatio
n of
the
stra
ight
line
gra
ph.
Ex
ampl
es o
f lin
ear g
raph
s
a)
Sket
ch a
nd c
ompa
re th
e gr
aphs
of:
y =
4 an
d x
= 4
b)
Sket
ch a
nd c
ompa
re th
e gr
aphs
of:
y =
x an
d y
= –x
c)
Sk
etch
and
com
pare
the
grap
hs o
f: y
= 2x
; y =
2x
+ 1;
y =
2x
– 1
d)
Sket
ch a
nd c
ompa
re th
e gr
aphs
of:
y =
3x; y
= 4
x; y
= 5
x
e)
Sket
ch th
e gr
aphs
of:
y =
–3x
+ 2;
usi
ng th
e ta
ble
met
hod
f) D
eter
min
e th
e eq
uatio
n of
the
stra
ight
line
pas
sing
thro
ugh
the
follo
win
g po
ints
:
LTSM
Res
ourc
es
Gra
ph p
aper
, cha
rts, r
uler
s, c
olou
r pen
s.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
2 Ac
tiviti
es (8
0 - 8
5)
WB
1 Ac
tiviti
es (R
9)
Gra
de 9
WB
Activ
ities
(R9)
W
B 1
Activ
ities
(R9)
G
rade
9 W
B Ac
tiviti
es (R
9)
51MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – te
st 1
5 m
arks
max
52 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LESS
ON
PLA
NS:
TER
M 2
53MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
1
TER
M 2
Dat
e: fr
om ..
......
to ..
......
Tim
e:
5
hour
s
Gra
de
7 8
9 To
pic
CO
NST
RU
CTI
ON
OF
GEO
MET
RIC
FIG
UR
ES
CAP
S pa
ges
45
95
134
Skills
and
Kn
owle
dge
Mea
surin
g an
gles
Ac
cura
tely
use
a p
rotra
ctor
to m
easu
re
and
clas
sify
ang
les:
< 90
° (a
cute
ang
les)
Rig
ht-a
ngle
s
> 90
° (o
btus
e an
gles
)
Stra
ight
ang
les
>
180°
(ref
lex
angl
es)
Con
stru
ctio
ns
Accu
rate
ly c
onst
ruct
geo
met
ric
figur
es a
ppro
pria
tely
usi
ng c
ompa
ss,
rule
r and
pro
tract
or, i
nclu
ding
:
angl
es, t
o on
e de
gree
of
accu
racy
circ
les
pa
ralle
l lin
es
pe
rpen
dicu
lar l
ines
Con
stru
ctio
ns
Accu
rate
ly c
onst
ruct
geo
met
ric fi
gure
s ap
prop
riate
ly u
sing
a c
ompa
ss, r
uler
and
pr
otra
ctor
, inc
ludi
ng:
Bi
sect
ing
lines
and
ang
les
Pe
rpen
dicu
lar l
ines
at a
giv
en p
oint
or
from
a g
iven
poi
nt
Tr
iang
les
Q
uadr
ilate
rals
Angl
es o
f 30°
, 45°
and
60°
and
thei
r m
ultip
les
with
out u
sing
a p
rotra
ctor
In
vest
igat
ing
prop
ertie
s of
geo
met
ric
figur
es
By c
onst
ruct
ion,
inve
stig
ate
the
angl
es in
a
trian
gle,
focu
sing
on:
the
sum
of t
he in
terio
r ang
les
of tr
iang
les
th
e si
ze o
f ang
les
in a
n eq
uila
tera
l tri
angl
e
the
side
s an
d ba
se a
ngle
s of
an
Con
stru
ctio
ns
Ac
cura
tely
con
stru
ct g
eom
etric
figu
res
appr
opria
tely
usi
ng a
com
pass
, rul
er a
nd
prot
ract
or, i
nclu
ding
bis
ectin
g an
gles
of
a tri
angl
e
C
onst
ruct
ang
les
of 4
5°, 3
0°, 6
0°an
d th
eir m
ultip
les
with
out u
sing
a p
rotra
ctor
In
vest
igat
ing
prop
ertie
s of
geo
met
ric
figur
es
By
con
stru
ctio
n, in
vest
igat
e th
e an
gles
in
a tri
angl
e, fo
cusi
ng o
n th
e re
latio
nshi
p be
twee
n th
e ex
terio
r ang
le o
f a tr
iang
le
and
its in
terio
r ang
les
By
con
stru
ctio
n, e
xplo
re th
e m
inim
um
54 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
isos
cele
s tri
angl
e By
con
stru
ctio
n, in
vest
igat
e si
des
and
angl
es in
qua
drila
tera
ls, f
ocus
ing
on:
th
e su
m o
f the
inte
rior a
ngle
s of
qu
adril
ater
als
th
e si
des
and
oppo
site
ang
les
of
para
llelo
gram
s
cond
ition
s fo
r tw
o tri
angl
es to
be
cong
ruen
t
By c
onst
ruct
ion,
inve
stig
ate
side
s,
angl
es a
nd d
iago
nals
in q
uadr
ilate
rals
, fo
cusi
ng o
n th
e di
agon
als
of re
ctan
gles
, sq
uare
s, p
aral
lelo
gram
s, rh
ombi
and
ki
tes
By
con
stru
ctio
n ex
plor
e th
e su
m o
f the
in
terio
r ang
les
of p
olyg
ons
Sugg
este
d m
etho
dolo
gy
In G
rade
7, l
earn
ers
M
easu
re a
ngle
s w
ith a
pro
tract
or
D
o ge
omet
ric c
onst
ruct
ions
usi
ng a
com
pass
, rul
er a
nd p
rotra
ctor
M
easu
ring
angl
es
Le
arne
rs h
ave
to b
e sh
own
how
to p
lace
the
prot
ract
or o
n th
e ar
m o
f the
ang
le to
be
mea
sure
d.
Le
arne
rs a
lso
have
to le
arn
how
to re
ad th
e si
ze o
f ang
les
on a
pro
tract
or.
Con
stru
ctio
ns
C
onst
ruct
ions
pro
vide
a u
sefu
l con
text
to e
xplo
re o
r con
solid
ate
know
ledg
e of
ang
les
and
shap
es.
Le
arne
rs h
ave
to b
e sh
own
how
to u
se a
com
pass
to d
raw
circ
les,
alth
ough
they
mig
ht h
ave
done
this
in G
rade
6.
Le
arne
rs s
houl
d be
aw
are
that
the
cent
re o
f the
circ
le is
at t
he fi
xed
poin
t of t
he c
ompa
ss a
nd th
e ra
dius
of t
he c
ircle
is
depe
nden
t on
how
wid
e th
e co
mpa
ss is
ope
ned
up.
M
ake
sure
lear
ners
und
erst
and
that
arc
s ar
e pa
rts o
f the
circ
les
of a
par
ticul
ar ra
dius
.
Initi
ally
, lea
rner
s ha
ve to
be
give
n ca
refu
l ins
truct
ions
abo
ut h
ow to
do
the
cons
truct
ions
of t
he v
ario
us s
hape
s
O
nce
they
are
com
forta
ble
with
the
appa
ratu
s an
d ca
n do
the
cons
truct
ions
, the
y ca
n pr
actis
e by
dra
win
g pa
ttern
s, fo
r ex
ampl
e of
circ
les
or p
aral
lel l
ines
. In
Gra
de 8
, all
the
cons
truct
ions
are
new
. In
Gra
de 7
, lea
rner
s on
ly c
onst
ruct
ed a
ngle
s, p
erpe
ndic
ular
and
par
alle
l lin
es. G
rade
8
lear
ners
als
o us
e co
nstru
ctio
ns to
exp
lore
pro
perti
es o
f tria
ngle
s an
d qu
adril
ater
als
Con
stru
ctio
ns
C
onst
ruct
ions
pro
vide
a u
sefu
l con
text
to e
xplo
re o
r con
solid
ate
know
ledg
e of
ang
les
and
shap
es
M
ake
sure
lear
ners
are
com
pete
nt a
nd c
omfo
rtabl
e w
ith th
e us
e of
a c
ompa
ss a
nd k
now
how
to m
easu
re a
nd re
ad a
ngle
55MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
size
s on
a p
rotra
ctor
Rev
ise
the
cons
truct
ions
of a
ngle
s if
nece
ssar
y be
fore
pro
ceed
ing
with
the
new
con
stru
ctio
ns
St
art w
ith th
e co
nstru
ctio
ns o
f lin
es, s
o th
at le
arne
rs c
an fi
rst e
xplo
re a
ngle
rela
tions
hips
on
stra
ight
line
s.
W
hen
cons
truct
ing
trian
gles
lear
ners
sho
uld
draw
on
know
n pr
oper
ties
and
cons
truct
ion
of c
ircle
s.
C
onst
ruct
ion
of s
peci
al a
ngle
s w
ithou
t pro
tract
ors
are
done
by:
o
bise
ctin
g a
right
ang
le to
get
45°
o
draw
ing
an e
quila
tera
l tria
ngle
to g
et 6
0°
o bi
sect
ing
the
angl
es o
f an
equi
late
ral t
riang
le to
get
30°
In
Gra
de 9
, lea
rner
s:
Bi
sect
ang
les
in a
tria
ngle
Con
stru
ct 3
0° w
ithou
t a p
rotra
ctor
Inve
stig
atio
n of
new
pro
perti
es o
f tria
ngle
s, q
uadr
ilate
rals
and
pol
ygon
s C
onst
ruct
ions
Sa
me
as fo
r Gra
de 8
. LT
SM
Res
ourc
es
Mat
hem
atic
al In
stru
men
ts s
et.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es
( 20
- 25
) W
B 1
Activ
ities
(45
- 51)
WB
1 Ac
tiviti
es (
39 -
47)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
56 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
2
TER
M 2
Dat
e : f
rom
....
to ..
....
T
ime:
10 h
ours
G
rade
7
8 9
Topi
c G
EOM
ETR
Y O
F 2D
SH
APES
C
APS
page
s 46
- 47
96
- 98
13
5 - 1
36
Skills
and
Kn
owle
dge
Cla
ssify
ing
2D s
hape
s D
escr
ibe,
sor
t, na
me
and
com
pare
tria
ngle
s ac
cord
ing
to th
eir s
ides
and
ang
les,
fo
cusi
ng o
n:
eq
uila
tera
l tria
ngle
s
isos
cele
s tri
angl
es
rig
ht-a
ngle
d tri
angl
es
Des
crib
e, s
ort,
nam
e an
d co
mpa
re
quad
rilat
eral
s in
term
s of
:
leng
th o
f sid
es
pa
ralle
l and
per
pend
icul
ar s
ides
size
of a
ngle
s (ri
ght-a
ngle
s or
not
) D
escr
ibe
and
nam
e pa
rts o
f a c
ircle
Si
mila
r and
con
grue
nt 2
D s
hape
s R
ecog
nize
and
des
crib
e si
mila
r and
co
ngru
ent f
igur
es b
y co
mpa
ring:
shap
e
size
So
lvin
g pr
oble
ms
Solv
e si
mpl
e ge
omet
ric p
robl
ems
invo
lvin
g un
know
n si
des
and
angl
es in
tria
ngle
s an
d
Cla
ssify
ing
2D s
hape
s Id
entif
y an
d w
rite
clea
r def
initi
ons
of
trian
gles
in te
rms
of th
eir s
ides
and
ang
les,
di
stin
guis
hing
bet
wee
n:
eq
uila
tera
l tria
ngle
s
isos
cele
s tri
angl
es
rig
ht-a
ngle
d tri
angl
es
Iden
tify
and
writ
e cl
ear d
efin
ition
s of
qu
adril
ater
als
in te
rms
of th
eir s
ides
and
an
gles
, dis
tingu
ishi
ng b
etw
een:
Para
llelo
gram
Rec
tang
le
S
quar
e
Rho
mbu
s
Tra
pezi
um
K
ite
Sim
ilar a
nd c
ongr
uent
2-D
sha
pes
Id
entif
y an
d de
scrib
e th
e pr
oper
ties
of
cong
ruen
t sha
pes
Id
entif
y an
d de
scrib
e th
e pr
oper
ties
of
sim
ilar s
hape
s So
lvin
g pr
oble
ms
Solv
e ge
omet
ric p
robl
ems
invo
lvin
g
Cla
ssify
ing
2D s
hape
s R
evis
e pr
oper
ties
and
defin
ition
s of
tria
ngle
s in
term
s of
thei
r sid
es a
nd a
ngle
s,
dist
ingu
ishi
ng b
etw
een:
equi
late
ral t
riang
les
is
osce
les
trian
gles
right
-ang
led
trian
gles
R
evis
e an
d w
rite
clea
r def
initi
ons
of
quad
rilat
eral
s in
term
s of
thei
r sid
es, a
ngle
s an
d di
agon
als,
dis
tingu
ishi
ng b
etw
een:
Para
llelo
gram
Rec
tang
le
S
quar
e
Rho
mbu
s
Tra
pezi
um
K
ite
Sim
ilar a
nd c
ongr
uent
tria
ngle
s
Thro
ugh
inve
stig
atio
n, e
stab
lish
the
min
imum
con
ditio
ns fo
r con
grue
nt
trian
gles
Thro
ugh
inve
stig
atio
n, e
stab
lish
the
min
imum
con
ditio
ns fo
r sim
ilar t
riang
les
Solv
ing
prob
lem
s So
lve
geom
etric
pro
blem
s in
volv
ing
57MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
quad
rilat
eral
s, u
sing
kno
wn
prop
ertie
s.
unkn
own
side
s an
d an
gles
in tr
iang
les
and
quad
rilat
eral
s, u
sing
kno
wn
prop
ertie
s an
d de
finiti
ons.
unkn
own
side
s an
d an
gles
in tr
iang
les
and
quad
rilat
eral
s, u
sing
kno
wn
prop
ertie
s of
tri
angl
es a
nd q
uadr
ilate
rals
, as
wel
l as
prop
ertie
s of
con
grue
nt a
nd s
imila
r tria
ngle
s
Sugg
este
d m
etho
dolo
gy
In G
rade
7, l
earn
ers
are:
•
Dis
tingu
ishi
ng a
nd n
amin
g tri
angl
es in
term
s of
thei
r sid
es a
nd a
ngle
s •
Dis
tingu
ishi
ng q
uadr
ilate
rals
in te
rms
of p
aral
lel a
nd p
erpe
ndic
ular
sid
es
• D
istin
guis
hing
sim
ilar a
nd c
ongr
uent
figu
res
• U
sing
kno
wn
prop
ertie
s of
sha
pes
to s
olve
geo
met
ric p
robl
ems
Tria
ngle
s Le
arne
rs s
houl
d be
abl
e to
dis
tingu
ish
betw
een
an e
quila
tera
l tria
ngle
(all
the
side
s ar
e eq
ual),
an
isos
cele
s tri
angl
e (a
t lea
st tw
o eq
ual
side
s) a
nd a
righ
t-ang
led
trian
gle
(one
righ
t-ang
le).
Qua
drila
tera
ls
Lear
ners
sho
uld
be a
ble
to s
ort a
nd g
roup
qua
drila
tera
ls in
the
follo
win
g w
ays:
•
all s
ides
equ
al (s
quar
e an
d rh
ombu
s)
• op
posi
te s
ides
equ
al (r
ecta
ngle
, par
alle
logr
am, s
quar
e, rh
ombu
s)
• at
leas
t one
pai
r of a
djac
ent s
ides
equ
al (s
quar
e, rh
ombu
s, k
ite)
• al
l fou
r ang
les
right
ang
les
(squ
are,
rect
angl
e)
• pe
rpen
dicu
lar s
ides
(squ
are,
rect
angl
e)
• tw
o pa
irs o
f opp
osite
sid
es p
aral
lel (
rect
angl
e, s
quar
e, p
aral
lelo
gram
) •
only
one
pai
r of o
ppos
ite s
ides
par
alle
l (tra
pezi
um)
Circ
les
Parts
of a
circ
le le
arne
rs s
houl
d kn
ow:
• ra
dius
•
circ
umfe
renc
e •
diam
eter
•
chor
d •
segm
ents
58 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
• se
ctor
s Si
mila
rity
and
cong
ruen
cy
Sim
ilarit
y an
d co
ngru
ency
can
be
expl
ored
with
any
2D
figu
res.
•
Lear
ners
sho
uld
reco
gniz
e th
at tw
o or
mor
e fig
ures
are
con
grue
nt if
they
are
equ
al in
all
resp
ects
i.e.
ang
les
and
side
s ar
e eq
ual.
• Le
arne
rs s
houl
d re
cogn
ize
that
two
or m
ore
figur
es a
re s
imila
r if t
hey
have
the
sam
e sh
ape,
but
diff
er in
siz
e i.e
. ang
les
are
the
sam
e, b
ut s
ides
are
pro
porti
onal
ly lo
nger
or s
horte
r. Si
mila
r fig
ures
are
furth
er e
xplo
red
whe
n do
ing
enla
rgem
ents
and
redu
ctio
ns.
Ref
er to
“Cla
rific
atio
n N
otes
” und
er 3
.4 T
rans
form
atio
n G
eom
etry
. So
lvin
g pr
oble
ms
At th
is s
tage
lear
ners
can
sol
ve s
impl
e ge
omet
ric p
robl
ems
to fi
nd u
nkno
wn
side
s in
equ
ilate
ral a
nd is
osce
les
trian
gles
, and
unk
now
n si
des
and
angl
es in
qua
drila
tera
ls. L
earn
ers
shou
ld g
ive
reas
ons
for t
heir
solu
tions
. Ex
ampl
es
a)
If ΔA
BC is
an
equi
late
ral t
riang
le, a
nd s
ide
AB is
3 c
m, w
hat i
s th
e le
ngth
of B
C?
Her
e le
arne
rs s
houl
d an
swer
: BC
= 3
cm
, be
caus
e th
e si
des
of a
n eq
uila
tera
l tria
ngle
are
equ
al.
b)
If AB
CD
is a
kite
, and
AB
= 2,
5 cm
and
BC
= 4
,5 c
m, w
hat i
s th
e le
ngth
of A
D a
nd D
C?
Lear
ners
sho
uld
use
the
prop
erty
for k
ites,
th
at a
djac
ent p
airs
of s
ides
are
equ
al, t
o fin
d th
e un
know
n si
des.
In G
rade
8:
• N
ew p
rope
rties
in te
rms
of a
ngle
s of
tria
ngle
s •
Lear
ners
writ
e cl
ear d
efin
ition
s of
the
prop
ertie
s of
tria
ngle
s an
d qu
adril
ater
als
• Th
ey u
se d
efin
ition
s to
sol
ve g
eom
etric
pro
blem
s •
They
inve
stig
ate
cond
ition
s fo
r 2D
sha
pes
to b
e co
ngru
ent o
r sim
ilar
Tria
ngle
s C
onst
ruct
ions
ser
ve a
s a
usef
ul c
onte
xt fo
r exp
lorin
g pr
oper
ties
of tr
iang
les
(See
not
es o
n C
onst
ruct
ions
). Pr
oper
ties
of tr
iang
les
lear
ners
sh
ould
kno
w in
clud
e:
• th
e su
m o
f the
inte
rior a
ngle
s of
tria
ngle
s =
180°
•
an e
quila
tera
l tria
ngle
has
all
side
s eq
ual a
nd a
ll in
terio
r ang
les
= 60
° •
an is
osce
les
trian
gle
has
at le
ast t
wo
equa
l sid
es a
nd it
s ba
se a
ngle
s ar
e eq
ual
• a
right
-ang
led
trian
gle
has
one
angl
e th
at is
a ri
ght-a
ngle
•
the
side
opp
osite
the
right
-ang
le in
a ri
ght-a
ngle
d tri
angl
e, is
cal
led
the
hypo
tenu
se
• in
a ri
ght-a
ngle
d tri
angl
e, th
e sq
uare
of t
he h
ypot
enus
e is
equ
al to
the
sum
of t
he s
quar
es o
f the
oth
er tw
o si
des
(The
orem
of
Pyth
agor
as).
59MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Qua
drila
tera
ls
Con
stru
ctio
ns s
erve
as
a us
eful
con
text
for e
xplo
ring
prop
ertie
s of
tria
ngle
s (S
ee n
otes
on
Con
stru
ctio
ns).
The
clas
sific
atio
n of
qu
adril
ater
als
shou
ld in
clud
e th
e re
cogn
ition
that
: •
rect
angl
es a
nd rh
ombi
are
spe
cial
kin
ds o
f par
alle
logr
ams
• a
squa
re is
a s
peci
al k
ind
of re
ctan
gle
and
rhom
bus.
Pr
oper
ties
of q
uadr
ilate
rals
lear
ners
sho
uld
know
incl
ude:
•
the
sum
of t
he in
terio
r ang
les
of q
uadr
ilate
rals
= 3
60°
• th
e op
posi
te s
ides
of p
aral
lelo
gram
s ar
e pa
ralle
l and
equ
al
• th
e op
posi
te a
ngle
s of
par
alle
logr
ams
are
equa
l •
the
oppo
site
ang
les
of a
rhom
bus
are
equa
l •
the
oppo
site
sid
es o
f a rh
ombu
s ar
e pa
ralle
l and
equ
al
• th
e si
ze o
f eac
h an
gle
of re
ctan
gles
and
squ
ares
is 9
0°
• a
trape
zium
has
one
pai
r of o
ppos
ite s
ides
par
alle
l •
a k
ite h
as tw
o pa
irs o
f adj
acen
t sid
es e
qual
Si
mila
rity
and
cong
ruen
cy
Ref
er to
“Cla
rific
atio
n N
otes
” und
er 3
.3 T
rans
form
atio
n G
eom
etry
and
not
es fo
r Gra
de 7
abo
ve..
Exam
ples
: •
Com
parin
g sq
uare
s to
oth
er re
ctan
gles
, lea
rner
s ca
n as
certa
in th
at h
avin
g co
rresp
ondi
ng a
ngle
s eq
ual d
oes
not n
eces
saril
y im
ply
that
the
side
s w
ill be
of p
ropo
rtion
al le
ngth
. Hen
ce h
avin
g eq
ual a
ngle
s al
one,
is n
ot a
suf
ficie
nt c
ondi
tion
for f
igur
es to
be
sim
ilar.
• C
ompa
ring
rhom
bii w
ith s
ides
pro
porti
onal
, lea
rner
s ca
n as
certa
in th
at h
avin
g si
des
prop
ortio
nal d
oes
not n
eces
saril
y im
ply
that
th
e co
rresp
ondi
ng a
ngle
s w
ill be
equ
al. S
o on
ly h
avin
g si
des
of p
ropo
rtion
al le
ngth
is n
ot a
suf
ficie
nt c
ondi
tion
for s
imila
rity
Solv
ing
prob
lem
s •
Lear
ners
can
sol
ve g
eom
etric
pro
blem
s to
find
unk
now
n si
des
and
angl
es in
tria
ngle
s an
d qu
adril
ater
als,
usi
ng k
now
n de
finiti
ons
as w
ell a
s an
gle
rela
tions
hips
on
stra
ight
line
s (s
ee n
otes
on
Geo
met
ry o
f Stra
ight
line
s.)
• Fo
r rig
ht-a
ngle
d tri
angl
es, l
earn
ers
can
also
use
the
Theo
rem
of P
ytha
gora
s to
find
unk
now
n le
ngth
s.
• Le
arne
rs s
houl
d gi
ve re
ason
s an
d ju
stify
thei
r sol
utio
ns fo
r eve
ry w
ritte
n st
atem
ent.
Not
e th
at s
olvi
ng g
eom
etric
pro
blem
s is
an
oppo
rtuni
ty to
pra
ctis
e so
lvin
g eq
uatio
ns.
Exam
ple:
60 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Â
1, Â
2, Â
3 an
d ar
e th
ree
angl
es o
n a
stra
ight
line
. Â2
= 75
°, Â
3 =
55°.
Wha
t is
the
size
of Â
1?
Lear
ners
can
find
Â1
by s
olvi
ng th
e fo
llow
ing
equa
tion:
Â
1 +
75°
+ 55
° =
180°
(bec
ause
the
sum
of a
ngle
s on
a s
traig
ht li
ne =
180
°)
Â1
= 18
0° –
130
°(ad
d –5
5° a
nd –
75° t
o bo
th s
ides
of t
he e
quat
ion)
Â
1 =
50°
In G
rade
9, l
earn
ers
revi
se a
nd w
rite
clea
r des
crip
tions
of a
ngle
rela
tions
hips
on
stra
ight
line
s. A
ngle
rela
tions
hips
lear
ners
sho
uld
know
incl
ude:
•
the
sum
of t
he a
ngle
s on
a s
traig
ht li
ne is
180
° •
If lin
es a
re p
erpe
ndic
ular
line
s, th
en a
djac
ent s
uppl
emen
tary
ang
les
are
each
equ
al to
90°
•
If lin
es in
ters
ect,
then
ver
tical
ly o
ppos
ite a
ngle
s ar
e eq
ual.
• if
para
llel l
ines
are
cut
by
a tra
nsve
rsal
, the
n co
rresp
ondi
ng a
ngle
s ar
e eq
ual
• if
para
llel l
ines
cut
by
a tra
nsve
rsal
, the
n al
tern
ate
angl
es a
re e
qual
•
if pa
ralle
l lin
es c
ut b
y a
trans
vers
al, t
hen
co-in
terio
r ang
les
are
supp
lem
enta
ry
The
abov
e an
gles
hav
e to
be
iden
tifie
d an
d na
med
by
lear
ners
. So
lvin
g pr
oble
ms
See
note
s in
the
abov
e se
ctio
ns
Con
grue
nt tr
iang
les
Con
stru
ctio
ns a
re a
use
ful c
onte
xt fo
r est
ablis
hing
the
min
imum
con
ditio
ns fo
r tw
o tri
angl
es to
be
cong
ruen
t. Se
e no
tes
on C
onst
ruct
ions
ab
ove.
Con
ditio
ns fo
r tw
o tri
angl
es to
be
cong
ruen
t: •
thre
e co
rresp
ondi
ng s
ides
are
equ
al (S
,S,S
) •
two
corre
spon
ding
sid
es a
nd th
e in
clud
ed a
ngle
are
equ
al (S
,A,S
) •
two
corre
spon
ding
ang
les
and
a co
rresp
ondi
ng s
ide
are
equa
l (A,
A,S)
•
right
-ang
le, h
ypot
enus
e an
d on
e ot
her c
orre
spon
ding
sid
e ar
e eq
ual (
R,H
,S)
61MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Sim
ilar t
riang
les
Con
stru
ctio
ns a
re a
use
ful c
onte
xt fo
r est
ablis
hing
the
min
imum
con
ditio
ns fo
r tw
o tri
angl
es to
be
sim
ilar.
See
note
s on
Con
stru
ctio
ns
abov
e.
Solv
ing
prob
lem
s Se
e no
tes
and
exam
ples
abo
ve.
LTSM
Res
ourc
es
2D s
hape
s an
d ch
arts
.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es (2
6 - 2
9)
WB
1 Ac
tiviti
es (R
11a&
b; R
13)
WB
2 Ac
tiviti
es (5
2 - 6
0)
WB
1 A
ctiv
ities
(48
- 52
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
62 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
3
TER
M 2
Dat
e: fr
om ..
.. to
.....
.
Tim
e:
9
hour
s
Gra
de
7 8
9 To
pic
GEO
MET
RY
OF
STR
AIG
HT
LIN
ES
CAP
S pa
ges
46
98
137
Skills
and
Kn
owle
dge
Def
ine:
•
Line
seg
men
t •
Ray
•
Stra
ight
line
•
Para
llel l
ines
•
Perp
endi
cula
r lin
es
Angl
e re
latio
nshi
ps
Rec
ogni
se a
nd d
escr
ibe
pairs
of a
ngle
s fo
rmed
by:
•
perp
endi
cula
r lin
es
• in
ters
ectin
g lin
es
• pa
ralle
l lin
es c
ut b
y a
trans
vers
al
Solv
ing
prob
lem
s So
lve
geom
etric
pro
blem
s us
ing
the
rela
tions
hips
bet
wee
n pa
irs o
f ang
les
desc
ribed
abo
ve
Angl
e re
latio
nshi
ps
Rev
ise
and
writ
e cl
ear d
escr
iptio
ns o
f the
re
latio
nshi
p be
twee
n an
gles
form
ed b
y:
• pe
rpen
dicu
lar l
ines
•
inte
rsec
ting
lines
•
para
llel l
ines
cut
by
a tra
nsve
rsal
So
lvin
g pr
oble
ms
Solv
e ge
omet
ric p
robl
ems
usin
g th
e re
latio
nshi
ps b
etw
een
pairs
of a
ngle
s de
scrib
ed a
bove
.
63MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Sugg
este
d m
etho
dolo
gy
Gra
de 7
lear
ners
sho
uld
know
that
: •
Line
seg
men
t is
a se
t of p
oint
s w
ith a
def
inite
sta
rting
-poi
nt a
nd a
n en
d-po
int.
• R
ay is
a s
et o
f poi
nts
with
a d
efin
ite s
tarti
ng-p
oint
and
no
defin
ite e
nd-p
oint
. •
Line
is a
set
of p
oint
s w
ith n
o de
finite
sta
rting
-poi
nt a
nd e
nd-p
oint
. •
If tw
o lin
es o
n th
e sa
me
plan
e ar
e a
cons
tant
dis
tanc
e ap
art,
then
the
lines
are
par
alle
l. E
xam
ple:
Thi
s is
writ
ten
as E
F ║
GH
•
If ve
rtica
l lin
e AO
mee
ts o
r int
erse
cts
with
hor
izon
tal l
ine
BC a
t rig
ht a
ngle
, the
n AO
is p
erpe
ndic
ular
to B
C.
Exam
ple:
T
his
is w
ritte
n as
AO
┴ B
C
In G
rade
7 le
arne
rs o
nly
defin
ed li
ne s
egm
ent,
ray,
stra
ight
line
, par
alle
l lin
es a
nd p
erpe
ndic
ular
line
s. In
Gra
de 8
, ang
le re
latio
nshi
ps
lear
ners
sho
uld
know
incl
ude:
•
the
sum
of t
he a
ngle
s on
a s
traig
ht li
ne is
180
° •
If lin
es a
re p
erpe
ndic
ular
, the
n ad
jace
nt s
uppl
emen
tary
ang
les
are
each
equ
al to
90°
. •
If lin
es in
ters
ect,
then
ver
tical
ly o
ppos
ite a
ngle
s ar
e eq
ual.
• if
para
llel l
ines
are
cut
by
a tra
nsve
rsal
, the
n co
rresp
ondi
ng a
ngle
s ar
e eq
ual
• if
para
llel l
ines
cut
by
a tra
nsve
rsal
, the
n al
tern
ate
angl
es a
re e
qual
•
if pa
ralle
l lin
es c
ut b
y a
trans
vers
al, t
hen
co-in
terio
r ang
les
are
supp
lem
enta
ry
The
abov
e an
gles
hav
e to
be
iden
tifie
d an
d na
med
by
lear
ner
Solv
ing
prob
lem
s Le
arne
rs s
houl
d so
lve
geom
etric
pro
blem
s to
find
unk
now
n an
gles
usi
ng th
e an
gle
rela
tions
hips
abo
ve, a
s w
ell a
s ot
her k
now
n pr
oper
ties
of tr
iang
les
and
quad
rilat
eral
s. L
earn
ers
shou
ld g
ive
reas
ons
and
just
ify th
eir s
olut
ions
for e
very
writ
ten
stat
emen
t.
64 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Not
e th
at s
olvi
ng g
eom
etric
pro
blem
s is
an
oppo
rtuni
ty to
pra
ctis
e so
lvin
g eq
uatio
ns.
Exa
mpl
e:
Â, B
and
Ĉ a
re th
ree
angl
es o
n a
stra
ight
line
. Â =
55°
, B =
75°
. Wha
t is
the
size
of
Ĉ ?
Le
arne
rs c
an fi
nd Ĉ
by
solv
ing
the
follo
win
g eq
uatio
n:
55°
+ 75
° +
Ĉ =
180
° (be
caus
e th
e su
m o
f ang
les
on a
stra
ight
line
= 1
80°)
Ĉ
= 1
80° –
130
° (ad
d –5
5° a
nd –
75° t
o bo
th s
ides
of t
he e
quat
ion)
Ĉ
= 5
0°
Gra
de 9
lear
ners
revi
se a
nd w
rite
clea
r des
crip
tions
of a
ngle
rela
tions
hips
on
stra
ight
line
s. A
ngle
rela
tions
hips
lear
ners
sho
uld
know
ar
e th
e sa
me
as fo
r Gra
de 8
. So
lvin
g pr
oble
ms
See
Gra
de 7
and
8 n
otes
. Ex
ampl
e:
B1
and
B2
are
two
angl
es o
n a
stra
ight
line
. B1
= 35
°. W
hat i
s th
e si
ze o
f B2
?
Le
arne
rs c
an fi
nd B
2 by
sol
ving
the
follo
win
g eq
uatio
n:
35°
+ B
2 =
180°
(bec
ause
the
sum
of a
ngle
s on
a s
traig
ht li
ne 1
80°)
B
2 =
180°
– 3
5° (a
dd –
35°
to b
oth
side
s of
the
equa
tion)
B
2 =
145°
LTSM
Res
ourc
es
Prot
ract
or, c
harts
, rul
ers,
col
our p
ens.
G
rade
7
Gra
de 8
G
rade
9
65MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es (2
4)
W
B 1
Activ
ities
(6
1 - 6
3)
WB
1 Ac
tiviti
es
(53
- 57
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
66 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTIG
RAD
E LE
SSO
N P
LAN
4
TER
M 2
Dat
e : f
rom
....
to ..
....
Tim
e:
9 ho
urs
Gra
de
7 8
9 To
pic
ALG
EBR
AIC
EXP
RES
SIO
NS
CAP
S pa
ges
63 &
69
90 &
(92
– 93
) (1
30 –
131
) & (1
42 –
144
) Sk
ills a
nd
Know
ledg
e Al
gebr
aic
lang
uage
•
Rec
ogni
ze a
nd in
terp
ret r
ules
or
rela
tions
hips
rep
rese
nted
in s
ymbo
lic
form
•
Iden
tify
varia
bles
and
con
stan
ts in
giv
en
form
ulae
and
equ
atio
ns
Alge
brai
c la
ngua
ge
• R
evis
e th
e fo
llow
ing
done
in G
rade
7:
o re
cogn
ize
and
inte
rpre
t rul
es o
r re
latio
nshi
ps re
pres
ente
d in
sym
bolic
fo
rm
o id
entif
y va
riabl
es a
nd c
onst
ants
in
give
n fo
rmul
ae a
nd e
quat
ions
•
Rec
ogni
ze a
nd id
entif
y co
nven
tions
for
writ
ing
alge
brai
c ex
pres
sion
s •
Iden
tify
and
clas
sify
like
and
unl
ike
term
s in
alg
ebra
ic e
xpre
ssio
ns
• R
ecog
nize
and
iden
tify
coef
ficie
nts
and
expo
nent
s in
alg
ebra
ic e
xpre
ssio
ns
Expa
nd a
nd s
impl
ify a
lgeb
raic
ex
pres
sion
s U
se c
omm
utat
ive,
ass
ocia
tive
and
dist
ribut
ive
law
s fo
r rat
iona
l num
bers
and
la
ws
of e
xpon
ents
to:
Ad
d an
d su
btra
ct li
ke te
rms
in a
lgeb
raic
ex
pres
sion
s.
M
ultip
ly in
tege
rs a
nd m
onom
ials
by:
♦♦
mon
omia
ls
♦♦
bino
mia
ls
♦♦
trin
omia
ls
D
eter
min
e th
e sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of s
ingl
e al
gebr
aic
Alge
brai
c la
ngua
ge
• R
evis
e th
e fo
llow
ing
done
in G
rade
8:
o R
ecog
nize
and
iden
tify
conv
entio
ns
for w
ritin
g al
gebr
aic
expr
essi
ons
o Id
entif
y an
d cl
assi
fy li
ke a
nd u
nlik
e te
rms
in a
lgeb
raic
exp
ress
ions
o
Rec
ogni
ze a
nd id
entif
y co
effic
ient
s an
d ex
pone
nts
in a
lgeb
raic
ex
pres
sion
s
Rec
ogni
ze a
nd d
iffer
entia
te b
etw
een
mon
omia
ls, b
inom
ials
and
trin
omia
ls
Expa
nd a
nd s
impl
ify a
lgeb
raic
ex
pres
sion
s R
evis
e th
e fo
llow
ing
done
in G
rade
8, u
sing
th
e co
mm
utat
ive,
ass
ocia
tive
and
dist
ribut
ive
law
s fo
r rat
iona
l num
bers
and
la
ws
of e
xpon
ents
to:
• ad
d an
d su
btra
ct li
ke te
rms
in a
lgeb
raic
ex
pres
sion
s •
mul
tiply
inte
gers
and
mon
omia
ls b
y:
♦♦
mon
omia
ls
♦♦
bin
omia
ls
♦♦
trin
omia
ls
67MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
term
s or
like
alg
ebra
ic te
rms
D
eter
min
e th
e nu
mer
ical
val
ue o
f al
gebr
aic
expr
essi
ons
by s
ubst
itutio
n
• di
vide
the
follo
win
g by
inte
gers
or
mon
omia
ls:
♦♦
mon
omia
ls
♦♦
bi
nom
ials
♦♦
trin
omia
ls
Si
mpl
ify a
lgeb
raic
exp
ress
ions
invo
lvin
g th
e ab
ove
oper
atio
ns
D
eter
min
e th
e sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of s
ingl
e al
gebr
aic
term
s or
like
a
lgeb
raic
term
s
Det
erm
ine
the
num
eric
al v
alue
of
alge
brai
c ex
pres
sion
s by
sub
stitu
tion
Ex
tend
the
abov
e al
gebr
aic
m
anip
ulat
ions
to in
clud
e:
o m
ultip
ly in
tege
rs a
nd m
onom
ials
by
poly
nom
ials
o
divi
de p
olyn
omia
ls b
y in
tege
rs o
r m
onom
ials
o
the
prod
uct o
f tw
o bi
nom
ials
o
the
squa
re o
f a b
inom
ial
Fact
oriz
e al
gebr
aic
expr
essi
ons
• Fa
ctor
ize
alge
brai
c ex
pres
sion
s th
at
invo
lve:
o
com
mon
fact
ors
o di
ffere
nce
of tw
o sq
uare
s o
trino
mia
ls o
f the
form
:
♦♦
cbx
x
2
♦♦
cbx
ax
2
, whe
re a
is a
com
mon
fa
ctor
•
Sim
plify
alg
ebra
ic e
xpre
ssio
ns th
at
invo
lve
the
abov
e fa
ctor
izat
ion
proc
esse
s •
Sim
plify
alg
ebra
ic fr
actio
ns u
sing
fa
ctor
izat
ion
68 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Sugg
este
d m
etho
dolo
gy
GR
ADE
7 Th
is is
an
intro
duct
ion
to fo
rmal
alg
ebra
ic la
ngua
ge a
nd is
new
in th
e Se
nior
Pha
se. T
he u
se o
f sym
bolic
lang
uage
hel
ps to
dev
elop
an
unde
rsta
ndin
g of
var
iabl
es.
Lear
ners
hav
e op
portu
nitie
s to
writ
e an
d in
terp
ret a
lgeb
raic
exp
ress
ions
whe
n th
ey w
rite
gene
ral r
ules
to d
escr
ibe
rela
tions
hips
bet
wee
n nu
mbe
rs in
num
ber p
atte
rns,
an
d w
hen
they
find
inpu
t and
out
put v
alue
s fo
r giv
en ru
les
in fl
ow d
iagr
ams,
tabl
es a
nd fo
rmul
ae.
Exam
ples
a)
W
hat d
oes
the
rule
2 x
n –
1 m
ean
for t
he fo
llow
ing
num
ber s
eque
nce:
1; 3
; 5; 7
; 9;..
.. H
ere
lear
ners
sho
uld
reco
gniz
e th
at 2
x n
–1
repr
esen
ts th
e ge
nera
l ter
m in
this
seq
uenc
e, w
here
n re
pres
ents
the
posi
tion
of th
e te
rm
in th
e se
quen
ce. T
hus
it is
the
rule
that
can
be
used
to fi
nd a
ny te
rm in
the
give
n se
quen
ce
b)
The
rela
tions
hip
betw
een
a bo
y’s
age
(x y
rs o
ld) a
nd h
is m
othe
r’s a
ge is
giv
en a
s 25
+ x
. How
can
this
rela
tions
hip
be u
sed
to fi
nd th
e m
othe
r’s a
ge w
hen
the
boy
is 1
1 ye
ars
old?
H
ere
lear
ners
sho
uld
reco
gniz
e th
at to
find
the
mot
her’s
age
, the
y sh
ould
sub
stitu
te th
e bo
y’s
give
n ag
e in
to th
e ru
le 2
5 +
x. T
hey
shou
ld a
lso
reco
gniz
e th
at th
e gi
ven
rule
mea
ns th
e m
othe
r is
25 y
ears
old
er th
an th
e bo
y. S
ee fu
rther
exa
mpl
es g
iven
for f
unct
ions
an
d re
latio
nshi
ps.
Alge
brai
c ex
pres
sion
s sh
ould
incl
ude
inte
gers
in th
e ru
les
or re
latio
nshi
ps re
pres
ente
d in
sym
bolic
form
. G
RAD
E 8
In g
rade
8 le
arne
rs w
ill be
doi
ng th
e fo
llow
ing:
Intro
duct
ion
to c
onve
ntio
ns o
f alg
ebra
ic la
ngua
ge.
M
anip
ulat
ing
alge
brai
c ex
pres
sion
s.
Lear
ners
hav
e op
portu
nitie
s to
writ
e an
d in
terp
ret a
lgeb
raic
exp
ress
ions
whe
n th
ey w
rite
gene
ral r
ules
to d
escr
ibe
rela
tions
hips
bet
wee
n nu
mbe
rs in
num
ber p
atte
rns,
and
whe
n th
ey fi
nd in
put o
r out
put v
alue
s fo
r giv
en ru
les
in fl
ow d
iagr
ams,
tabl
es, f
orm
ulae
and
equ
atio
ns.
Exam
ples
of i
nter
pret
ing
alge
brai
c ex
pres
sion
s a)
W
hat d
oes
the
rule
2n m
ean
for t
he fo
llow
ing
num
ber s
eque
nce:
2; 4
; 8; 1
6; 3
2...
Her
e le
arne
rs s
houl
d re
cogn
ize
that
2n r
epre
sent
s th
e ge
nera
l ter
m in
this
seq
uenc
e, w
here
n re
pres
ents
the
posi
tion
of th
e te
rm in
the
sequ
ence
. Thu
s, it
is th
e ru
le th
at c
an b
e us
ed to
find
any
term
in th
e gi
ven
sequ
ence
. b)
Th
e re
latio
nshi
p be
twee
n a
boy’
s ag
e (x
yrs
old
) and
his
mot
her’s
age
is g
iven
as
25 +
x. H
ow c
an th
is re
latio
nshi
p be
use
d to
find
th
e m
othe
r’s a
ge w
hen
the
boy
is 1
1 ye
ars
old?
Her
e le
arne
rs s
houl
d re
cogn
ize
that
to fi
nd th
e m
othe
r’s a
ge, t
hey
mus
t sub
stitu
te
the
boy’
s gi
ven
age
into
the
rule
25
+ x.
The
y sh
ould
als
o re
cogn
ize
that
the
give
n ru
le m
eans
the
mot
her i
s 25
yea
rs o
lder
than
69MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
the
boy
GR
ADE
9 G
rade
lear
ners
will
do th
e fo
llow
ing
alge
brai
c m
anip
ulat
ions
whi
ch in
clud
e:
• m
ultip
ly in
tege
rs a
nd m
onom
ials
by
poly
nom
ials
•
divi
de p
olyn
omia
ls b
y in
tege
rs o
r mon
omia
ls
• th
e pr
oduc
t of t
wo
bino
mia
ls
• th
e sq
uare
of a
bin
omia
l M
anip
ulat
ing
alge
brai
c ex
pres
sion
s M
ake
sure
lear
ners
und
erst
and
that
the
rule
s fo
r ope
ratin
g w
ith in
tege
rs a
nd ra
tiona
l num
bers
, inc
ludi
ng la
ws
of e
xpon
ents
, app
ly e
qual
ly
whe
n nu
mbe
rs a
re re
plac
ed w
ith v
aria
bles
. The
var
iabl
es a
re n
umbe
rs o
f a g
iven
type
(e.g
. int
eger
s or
ratio
nal n
umbe
rs) i
n ge
nera
lized
fo
rm. W
hen
mul
tiply
ing
or d
ivid
ing
expr
essi
ons,
mak
e su
re le
arne
rs u
nder
stan
d ho
w th
e di
strib
utiv
e ru
le w
orks
. The
ass
ocia
tive
rule
allo
ws
for g
roup
ing
of li
ke te
rms
whe
n ad
ding
. Loo
k ou
t for
the
follo
win
g co
mm
on m
isco
ncep
tions
: •
x +
x =
2x a
nd N
OT
x2 . N
ote
the
conv
entio
n is
to w
rite
2x ra
ther
than
x2
• x2 +
x2 =
2x2 a
nd N
OT
2x4
• a
+b =
a +
b a
nd N
OT
ab
• (–
2x2 )
3 = –
8x6 a
nd N
OT
–6x5
• -x
(3x
+ 1)
= –
3x2 –
x a
nd N
OT
–3x2 +
1
• 6𝑥𝑥
2 +1
𝑥𝑥2 =
6 +
1 𝑥𝑥2 a
nd N
OT
6 +
1
• If
x =
2 th
en –
3x2
= –3
(2)2
= 3
x 4
= 1
2 an
d N
OT
(–6)
2 •
If x
= –2
then
–x2 –
x =
–(–
2)2 –
(–2)
= –4
+ 2
= –
2 an
d N
OT
4 +
2 =
6 •
√25𝑥𝑥
2−
9𝑥𝑥2
= √16𝑥𝑥2
= 4
x an
d N
OT
5x –
3x
= 2x
•
(x +
2)2 =
x2 +
4x
+ 4
and
NO
T x2 +
4
See
CAP
S p.
131
for e
xam
ples
of e
xpre
ssio
ns a
Gra
de 9
lear
ner s
houl
d be
abl
e to
sim
plify
. LT
SM
70 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Res
ourc
es
Cha
rts.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
W
B 2
Activ
ities
(74
– 76
; 120
- 12
2)
W
B 1
Activ
ities
(R
8; 2
9 –
31; 4
0 - 4
1)
WB
Activ
ities
(R
8; 3
5 - 3
6)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
71MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTIG
RAD
E LE
SSO
N P
LAN
5
TER
M 2
Dat
e : f
rom
....
to ..
....
Tim
e:
8 ho
urs
Gra
de
7 8
9 To
pic
ALG
EBR
AIC
EQ
UAT
ION
S C
APS
page
s 64
& 6
9 91
& 9
4 13
2 - 1
33 &
144
Sk
ills a
nd
Know
ledg
e N
umbe
r sen
tenc
es
W
rite
num
ber s
ente
nces
to d
escr
ibe
prob
lem
situ
atio
ns
An
alys
e an
d in
terp
ret n
umbe
r sen
tenc
es
that
des
crib
e a
give
n si
tuat
ion
Solv
e an
d co
mpl
ete
num
ber s
ente
nces
by
: o
insp
ectio
n o
trial
and
impr
ovem
ent
Id
entif
y va
riabl
es a
nd c
onst
ants
in g
iven
fo
rmul
ae o
r equ
atio
ns
D
eter
min
e th
e nu
mer
ical
val
ue o
f an
expr
essi
on b
y su
bstit
utio
n.
Equa
tions
Rev
ise
the
follo
win
g do
ne in
Gra
de 7
: o
set u
p eq
uatio
ns to
des
crib
e pr
oble
m
situ
atio
ns
o an
alys
e an
d in
terp
ret e
quat
ions
that
de
scrib
e a
give
n si
tuat
ion
o so
lve
equa
tions
by
insp
ectio
n o
dete
rmin
e th
e nu
mer
ical
val
ue o
f an
expr
essi
on b
y su
bstit
utio
n.
o id
entif
y va
riabl
es a
nd c
onst
ants
in
give
n fo
rmul
ae o
r equ
atio
ns
Ex
tend
sol
ving
equ
atio
ns to
incl
ude:
o
Usi
ng a
dditi
ve a
nd m
ultip
licat
ive
inve
rses
o
Usi
ng la
ws
of e
xpon
ents
Use
sub
stitu
tion
in e
quat
ions
to g
ener
ate
tabl
es o
f ord
ered
pai
rs
Equa
tions
Rev
ise
the
follo
win
g do
ne in
Gra
de 8
: o
Set u
p eq
uatio
ns to
des
crib
e pr
oble
m
situ
atio
ns
o An
alys
e an
d in
terp
ret e
quat
ions
that
de
scrib
e a
give
n si
tuat
ion
o So
lve
equa
tions
by:
♦♦
insp
ectio
n
♦♦
usi
ng a
dditi
ve a
nd m
ultip
licat
ive
inv
erse
s
♦♦
usi
ng la
ws
of e
xpon
ents
o
Det
erm
ine
the
num
eric
al v
alue
of a
n ex
pres
sion
by
subs
titut
ion.
o
Use
sub
stitu
tion
in e
quat
ions
to
gene
rate
tabl
es o
f ord
ered
pai
rs
Ex
tend
sol
ving
equ
atio
ns to
incl
ude:
o
usin
g fa
ctor
isat
ion
o eq
uatio
ns o
f the
form
: a p
rodu
ct o
f fa
ctor
s =
0
Sugg
este
d m
etho
dolo
gy
GR
ADE
7 In
Gra
de 7
, the
num
ber s
ente
nces
that
lear
ners
can
sol
ve a
re e
xten
ded
to in
clud
e nu
mbe
r sen
tenc
es w
ith in
tege
rs, s
quar
e nu
mbe
rs a
nd
cubi
c nu
mbe
rs.
Num
ber s
ente
nces
are
use
d he
re a
s a
mor
e fa
milia
r ter
m fo
r Gra
de 7
lear
ners
than
equ
atio
ns. H
owev
er, t
he te
rm e
quat
ion
will
be u
sed
inst
ead
of n
umbe
r sen
tenc
es in
late
r gra
des.
72 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Lear
ners
hav
e op
portu
nitie
s to
writ
e, s
olve
and
com
plet
e nu
mbe
r sen
tenc
es w
hen
they
writ
e ge
nera
l rul
es to
des
crib
e re
latio
nshi
ps
betw
een
num
bers
in n
umbe
r pa
ttern
s, a
nd w
hen
they
find
inpu
t and
out
put v
alue
s fo
r giv
en ru
les
in fl
ow d
iagr
ams,
tabl
es a
nd fo
rmul
ae. R
athe
r tha
n us
e fo
rmal
al
gebr
aic
proc
esse
s, le
arne
rs s
olve
num
ber s
ente
nces
by
insp
ectio
n or
det
erm
ine
the
num
eric
al v
alue
of e
xpre
ssio
ns b
y su
bstit
utio
n. In
th
is p
hase
, it i
s us
eful
whe
n so
lvin
g eq
uatio
ns to
beg
in to
spe
cify
whe
ther
x is
a n
atur
al n
umbe
r, in
tege
r or r
atio
nal n
umbe
r. Th
is b
uild
s le
arne
rs’ a
war
enes
s of
the
dom
ain
of x
. Ex
ampl
es
a)
Solv
e 𝑥𝑥
if 𝑥𝑥
+4
=7,
whe
re x
is
a na
tura
l num
ber.
(Wha
t mus
t be
adde
d to
4 to
giv
e 7?
) b)
So
lve
x if
𝑥𝑥+
4=
−7,
whe
re x
is a
n in
tege
r. (W
hat m
ust b
e ad
ded
to 4
to g
ive 7
?)
c)
Solv
e x
if 2𝑥𝑥
=30
, whe
re x
is a
nat
ural
num
ber.
(Wha
t mus
t be
mul
tiplie
d by
2 to
giv
e 30
?)
d)
Writ
e a
num
ber s
ente
nces
to fi
nd th
e ar
ea o
f a re
ctan
gle
with
leng
th 4
,5 c
m a
nd b
reat
h 2
cm.
e)
If 𝑦𝑦
=𝑥𝑥2
+1,
cal
cula
te y
whe
n x
= 3.
G
RAD
E 8
Up
to a
nd in
clud
ing
Gra
de 7
lear
ners
use
the
term
'num
bers
sen
tenc
es'.
From
Gra
de 8
the
term
'equ
atio
ns' i
s us
ed. S
olvi
ng e
quat
ions
us
ing
addi
tive
and
mul
tiplic
ativ
e in
vers
es a
s w
ell a
s la
ws
of e
xpon
ents
Ag
ain,
lear
ners
hav
e op
portu
nitie
s to
writ
e an
d so
lve
equa
tions
whe
n th
ey w
rite
gene
ral r
ules
to d
escr
ibe
rela
tions
hips
bet
wee
n nu
mbe
rs
in n
umbe
r pat
tern
s, a
nd w
hen
they
find
inpu
t or o
utpu
t val
ues
for g
iven
rule
s in
flow
dia
gram
s, ta
bles
and
form
ulae
. In
this
Gra
de, l
earn
ers
are
intro
duce
d to
:
Solv
ing
equa
tions
usi
ng a
dditi
ve a
nd m
ultip
licat
ive
inve
rses
as
wel
l as
law
s of
exp
onen
ts
Lear
ners
hav
e op
portu
nitie
s to
writ
e an
d so
lve
equa
tions
whe
n th
ey w
rite
gene
ral r
ules
to d
escr
ibe
rela
tions
hips
bet
wee
n nu
mbe
rs in
nu
mbe
r pat
tern
s, a
nd w
hen
they
find
inpu
t or o
utpu
t val
ues
for g
iven
rule
s in
flow
dia
gram
s, ta
bles
and
form
ulae
.
73MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Exam
ples
of e
quat
ions
EQU
ATIO
N
HIN
T a)
So
lve 𝑥𝑥
if 𝑥𝑥
+6
= −
9 Ad
d −6
to b
oth
side
s of
the
equa
tion.
b)
So
lve 𝑥𝑥
if −2
𝑥𝑥=
8 D
ivid
e bo
th s
ides
of t
he e
quat
ion
by −
2.
c)
Solv
e x
if −𝑥𝑥
= −
5 D
ivid
e bo
th s
ides
of t
he e
quat
ion
by −
1 d)
So
lve
x if
3𝑥𝑥+
1=
7 Ad
d −1
to b
oth
side
s of
the
equa
tion
and
then
div
ide
both
sid
es o
f the
equ
atio
n by
3.
e)
Pr
ovid
e an
equ
atio
n to
find
the
area
of a
rect
angl
e w
ith le
ngth
2x
cm a
nd w
idth
( 2𝑥𝑥
+1)
cm
f) If
the
area
of a
rect
angl
e is
( 4𝑥𝑥2
−6𝑥𝑥
) 𝑐𝑐𝑐𝑐
2 , a
nd it
s w
idth
is 2𝑥𝑥 𝑐𝑐𝑐𝑐
, wha
t will
be it
s le
ngth
in te
rms
of 𝑥𝑥
? g)
If 𝑦𝑦
= 𝑥𝑥
2 +
1, C
alcu
late
y w
hen
x =
4.
h)
Than
di is
6 y
ears
old
er th
an S
ophi
e. In
3 y
ears
’ tim
e Th
andi
will
be tw
ice
as o
ld a
s So
phie
. How
old
is T
hand
i now
? G
RAD
E 9
Gra
de 9
lear
ners
are
taug
ht:
• So
lvin
g eq
uatio
ns u
sing
fact
oriz
atio
n •
Solv
ing
equa
tions
of t
he fo
rm: a
pro
duct
of f
acto
rs =
0
Agai
n le
arne
rs h
ave
oppo
rtuni
ties
to w
rite
and
solv
e eq
uatio
ns w
hen
they
writ
e ge
nera
l rul
es to
des
crib
e re
latio
nshi
ps b
etw
een
num
bers
in
num
ber p
atte
rns,
and
whe
n th
ey fi
nd in
put o
r out
put v
alue
s fo
r giv
en ru
les
in fl
ow d
iagr
ams,
tabl
es a
nd fo
rmul
ae.
In G
rade
9, l
earn
ers
can
be g
iven
equ
atio
ns w
here
they
hav
e to
exp
and,
sim
plify
or f
acto
rize
expr
essi
ons
first
, bef
ore
solv
ing
the
equa
tion.
Fo
r equ
atio
ns o
f the
form
: a p
rodu
ct o
f tw
o fa
ctor
s =
0, le
arne
rs h
ave
to u
nder
stan
d th
at if
the
prod
uct o
f tw
o fa
ctor
s eq
uals
0, t
hen
at
leas
t one
of t
he fa
ctor
s m
ust b
e eq
ual t
o 0.
Hen
ce to
sol
ve th
e eq
uatio
n, e
ach
fact
or m
ust b
e w
ritte
n as
an
equa
tion
equa
l to
0, a
nd
ther
efor
e m
ore
than
one
sol
utio
n fo
r x is
pos
sibl
e. W
hen
wor
king
with
alg
ebra
ic fr
actio
ns, l
earn
ers
mus
t be
rem
inde
d th
at th
e de
nom
inat
or
cann
ot e
qual
0, s
o an
y va
lue
of x
that
mak
es th
e de
nom
inat
or 0
can
not b
e a
solu
tion
to th
e eq
uatio
n.
Exam
ples
of e
quat
ions
Se
e C
APS
page
132
and
144
LTSM
74 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Res
ourc
es
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
W
B 2
Activ
ities
(77
– 79
; 123
- 12
5)
WB
1 Ac
tiviti
es (3
2 - 3
8)
W
B 2
Activ
ities
(111
-113
)
WB
Act
iviti
es (8
1 - 8
6)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
75MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LE
SSO
N P
LAN
S: T
ERM
3
76 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
1
TER
M 3
Dat
e : f
rom
....
to ..
....
Tim
e:
8
hou
rs
Gra
de
7 8
9 To
pic
CO
MM
ON
FR
ACTI
ON
S C
APS
page
s 49
- 51
10
0 - 1
02
122
Skills
and
Kn
owle
dge
Orde
ring,
com
parin
g an
d si
mpl
ifyin
g fra
ctio
ns
C
ompa
re a
nd o
rder
com
mon
fra
ctio
ns, i
nclu
ding
spe
cific
ally
te
nths
and
hun
dred
ths
Ex
tend
to th
ousa
ndth
s Ca
lculat
ions w
ith fr
actio
ns
R
evis
e th
e fo
llow
ing
done
in G
rade
6:
o ad
ditio
n an
d su
btra
ctio
n of
co
mm
on fr
actio
ns, i
nclu
ding
m
ixed
num
bers
, lim
ited
to
fract
ions
with
the
sam
e de
nom
inat
or o
r whe
re o
ne
deno
min
ator
is a
mul
tiple
of
anot
her
o fin
ding
frac
tions
of w
hole
nu
mbe
rs
Ex
tend
o
addi
tion
and
subt
ract
ion
to
fract
ions
whe
re o
ne
deno
min
ator
is n
ot a
mul
tiple
of
the
othe
r
Calcu
lation
s with
frac
tions
Rev
ise:
-
addi
tion
and
subt
ract
ion
of
com
mon
frac
tions
, in
clud
ing
mix
ed n
umbe
rs
- fin
ding
frac
tions
of w
hole
nu
mbe
rs
- m
ultip
licat
ion
of c
omm
on
fract
ions
, inc
ludi
ng
mix
ed n
umbe
rs
D
ivid
e w
hole
num
bers
and
co
mm
on fr
actio
ns b
y co
mm
on
fract
ions
Cal
cula
te th
e sq
uare
s, c
ubes
, sq
uare
root
s an
d cu
be ro
ots
of
com
mon
frac
tions
Calcu
lation
s with
frac
tions
All f
our o
pera
tions
with
com
mon
frac
tions
an
d m
ixed
num
bers
All f
our o
pera
tions
, with
num
bers
that
in
volv
e th
e sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of c
omm
on fr
actio
ns
77MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
o m
ultip
licat
ion
of c
omm
on
fract
ions
, inc
ludi
ng m
ixed
nu
mbe
rs, n
ot li
mite
d to
fra
ctio
ns w
here
one
de
nom
inat
or is
a m
ultip
le o
f an
othe
r Ca
lculat
ion te
chniq
ues
C
onve
rt m
ixed
num
bers
to c
omm
on
fract
ions
in
orde
r to
perfo
rm c
alcu
latio
ns w
ith th
em
U
se k
now
ledg
e of
mul
tiple
s an
d fa
ctor
s to
writ
e fra
ctio
ns in
the
sim
ples
t for
m
befo
re o
r afte
r cal
cula
tions
Use
kno
wle
dge
of e
quiv
alen
t fra
ctio
ns to
ad
d an
d su
btra
ct c
omm
on fr
actio
ns
Solvi
ng p
robl
ems
So
lve
prob
lem
s in
con
text
s in
volv
ing
com
mon
frac
tions
and
mix
ed n
umbe
rs,
incl
udin
g gr
oupi
ng, s
harin
g an
d fin
ding
fra
ctio
ns o
f who
le n
umbe
rs
Calcu
lation
tech
nique
s
Rev
ise:
-
conv
ert m
ixed
num
bers
to
com
mon
frac
tions
in
orde
r to
perfo
rm c
alcu
latio
ns w
ith
them
-
use
know
ledg
e of
mul
tiple
s an
d fa
ctor
s to
writ
e fra
ctio
ns in
the
sim
ples
t for
m b
efor
e or
afte
r ca
lcul
atio
ns
- us
e kn
owle
dge
of e
quiv
alen
t fra
ctio
ns to
add
and
sub
tract
co
mm
on fr
actio
ns
U
se k
now
ledg
e of
reci
proc
al
rela
tions
hips
to d
ivid
e co
mm
on
fract
ions
So
lving
pro
blem
s
Solv
e pr
oble
ms
in c
onte
xts
invo
lvin
g co
mm
on fr
actio
ns a
nd
mix
ed n
umbe
rs, i
nclu
ding
gr
oupi
ng, s
harin
g an
d fin
ding
fra
ctio
ns o
f who
le n
umbe
rs
Calcu
lation
tech
nique
s
Rev
ise:
-
conv
ert m
ixed
num
bers
to c
omm
on
fract
ions
in
orde
r to
perfo
rm c
alcu
latio
ns w
ith th
em
- us
e kn
owle
dge
of m
ultip
les
and
fact
ors
to w
rite
fract
ions
in th
e si
mpl
est f
orm
be
fore
or a
fter c
alcu
latio
ns
- us
e kn
owle
dge
of e
quiv
alen
t fra
ctio
ns
to a
dd a
nd s
ubtra
ct c
omm
on fr
actio
ns
- us
e kn
owle
dge
of re
cipr
ocal
re
latio
nshi
ps to
div
ide
com
mon
fra
ctio
ns
Solvi
ng p
robl
ems
So
lve
prob
lem
s in
con
text
s in
volv
ing
com
mon
frac
tions
, mix
ed n
umbe
rs
and
perc
enta
ges
Perc
entag
es
R
evis
e th
e fo
llow
ing
done
in G
rade
6:
- Fi
ndin
g pe
rcen
tage
s of
who
le n
umbe
rs
C
alcu
late
the
perc
enta
ge o
f par
t of a
who
le
C
alcu
late
per
cent
age
incr
ease
or
decr
ease
of w
hole
num
bers
Solv
e pr
oble
ms
in c
onte
xts
invo
lvin
g pe
rcen
tage
s
Perc
entag
es
R
evis
e:
- fin
ding
per
cent
ages
of w
hole
nu
mbe
rs
- ca
lcul
atin
g th
e pe
rcen
tage
of p
art
of a
who
le
- ca
lcul
atin
g pe
rcen
tage
incr
ease
or
dec
reas
e
Cal
cula
te a
mou
nts
if gi
ven
78 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Equiv
alent
form
s R
evis
e th
e fo
llow
ing
done
in G
rade
6:
re
cogn
ize
and
use
equi
vale
nt fo
rms
of
com
mon
frac
tions
with
1-d
igit
or 2
-dig
it de
nom
inat
ors
(frac
tions
whe
re o
ne
deno
min
ator
is a
mul
tiple
of t
he o
ther
)
reco
gniz
e eq
uiva
lenc
e be
twee
n co
mm
on
fract
ion
and
deci
mal
frac
tion
form
s of
the
sam
e nu
mbe
r
reco
gniz
e eq
uiva
lenc
e be
twee
n co
mm
on
fract
ion,
dec
imal
frac
tion
and
perc
enta
ge
form
s of
the
sam
e nu
mbe
r
perc
enta
ge in
crea
se o
r de
crea
se
So
lve
prob
lem
s in
con
text
s in
volv
ing
perc
enta
ges
Equiv
alent
form
s R
evis
e eq
uiva
lent
form
s be
twee
n:
co
mm
on fr
actio
ns
(frac
tions
whe
re o
ne
deno
min
ator
is a
mul
tiple
of
the
othe
r)
com
mon
frac
tion
and
deci
mal
fra
ctio
n fo
rms
of th
e sa
me
num
ber
co
mm
on fr
actio
n,
deci
mal
frac
tion
and
perc
enta
ge fo
rms
of th
e sa
me
num
ber
Equiv
alent
form
s R
evis
e eq
uiva
lent
form
s be
twee
n:
co
mm
on fr
actio
ns w
here
one
de
nom
inat
or is
a m
ultip
le o
f ano
ther
com
mon
frac
tion
and
deci
mal
frac
tion
form
s of
the
sam
e nu
mbe
r
com
mon
frac
tion,
dec
imal
fra
ctio
n an
d pe
rcen
tage
form
s of
th
e sa
me
num
ber
Sugg
este
d m
etho
dolo
gy
Calcu
lation
s with
frac
tions
Le
arne
rs s
houl
d do
con
text
free
cal
cula
tions
and
sol
ve p
robl
ems
in c
onte
xts.
It
is n
ot e
xpec
ted
that
lear
ners
kno
w ru
les
for s
impl
ifyin
g fra
ctio
ns o
r for
con
verti
ng b
etw
een
mix
ed n
umbe
rs a
nd fr
actio
n fo
rms.
Lear
ners
sho
uld
know
from
wor
king
with
equ
ival
ence
, whe
n a
fract
ion
is e
qual
to o
r gre
ater
than
1.
LC
M’s
hav
e to
be
foun
d w
hen
addi
ng a
nd s
ubtra
ctin
g fra
ctio
ns o
f diff
eren
t den
omin
ator
s. H
ere
lear
ners
use
kno
wle
dge
of c
omm
on
mul
tiple
s to
find
the
LCM
i.e.
wha
t num
ber c
an b
oth
deno
min
ator
s be
div
ided
into
.
To
sim
plify
frac
tions
, lea
rner
s us
e kn
owle
dge
of c
omm
on fa
ctor
s i.e
. wha
t can
div
ide
equa
lly in
to th
e nu
mer
ator
and
den
omin
ator
of a
fract
ion.
Em
phas
ize
that
whe
n si
mpl
ifyin
g, th
e fra
ctio
ns m
ust r
emai
n eq
uiva
lent
.
Le
arne
rs s
houl
d re
cogn
ize
that
find
ing
a ‘fr
actio
n of
a w
hole
num
ber’
or ‘f
indi
ng a
frac
tion
of a
frac
tion’
mea
ns m
ultip
lyin
g th
e fra
ctio
n
and
the
who
le n
umbe
r or t
he fr
actio
n w
ith th
e fra
ctio
n.
W
hen
lear
ners
find
frac
tions
of w
hole
num
bers
, the
exa
mpl
es c
an b
e ch
osen
to re
sult
eith
er in
who
le n
umbe
rs o
r fra
ctio
ns o
r bot
h.
Le
arne
rs s
houl
d al
so u
se th
e co
nven
tion
of w
ritin
g th
e w
hole
num
ber a
s a
fract
ion
over
whe
n m
ultip
lyin
g.
79MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Calcu
lation
usin
g pe
rcen
tages
Le
arne
rs s
houl
d do
con
text
free
cal
cula
tions
and
sol
ve p
robl
ems
in c
onte
xts.
W
hen
doin
g ca
lcul
atio
ns u
sing
per
cent
ages
, lea
rner
s ha
ve to
use
the
equi
vale
nt c
omm
on fr
actio
n fo
rm, w
hich
is a
frac
tion
with
deno
min
ator
100
.
Lear
ners
sho
uld
beco
me
fam
iliar w
ith th
e eq
uiva
lent
frac
tion
and
deci
mal
form
s of
com
mon
per
cent
ages
like
To
cal
cula
te p
erce
ntag
e of
par
t of a
who
le, o
r per
cent
age
incr
ease
or d
ecre
ase,
lear
ners
hav
e to
lear
n th
e st
rate
gy o
f
mul
tiply
ing
by 1
00. I
t is
usef
ul fo
r lea
rner
s to
lear
n to
use
cal
cula
tors
for s
ome
of th
ese
calc
ulat
ions
whe
re th
e fra
ctio
ns
are
not e
asily
sim
plifi
ed.
W
hen
usin
g ca
lcul
ator
s, le
arne
rs c
an a
lso
use
the
equi
vale
nt d
ecim
al fr
actio
n fo
rm fo
r per
cent
ages
to d
o th
e
calc
ulat
ions
.
Grad
e 8
Wha
t is di
ffere
nt to
Gra
de 7?
D
ivid
e by
com
mon
fra
ctio
ns
Sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of c
omm
on fr
actio
ns
Multip
licati
on
Fo
r mul
tiplic
atio
n of
frac
tions
, lea
rner
s sh
ould
be
enco
urag
ed to
sim
plify
frac
tions
by
divi
ding
num
erat
ors
and
deno
min
ator
s by
com
mon
fact
ors.
Le
arne
rs s
houl
d no
te th
e di
ffere
nce
betw
een
addi
ng o
r sub
tract
ing
fract
ions
, and
mul
tiply
ing
fract
ions
W
hen
lear
ners
find
frac
tions
of w
hole
num
bers
, the
exa
mpl
es c
an b
e ch
osen
to re
sult
eith
er in
who
le n
umbe
rs o
r fra
ctio
ns o
r bot
h.
Le
arne
rs s
houl
d al
so u
se th
e co
nven
tion
of w
ritin
g th
e w
hole
num
ber a
s a
fract
ion
over
whe
n m
ultip
lyin
g.
80 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Divi
sion
Th
e te
chni
que
of ‘i
nver
t and
mul
tiply
’ app
lies
to d
ivis
ion
in g
ener
al a
nd n
ot ju
st to
div
isio
n by
frac
tions
. Hen
ce, a
use
ful w
ay o
f mak
ing
lear
ners
com
forta
ble
with
div
isio
n by
frac
tions
is to
sta
rt w
ith e
xam
ples
of d
ivis
ion
by w
hole
num
bers
.
Le
arne
rs h
ave
to u
nder
stan
d th
at d
ivid
ing
by a
num
ber i
s th
e sa
me
as m
ultip
lyin
g by
the
reci
proc
al o
f the
num
ber i
.e. t
he re
cipr
ocal
of n
is 1
Kn
owin
g th
e ru
les
of o
pera
tions
for c
alcu
latin
g sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of c
omm
on fr
actio
ns is
impo
rtant
, onc
e
lear
ners
are
com
forta
ble
doin
g al
l the
ope
ratio
ns w
ith fr
actio
ns, c
alcu
latio
ns d
o no
t hav
e to
be
rest
ricte
d to
pos
itive
frac
tions
.
Grad
e 9
Wha
t is di
ffere
nt to
Gra
de 8?
In G
rade
9, l
earn
ers
wor
k w
ith d
ecim
al fr
actio
ns m
ostly
as
coef
ficie
nts
in a
lgeb
raic
exp
ress
ions
and
equ
atio
ns. T
hey
are
expe
cted
to b
e
com
pete
nt in
per
form
ing
mul
tiple
ope
ratio
ns u
sing
dec
imal
frac
tions
and
mix
ed n
umbe
rs, a
pply
ing
prop
er- t
ies
of ra
tiona
l num
bers
appr
opria
tely.
The
y ar
e al
so e
xpec
ted
to re
cogn
ize
and
use
equi
vale
nt fo
rms
for d
ecim
al fr
actio
ns a
ppro
pria
tely
in c
alcu
latio
ns.
LTSM
Res
ourc
es
N
umbe
r lin
es: 0
– 1
20, S
truct
ured
-, s
emi-
stru
ctur
ed -
, uns
truct
ured
num
ber l
ines
; Cou
nter
s, P
ictu
res,
arra
ys/ d
iagr
ams,
Fla
sh c
ards
, Bas
e 10
bloc
ks, 1
00 c
harts
, mul
tiplic
atio
n ta
ble
G
rade
7
Gra
de 8
Gra
de 9
Wor
kboo
k re
fere
nce
1
Activ
ities
(30
- 43)
WB
1 Ac
tiviti
es (R
5; R
6)
WB
2 Ac
tiviti
es (6
5 - 6
8)
WB
1 Ac
tiviti
es (R
5; 1
6
Text
book
refe
renc
e
81MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
ass
essm
ent
– Te
st: m
arks
max
82 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
2
TER
M 3
Dat
e : f
rom
....
to ..
....
Tim
e: 8
hou
rs
Gra
de
7 8
9 To
pic
DEC
IMAL
FR
ACTI
ON
S C
APS
page
s 51
- 53
10
3 - 1
05
123
Skills
and
Kn
owle
dge
Ord
erin
g an
d co
mpa
ring
deci
mal
fr
actio
ns
R
evis
e th
e fo
llow
ing
done
in G
rade
6:
-
coun
t for
war
ds a
nd b
ackw
ards
in
deci
mal
frac
tions
to a
t lea
st tw
o de
cim
al p
lace
s
- co
mpa
re a
nd o
rder
dec
imal
frac
tions
to
at le
ast t
wo
deci
mal
pla
ces
-
plac
e va
lue
of d
igits
to a
t lea
st tw
o de
cim
al p
lace
s
- ro
undi
ng o
ff de
cim
al fr
actio
ns to
at l
east
1
deci
mal
pla
ce
Ex
tend
al
l of
th
e ab
ove
to
deci
mal
fra
ctio
ns to
at l
east
thre
e de
cim
al p
lace
s an
d ro
undi
ng o
ff to
at
leas
t 2
deci
mal
pl
aces
Ord
erin
g an
d co
mpa
ring
deci
mal
fr
actio
ns
R
evis
e:
-
orde
ring,
com
parin
g an
d pl
ace
valu
e of
dec
imal
fra
ctio
ns to
at l
east
3
deci
mal
pla
ces
-
roun
ding
off
deci
mal
fra
ctio
ns to
at l
east
2
deci
mal
pla
ce
Ord
erin
g an
d co
mpa
ring
deci
mal
fr
actio
ns
R
evis
e:
-
orde
ring,
com
parin
g an
d pl
ace
valu
e of
dec
imal
fra
ctio
ns to
at l
east
3
deci
mal
pla
ces
-
roun
ding
off
deci
mal
fra
ctio
ns to
at l
east
2
deci
mal
pla
ce
83MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Calcu
lation
s with
dec
imal
fracti
ons
R
evis
e th
e fo
llow
ing
done
in G
rade
6:
-
addi
tion
and
subt
ract
ion
of d
ecim
al
fract
ions
of a
t lea
st tw
o de
cim
al p
lace
s
- m
ultip
licat
ion
of d
ecim
al fr
actio
ns b
y 10
and
100
Exte
nd a
dditi
on a
nd s
ubtra
ctio
n to
dec
imal
fra
ctio
ns o
f at l
east
thre
e de
cim
al p
lace
s
mul
tiply
dec
imal
frac
tions
to in
clud
e:
-
deci
mal
frac
tions
to a
t lea
st 3
dec
imal
pl
aces
by
who
le n
umbe
rs
-
deci
mal
frac
tions
to a
t lea
st 2
dec
imal
pl
aces
by
deci
mal
frac
tions
to a
t lea
st 1
de
cim
al p
lace
Div
ide
deci
mal
frac
tions
to in
clud
e de
cim
al fr
actio
ns to
at l
east
3 d
ecim
al
plac
es b
y w
hole
num
bers
Ca
lculat
ion te
chniq
ues
U
se
know
ledg
e of
pl
ace
valu
e to
es
timat
e th
e nu
mbe
r of
de
cim
al
plac
es in
the
resu
lt be
fore
per
form
ing
calc
ulat
ions
Use
roun
ding
off
and
a ca
lcul
ator
to c
heck
re
sults
whe
re a
ppro
pria
te
Calcu
lation
s with
dec
imal
fracti
ons
R
evis
e:
-
addi
tion,
sub
tract
ion,
mul
tiplic
atio
n an
d of
de
cim
al fr
actio
ns to
at l
east
3 d
ecim
al
plac
es
-
divi
sion
of d
ecim
al fr
actio
ns b
y w
hole
nu
mbe
rs
Ex
tend
mul
tiplic
atio
n to
'mul
tiplic
atio
n by
dec
imal
frac
tions
' not
lim
ited
to
one
deci
mal
pla
ce
Ex
tend
div
isio
n to
'div
isio
n of
dec
imal
fra
ctio
ns b
y de
cim
al fr
actio
ns'
C
alcu
late
the
squa
res,
cub
es,
squa
re ro
ots
and
cube
root
s of
de
cim
al fr
actio
ns
Calcu
lation
tech
nique
s
Use
kno
wle
dge
of p
lace
val
ue t
o es
timat
e th
e nu
mbe
r of
dec
imal
pl
aces
in
th
e re
sult
befo
re
perfo
rmin
g ca
lcul
atio
ns
U
se ro
undi
ng o
ff an
d a
calc
ulat
or to
ch
eck
resu
lts w
here
app
ropr
iate
Calcu
lation
s with
dec
imal
fracti
ons
m
ultip
le
oper
atio
ns
with
de
cim
al
fract
ions
, us
ing
a ca
lcul
ator
whe
re
appr
opria
te
m
ultip
le o
pera
tions
with
or
with
out
brac
kets
, with
num
bers
that
invo
lve
the
squa
res,
cub
es,
squa
re r
oots
an
d cu
be ro
ots
of d
ecim
al fr
actio
ns
Calcu
lation
tech
nique
s
Use
kno
wle
dge
of p
lace
val
ue t
o es
timat
e th
e nu
mbe
r of
dec
imal
pl
aces
in
th
e re
sult
befo
re
perfo
rmin
g ca
lcul
atio
ns
U
se ro
undi
ng o
ff an
d a
calc
ulat
or to
ch
eck
resu
lts w
here
app
ropr
iate
So
lving
pro
blem
s
Solv
e pr
oble
ms
in c
onte
xt in
volv
ing
deci
mal
Fr
actio
ns
Solvi
ng p
robl
ems
So
lve
prob
lem
s in
con
text
invo
lvin
g de
cim
al
fract
ions
Solvi
ng p
robl
ems
So
lve
prob
lem
s in
con
text
invo
lvin
g de
cim
al
fract
ions
84 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Equiv
alent
form
s
Rev
ise
the
follo
win
g do
ne in
Gra
de 6
:
- re
cogn
ize
equi
vale
nce
betw
een
com
mon
frac
tion
and
deci
mal
frac
tion
form
s of
the
sam
e nu
mbe
r
- re
cogn
ize
equi
vale
nce
betw
een
com
mon
fra
ctio
n, d
ecim
al fr
actio
n an
d pe
rcen
tage
form
s of
the
sam
e nu
mbe
r
Equiv
alent
form
s
Rev
ise
equi
vale
nt fo
rms
betw
een:
- co
mm
on fr
actio
n an
d de
cim
al
fract
ion
form
s of
the
sam
e nu
mbe
r
- co
mm
on fr
actio
n, d
ecim
al
fract
ion
and
perc
enta
ge
form
s of
the
sam
e nu
mbe
r
Equiv
alent
form
s
Rev
ise
equi
vale
nt fo
rms
betw
een:
- co
mm
on fr
actio
n an
d de
cim
al
fract
ion
form
s of
the
sam
e nu
mbe
r
- co
mm
on fr
actio
n, d
ecim
al
fract
ion
and
perc
enta
ge
form
s of
the
sam
e nu
mbe
r
Sugg
este
d m
etho
dolo
gy
Orde
ring,
coun
ting a
nd co
mpa
ring d
ecim
al fra
ctio
ns
C
ount
ing
shou
ld n
ot o
nly
be th
ough
t of a
s ve
rbal
cou
ntin
g. L
earn
ers
can
coun
t in
deci
mal
inte
rval
s us
ing:
- st
ruct
ured
, sem
i-stru
ctur
ed o
r em
pty
num
ber l
ines
- ch
ain
diag
ram
s fo
r cou
ntin
g
Lear
ners
sho
uld
be g
iven
a ra
nge
of e
xerc
ises
suc
h as
:
- ar
rang
e gi
ven
num
bers
from
the
smal
lest
to th
e bi
gges
t: or
big
gest
to s
mal
lest
- fil
l in
mis
sing
num
bers
in
◆
a s
eque
nce
◆
on
a nu
mbe
r grid
◆ o
n a
num
ber l
ine
◆
fill i
n <,
= o
r > e
xam
ple:
0,4
* 0.
04∗0
,004
( no
te g
rade
8 d
o up
tp 3
dec
imal
pla
ces)
C
ount
ing
exer
cise
s in
cha
in d
iagr
ams
can
be c
heck
ed u
sing
cal
cula
tors
and
lear
ners
can
exp
lain
any
diff
eren
ces
betw
een
thei
r ans
wer
s an
d th
ose
show
n by
the
calc
ulat
or.
85MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Calcu
lating
with
dec
imal
fracti
ons
Le
arne
rs s
houl
d do
con
text
free
cal
cula
tions
and
sol
ve p
robl
ems
in c
onte
xts.
Lear
ners
sho
uld
estim
ate
thei
r ans
wer
s be
fore
cal
cula
ting,
esp
ecia
lly w
ith m
ultip
licat
ion
by d
ecim
al fr
actio
ns. T
hey
shou
ld b
e ab
le to
ju
dge
the
reas
onab
lene
ss o
f ans
wer
s re
latin
g to
how
man
y de
cim
al p
lace
s an
d al
so c
heck
thei
r ow
n an
swer
s.
m
ultip
licat
ion
by d
ecim
al fr
actio
ns s
houl
d st
art w
ith fa
milia
r num
bers
that
lear
ners
can
cal
cula
te b
y in
spec
tion,
so
that
lear
ners
ge
t a s
ense
of h
ow d
ecim
al p
lace
s ar
e af
fect
ed b
y m
ultip
licat
ion.
Eq
uiva
lence
betw
een c
omm
on fr
actio
ns an
d dec
imal
frac
tions
Lear
ners
are
not
exp
ecte
d to
be
able
to c
onve
rt an
y co
mm
on fr
actio
n in
to it
s de
cim
al fo
rm, m
erel
y to
see
the
rela
tions
hip
betw
een
tent
hs, h
undr
edth
s an
d th
ousa
ndth
s in
thei
r dec
imal
form
s.
Le
arne
rs s
houl
d st
art b
y re
writ
ing
and
conv
ertin
g te
nths
, hun
dred
ths
and
thou
sand
ths
in c
omm
on fr
actio
n fo
rm to
dec
imal
frac
tions
. W
here
den
omin
ator
s of
oth
er fr
actio
ns a
re fa
ctor
s of
10
e.g.
2,5
or f
acto
rs o
f 100
e.g
. 2, 4
, 20,
25
lear
ners
can
con
vert
thes
e to
hu
ndre
dths
usi
ng w
hat t
hey
know
abo
ut e
quiv
alen
ce.
It is
use
ful t
o us
e ca
lcul
ator
s to
hel
p le
arne
rs c
onve
rt be
twee
n co
mm
on fr
actio
ns a
nd d
ecim
al fr
actio
ns (h
ere
lear
ners
will
use
wha
t the
y kn
ow a
bout
the
rela
tions
hip
betw
een
fract
ions
and
div
isio
n).
- D
ivid
ing
who
le n
umbe
rs b
y 10
, 100
, 1 0
00, e
tc. c
an h
elp
to b
uild
lear
ners
’ und
erst
andi
ng o
f pla
ce v
alue
with
dec
imal
s. T
his
is a
lso
usef
ul to
do
on th
e ca
lcul
ator
– le
arne
rs c
an d
iscu
ss th
e pa
ttern
s th
ey s
ee w
hen
divi
ding
.
- Si
mila
rly c
alcu
lato
rs c
an b
e us
eful
tool
s fo
r lea
rner
s to
lear
n ab
out p
atte
rns
whe
n m
ultip
lyin
g de
cim
als
by 1
0, 1
00 o
r,1 0
00 e
tc.
n
Gra
de 8
W
hat is
diffe
rent
to G
rade
7?
M
ultip
licat
ion
by d
ecim
al fr
actio
ns n
ot li
mite
d to
one
dec
imal
pla
ce
D
ivis
ion
of d
ecim
al fr
actio
ns b
y de
cim
al fr
actio
ns
Sq
uare
s, c
ubes
, squ
are
root
s an
d cu
be ro
ots
of d
ecim
al fr
actio
ns
86 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Calcu
lating
usin
g de
cimal
fracti
ons
Fo
r div
isio
n by
dec
imal
frac
tions
with
out c
alcu
lato
rs, l
earn
ers
have
to u
se th
eir k
now
ledg
e of
mul
tiplic
atio
n by
10
or m
ultip
les
of 1
0 to
m
ake
the
divi
sor a
who
le n
umbe
r. H
ence
sta
rt w
ith fa
milia
r num
bers
that
lear
ners
can
cal
cula
te b
y in
spec
tion,
so
that
lear
ners
get
a
sens
e of
how
dec
imal
pla
ces
are
affe
cted
by
divi
sion
.
For b
igge
r and
unf
amilia
r dec
imal
frac
tions
, lea
rner
s sh
ould
use
cal
cula
tors
for m
ultip
licat
ion
and
divi
sion
, but
stil
l jud
ge th
e re
ason
able
ness
of t
heir
solu
tions
.
Sim
ilarly
, fin
ding
squ
ares
, cu
bes,
squ
are
root
s an
d cu
be r
oots
for
dec
imal
fra
ctio
ns s
houl
d st
art
with
fam
iliar
num
bers
tha
t le
arne
rs c
an c
alcu
late
by
insp
ectio
n.
O
nce
lear
ners
are
com
forta
ble
with
all
the
oper
atio
ns u
sing
dec
imal
frac
tions
, cal
cula
tions
sho
uld
not b
e re
stric
ted
to p
ositi
ve
deci
mal
frac
tions
. G
rade
9
Wha
t is di
ffere
nt to
Gra
de 8?
In
Gra
de 9
, lea
rner
s w
ork
with
dec
imal
frac
tions
mos
tly a
s co
effic
ient
s in
alg
ebra
ic e
xpre
ssio
ns a
nd e
quat
ions
. The
y ar
e ex
pect
ed to
be
com
pete
nt in
per
form
ing
mul
tiple
ope
ratio
ns u
sing
dec
imal
frac
tions
and
mix
ed n
umbe
rs, a
pply
ing
prop
er- t
ies
of ra
tiona
l num
bers
appr
opria
tely.
The
y ar
e al
so e
xpec
ted
to re
cogn
ize
and
use
equi
vale
nt fo
rms
for d
ecim
al fr
actio
ns a
ppro
pria
tely
in c
alcu
latio
ns.
LTSM
Res
ourc
es
Num
ber l
ines
, fla
sh c
ards
, cha
rts.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es (4
4 - 4
6)
W
B 1
Activ
ities
(R6a
&b )
WB
2 Ac
tiviti
es (7
1 - 7
4)
WB
1 Ac
tiviti
es
(R6;
16
- 17)
Text
book
refe
renc
e
87MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
ass
essm
ent
– Te
st: m
arks
max
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
3
TER
M 3
Dat
e : f
rom
....
to ..
....
Tim
e:
7 ho
urs
Gra
de
7 8
9 To
pic
DAT
A H
AND
LIN
G
CAP
S pa
ges
70 –
72
109
- 111
14
9 - 1
51
Or
gani
ze an
d su
mm
arize
data
Org
aniz
e (in
clud
ing
grou
ping
whe
re
appr
opria
te)
and
reco
rd d
ata
usin
g -
tally
mar
ks
- ta
bles
-
stem
-and
-leaf
dis
play
s
Gro
up d
ata
into
inte
rval
s
Sum
mar
ize
and
dist
ingu
ishi
ng b
etw
een
ungr
oupe
d nu
mer
ical
dat
a by
det
erm
inin
g:
- m
ean
Orga
nize
and
sum
mar
ize da
ta
O
rgan
ize
(incl
udin
g gr
oupi
ng w
here
ap
prop
riate
) and
reco
rd d
ata
usin
g -
tally
mar
ks
- ta
bles
-
stem
-and
-leaf
dis
play
s
Gro
up d
ata
into
inte
rval
s
Sum
mar
ize
data
usi
ng m
easu
res
of
cent
ral
tend
ency
, incl
udin
g:
- m
ean
- m
edia
n
Orga
nize
and
sum
mar
ize da
ta
O
rgan
ize
num
eric
al d
ata
in d
iffer
ent
way
s in
ord
er to
sum
mar
ize
by
dete
rmin
ing:
-
mea
sure
s of
cen
tral t
ende
ncy
- m
easu
res
of d
ispe
rsio
n, in
clud
ing
extre
mes
and
ou
tlier
s
Org
aniz
e da
ta a
ccor
ding
to m
ore
than
on
e cr
iteria
88 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
-
med
ian
- m
ode
Id
entif
y th
e la
rges
t and
sm
alle
st s
core
s in
a d
ata
set a
nd d
eter
min
e th
e di
ffere
nce
betw
een
them
in o
rder
to
dete
rmin
e th
e sp
read
of t
he d
ata
(rang
e)
Repr
esen
t da
ta •
Dra
w a
var
iety
of g
raph
s by
ha
nd/te
chno
logy
to d
ispl
ay a
nd
inte
rpre
t dat
a (g
roup
ed a
nd
ungr
oupe
d) i
nclu
ding
:
- ba
r gra
phs
and
doub
le b
ar g
raph
s
- hi
stog
ram
s w
ith g
iven
inte
rval
s
- pi
e ch
arts
Int
erpret
data
•
Crit
ical
ly re
ad a
nd in
terp
ret d
ata
repr
esen
ted
in:
-
wor
ds
-
bar g
raph
s
- do
uble
bar
gra
phs
-
pie
char
ts
-
hist
ogra
ms
- m
ode
Su
mm
ariz
e da
ta u
sing
mea
sure
s of
di
sper
sion
, inc
ludi
ng:
- ra
nge
- ex
trem
es
Repr
esen
t da
ta •
Dra
w a
var
iety
of g
raph
s by
ha
nd/te
chno
logy
to d
ispl
ay a
nd
inte
rpre
t dat
a in
clud
ing:
- ba
r gra
phs
and
doub
le b
ar g
raph
s
- hi
stog
ram
s w
ith g
iven
and
ow
n in
terv
als
-
pie
char
ts
-
brok
en-li
ne g
raph
s Int
erpret
data
•
Crit
ical
ly re
ad a
nd in
terp
ret d
ata
repr
esen
ted
in:
-
wor
ds
-
bar g
raph
s
- do
uble
bar
gra
phs
-
pie
char
ts
-
hist
ogra
ms
-
brok
en-li
ne g
raph
s
Repr
esen
t da
ta •
Dra
w a
var
iety
of g
raph
s by
ha
nd/te
chno
logy
to d
ispl
ay a
nd
inte
rpre
t dat
a in
clud
ing:
- ba
r gra
phs
and
doub
le b
ar g
raph
s
- hi
stog
ram
s w
ith g
iven
and
ow
n in
terv
als
-
pie
char
ts
-
brok
en-li
ne g
raph
s -
scat
ter p
lots
Int
erpret
data
•
Crit
ical
ly re
ad a
nd in
terp
ret d
ata
repr
esen
ted
in a
var
iety
of w
ays
• C
ritic
ally
com
pare
two
sets
of d
ata
rela
ted
to th
e sa
me
issu
e
89MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Analy
se d
ata
• C
ritic
ally
ana
lyse
dat
a by
ans
wer
ing
ques
tions
rela
ted
to:
-
data
cat
egor
ies,
incl
udin
g da
ta
inte
rval
s
- da
ta s
ourc
es a
nd c
onte
xts
-
cent
ral t
ende
ncie
s (m
ean,
mod
e,
med
ian)
- sc
ales
use
d on
gra
phs
Repo
rt da
ta •
Sum
mar
ize
data
in s
hort
para
grap
hs th
at
incl
ude
-
draw
ing
conc
lusi
ons
abou
t the
dat
a
- m
akin
g pr
edic
tions
bas
ed o
n th
e da
ta
-
iden
tifyi
ng s
ourc
es o
f erro
r and
bia
s in
th
e da
ta
-
choo
sing
app
ropr
iate
sum
mar
y st
atis
tics
for t
he
data
(mea
n, m
edia
n, m
ode)
Analy
se d
ata
• C
ritic
ally
ana
lyse
dat
a by
ans
wer
ing
ques
tions
rela
ted
to:
-
data
cat
egor
ies,
incl
udin
g da
ta
inte
rval
s
- da
ta s
ourc
es a
nd c
onte
xts
-
cent
ral t
ende
ncie
s (m
ean,
mod
e,
med
ian)
- sc
ales
use
d on
gra
phs
-
sam
ples
and
pop
ulat
ions
- di
sper
sion
of d
ata
-
erro
r and
bia
s in
the
data
Re
port
data
• Su
mm
ariz
e da
ta in
sho
rt pa
ragr
aphs
that
in
clud
e
- dr
awin
g co
nclu
sion
s ab
out t
he d
ata
-
mak
ing
pred
ictio
ns b
ased
on
the
data
- id
entif
ying
sou
rces
of e
rror a
nd b
ias
in
the
data
- ch
oosi
ng a
ppro
pria
te s
umm
ary
stat
istic
s fo
r the
dat
a (m
ean,
med
ian,
m
ode,
rang
e)
-
the
role
of e
xtre
mes
in th
e da
ta
Analy
se d
ata
• C
ritic
ally
ana
lyse
dat
a by
ans
wer
ing
ques
tions
rela
ted
to:
-
data
col
lect
ion
met
hods
- su
mm
ary o
f dat
a
- so
urce
s of
erro
r and
bia
s in
the
data
Re
port
data
• Su
mm
ariz
e da
ta in
sho
rt pa
ragr
aphs
that
in
clud
e
- dr
awin
g co
nclu
sion
s ab
out t
he d
ata
-
mak
ing
pred
ictio
ns b
ased
on
the
data
- m
akin
g co
mpa
rison
s be
twee
n tw
o se
ts
of d
ata
-
iden
tifyi
ng s
ourc
es o
f erro
r and
bia
s in
th
e da
ta
-
choo
sing
app
ropr
iate
sum
mar
y st
atis
tics
for t
he
data
(mea
n, m
edia
n, m
ode,
ra
nge)
- th
e ro
le o
f ext
rem
es a
nd o
utlie
rs in
the
data
90 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Sugg
este
d m
etho
dolo
gy
The
follo
win
g ar
e ne
w in
Gra
de 7
•
sam
ples
and
pop
ulat
ions
•
mul
tiple
cho
ice
ques
tionn
aire
s •
stem
-and
-leaf
dis
play
s •
grou
ping
dat
a in
inte
rval
s •
mea
n •
rang
e •
hist
ogra
ms
• sc
ales
on
grap
hs
Data
sets
and c
ontex
ts Le
arne
rs s
houl
d be
exp
osed
to a
var
iety
of c
onte
xts
that
dea
l with
soc
ial a
nd e
nviro
nmen
tal i
ssue
s, a
nd s
houl
d w
ork
with
giv
en d
ata
sets
, rep
rese
nted
in a
var
iety
of w
ays,
that
incl
ude
big
num
ber r
ange
s, p
erce
ntag
es a
nd d
ecim
al fr
actio
ns. L
earn
ers
shou
ld th
en
prac
tice
orga
nizi
ng a
nd s
umm
ariz
ing
this
dat
a, a
naly
sing
and
inte
rpre
ting
the
data
, and
writ
ing
a re
port
abou
t the
dat
a Co
mplet
e a d
ata cy
cle
Lear
ners
sho
uld
com
plet
e at
leas
t one
dat
a cy
cle
for t
he y
ear,
star
ting
with
pos
ing
thei
r ow
n qu
estio
ns, s
elec
ting
the
sour
ces
and
met
hod
for c
olle
ctin
g da
ta, r
ecor
ding
the
data
, org
aniz
ing
the
data
, rep
rese
ntin
g th
e da
ta, t
hen
anal
ysin
g,
sum
mar
izin
g, in
terp
retin
g an
d re
porti
ng th
e da
ta. C
halle
nge
lear
ners
to th
ink
abou
t wha
t kin
ds o
f que
stio
ns a
nd d
ata
need
to b
e co
llect
ed
to b
e re
pres
ente
d in
a h
isto
gram
, pie
cha
rt, b
ar g
raph
or a
line
grap
h.
Repr
esen
ting
data
• D
raw
ing
pie
char
ts to
repr
esen
t dat
a do
not
hav
e to
be
accu
rate
ly d
raw
n w
ith a
com
pass
and
pro
tract
or, e
tc. L
earn
ers
can
use
any
roun
d ob
ject
to d
raw
a c
ircle
, the
n di
vide
the
circ
le in
to h
alve
s an
d qu
arte
rs a
nd e
ight
hs if
nee
ded,
as
a gu
ide
to e
stim
ate
the
prop
ortio
ns o
f the
circ
le th
at n
eed
to b
e sh
own
to re
pres
ent t
he d
ata.
Wha
t is
impo
rtant
is th
at th
e va
lues
or p
erce
ntag
es a
ssoc
iate
d w
ith th
e da
ta, a
re s
how
n pr
opor
tiona
lly o
n th
e pi
e ch
art.
D
raw
ing,
read
ing
and
inte
rpre
ting
pie
char
ts is
a u
sefu
l con
text
to re
-vis
it eq
uiva
lenc
e be
twee
n fra
ctio
ns a
nd p
erce
ntag
es, e
.g. 2
5% o
f th
e da
ta is
repr
esen
ted
by a
1 4 se
ctor
of t
he c
ircle
.
91MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
• It
is a
lso
a co
ntex
t in
whi
ch le
arne
rs c
an fi
nd p
erce
ntag
es o
f who
le n
umbe
rs e
.g. i
f 25%
of 3
00 le
arne
rs li
ke ru
gby,
how
man
y (a
ctua
l nu
mbe
r) le
arne
rs li
ke ru
gby?
•
His
togr
ams
are
used
to re
pres
ent g
roup
ed d
ata
show
n in
inte
rval
s on
the
horiz
onta
l axi
s of
the
grap
h. P
oint
out
the
diffe
renc
es
betw
een
hist
ogra
ms
and
bar g
raph
s, in
par
ticul
ar b
ar g
raph
s th
at re
pres
ent d
iscr
ete
data
(e.g
. fav
ourit
e sp
orts
) com
pare
d to
hi
stog
ram
s th
at s
how
cat
egor
ies
in c
onse
cutiv
e, n
on- o
verla
ppin
g in
terv
als,
(e.g
. tes
t sco
res
out o
f 100
sho
wn
in in
terv
als
of 1
0). T
he
bars
on
bar g
raph
s do
not
hav
e to
touc
h ea
ch o
ther
, whi
le in
a h
isto
gram
they
hav
e to
touc
h si
nce
they
sho
w c
onse
cutiv
e in
terv
als.
De
velop
ing c
ritica
l ana
lysis
skills
•
Lear
ners
sho
uld
com
pare
the
sam
e da
ta re
pres
ente
d in
diff
eren
t way
s e.
g. in
a p
ie c
hart
or a
bar
gra
ph o
r a ta
ble,
and
dis
cuss
wha
t in
form
atio
n is
sho
wn
and
wha
t is
hidd
en; t
hey
shou
ld e
valu
ate
wha
t for
m o
f rep
rese
ntat
ion
wor
ks b
est f
or th
e gi
ven
data
. •
Lear
ners
sho
uld
com
pare
gra
phs
on th
e sa
me
topi
c bu
t whe
re d
ata
has
been
col
lect
ed fr
om d
iffer
ent g
roup
s of
peo
ple,
at d
iffer
ent
times
, in
diffe
rent
pla
ces
or in
diff
eren
t way
s. H
ere
lear
ners
sho
uld
disc
uss
diffe
renc
es b
etw
een
the
data
with
an
awar
enes
s of
bia
s re
late
d to
the
impa
ct o
f dat
a so
urce
s an
d m
etho
ds o
f dat
a co
llect
ion
on th
e in
terp
reta
tion
of th
e da
ta.
• Le
arne
rs s
houl
d co
mpa
re d
iffer
ent w
ays
of s
umm
aris
ing
the
sam
e da
ta s
ets,
dev
elop
ing
an a
war
enes
s of
how
dat
a re
porti
ng c
an b
e m
anip
ulat
ed; e
valu
atin
g w
hich
sum
mar
y st
atis
tics
best
repr
esen
t the
dat
a.
• Le
arne
rs s
houl
d co
mpa
re g
raph
s of
the
sam
e da
ta, w
here
the
scal
es o
f the
gra
phs
are
diffe
rent
. Her
e le
arne
rs s
houl
d di
scus
s di
ffere
nces
with
an
awar
enes
s of
how
repr
esen
tatio
n of
dat
a ca
n be
man
ipul
ated
; the
y sh
ould
eva
luat
e w
hich
form
of
repr
esen
tatio
n w
orks
bes
t for
the
give
n da
ta.
Lear
ners
sho
uld
writ
e re
ports
on
the
data
in s
hort
para
grap
hs.
Gra
de 8
W
hat is
diffe
rent
to G
rade
7?
The
follo
win
g ar
e ne
w in
Gra
de 8
•
extre
mes
•
brok
en li
ne g
raph
s •
disp
ersi
on o
f dat
a •
erro
r and
bia
s in
dat
a
.
92 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Repr
esen
ting
data
• Br
oken
-line
gra
phs
refe
r to
dat
a gr
aphs
that
rep
rese
nt d
ata
poin
ts jo
ined
by
a lin
e an
d ar
e no
t the
sam
e as
stra
ight
line
gra
phs
that
are
dra
wn
usin
g th
e eq
uatio
n of
the
line.
•
Brok
en-li
ne g
raph
s ar
e us
ed to
repr
esen
t dat
a th
at c
hang
es c
ontin
uous
ly o
ver t
ime,
e.g
. ave
rage
dai
ly te
mpe
ratu
re fo
r a m
onth
. Ea
ch d
ay’s
tem
pera
ture
is re
pres
ente
d w
ith a
poi
nt o
n th
e gr
aph,
and
onc
e th
e w
hole
mon
th h
as b
een
plot
ted,
the
poin
ts a
re jo
ined
to
sho
w a
bro
ken-
line
grap
h.
• Br
oken
-line
gra
phs
are
usef
ul to
read
‘tre
nds’
and
pat
tern
s in
the
data
, for
pre
dict
ive
purp
oses
e.g
. will
the
tem
pera
ture
s go
up
or
dow
n in
the
next
mon
th.
Deve
loping
crit
ical a
nalys
is sk
ills
. • Le
arne
rs s
houl
d co
mpa
re d
ata
on th
e sa
me
topi
c, w
here
one
set
of d
ata
has
extre
mes
, and
dis
cuss
diff
eren
ces
with
an
awar
enes
s of
the
effe
ct o
f the
ext
rem
es o
n th
e in
terp
reta
tion
of th
e da
ta, i
n pa
rticu
lar,
extre
mes
affe
ct th
e ra
nge.
•
Lear
ners
sho
uld
writ
e re
ports
on
the
data
in s
hort
para
grap
hs.
Gra
de 9
W
hat is
diffe
rent
to G
rade
8?
• O
rgan
izin
g da
ta a
ccor
ding
to m
ore
than
one
crit
eria
•
Out
liers
•
Scat
ter p
lots
Co
mplet
e a d
ata cy
cle
Al
l wha
t is
done
gra
de 8
plu
s sc
atte
r plo
t. Re
pres
entin
g da
ta
93MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
All w
hat i
s do
ne in
gra
de 7
and
8 in
clud
ing
wha
t is
men
tione
d be
low
•
A sc
atte
r plo
t is
used
to re
pres
ent d
ata
that
invo
lves
two
diffe
rent
crit
eria
and
the
grap
h is
use
d to
look
at t
he re
latio
nshi
p be
twee
n th
e tw
o cr
iteria
. e.g
. How
doe
s th
e pe
rform
ance
of l
earn
ers
in m
athe
mat
ics
com
pare
to th
eir p
erfo
rman
ce in
Eng
lish?
Eac
h po
int o
n th
e gr
aph
repr
esen
ts th
e re
sults
of o
ne le
arne
r in
Mat
hem
atic
s an
d En
glis
h. A
fter a
ll th
e re
sults
hav
e be
en p
lotte
d, y
ou c
an c
ompa
re th
e re
latio
nshi
p be
twee
n pe
rform
ance
in E
nglis
h an
d m
athe
mat
ics
for a
ll th
e le
arne
rs i.
e. if
they
sco
re h
igh
in M
athe
mat
ics,
do
they
als
o sc
ore
high
in E
nglis
h? o
r, if
they
sco
re h
igh
in m
athe
mat
ics
do th
ey s
core
low
in E
nglis
h, o
r is
ther
e no
rela
tions
hip
betw
een
wha
t the
y sc
ore
on m
athe
mat
ics
to w
hat t
hey
scor
e in
Eng
lish?
•
The
scat
ter p
lot a
llow
s on
e to
see
tren
ds a
nd m
ake
pred
ictio
ns, a
s w
ell a
s id
entif
y ou
tlier
s in
the
data
.
LT
SM
Res
ourc
es
Cha
rts, s
quar
e pa
pers
, col
our p
ens.
G
rade
7
G
rade
8
Gra
de 9
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es (R
15,1
6)
WB
2 A
ctiv
ities
(126
- 14
0)
WB
1 Ac
tiviti
es (R
16a&
b)
WB
2 Ac
tiviti
es (9
2 - 1
04)
WB
1 Ac
tiviti
es (R
16a&
b)
WB
2 Ac
tiviti
es
(123
– 1
43)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
94 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
95MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
4
TER
M 3
Dat
e : f
rom
....
to ..
....
T
ime:
5 h
ours
G
rade
7
8 9
Topi
c 2D
GEO
MET
RY
& P
RO
PER
TIES
OF
TRIA
NG
LES
AND
TH
EOR
EM O
F PY
THAG
OR
AS
CAP
S pa
ges
46 a
nd 5
5 10
5 13
8 Sk
ills a
nd K
now
ledg
e Cl
assif
ying
2D s
hape
s
Des
crib
e, s
ort,
nam
e an
d co
mpa
re
trian
gles
acc
ordi
ng to
thei
r sid
es
and
angl
es, f
ocus
ing
on:
- eq
uila
tera
l tria
ngle
s -
isos
cele
s tri
angl
es
- rig
ht-a
ngle
d tri
angl
es
U
se a
ppro
pria
te fo
rmul
ae to
cal
cula
te
the
area
of:
- sq
uare
s -
rect
angl
es
- tri
angl
es
Deve
lop
and
use
the
theo
rem
of P
ythag
oras
Inve
stig
ate
the
rela
tions
hip
betw
een
the
leng
ths
of th
e si
des
of a
rig
ht-
angl
ed
trian
gle
to
deve
lop
the
Theo
rem
of P
ytha
gora
s
Det
erm
ine
whe
ther
a tr
iang
le is
a
right
-ang
led
trian
gle
or n
ot if
the
leng
th o
f the
thre
e si
des
of th
e tri
angl
e ar
e kn
own
U
se th
e Th
eore
m o
f Pyt
hago
ras
to
calc
ulat
e a
mis
sing
leng
th in
a ri
ght-
angl
ed tr
iang
le, l
eavi
ng ir
ratio
nal
answ
ers
in s
urd
form
Solve
pro
blem
s us
ing
the
theo
rem
of
Pyth
agor
as
U
se th
e Th
eore
m o
f Pyt
hago
ras
to
solv
e pr
oble
ms
invo
lvin
g un
know
n le
ngth
s in
geo
met
ric fi
gure
s th
at
cont
ain
right
-ang
led
trian
gles
Sugg
este
d m
etho
dolo
gy
Trian
gles
Le
arne
rs s
houl
d be
abl
e to
dis
tingu
ish
betw
een
an e
quila
tera
l tria
ngle
(all
the
side
s ar
e eq
ual),
an
isos
cele
s tri
angl
e (a
t lea
st
two
equa
l sid
es) a
nd a
righ
t- an
gled
tria
ngle
(one
righ
t-ang
le).
Form
ulae
lear
ners
sho
uld
know
and
use
are
:
- ar
ea o
f a s
quar
e =
l2
- ar
ea o
f a re
ctan
gle
= l x
b
- Ar
ea of
a tri
angle
= 1 2
(b×
h)
96 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Gra
de 8
and
9
Th
e th
eore
m o
f Pyt
hago
ras
is n
ew in
Gra
de 8
.
It
is im
porta
nt th
at le
arne
rs u
nder
stan
d th
at th
e Th
eore
m o
f Pyt
hago
ras
appl
ies
only
to ri
ght-a
ngle
d tri
angl
es.
Th
e Th
eore
m o
f Pyt
hago
ras
is b
asic
ally
a fo
rmul
a to
cal
cula
te u
nkno
wn
leng
th o
f sid
es in
righ
t-ang
led
trian
gles
.
In
the
FET
phas
e, th
e Th
eore
m o
f Pyt
hago
ras
is c
ruci
al to
the
furth
er s
tudy
of G
eom
etry
and
Trig
onom
etry
In
par
ticul
ar, t
he T
heor
em o
f Pyt
hago
ras
can
be th
e fir
st s
tep
in c
alcu
latio
ns o
f per
imet
ers
or a
reas
of c
ompo
site
figu
res,
whe
n
one
of th
e fig
ures
is a
righ
t-ang
led
trian
gle
with
an
unkn
own
leng
th.
G
rade
9 w
ill so
lve
prob
lem
s in
volv
ing
geom
etric
figu
res
LT
SM
Res
ourc
es
Cha
rts w
ith p
ictu
res
and
shap
es.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es (R
10)
WB
2 (1
42)
WB
2 Ac
tiviti
es (7
7- 8
1)
W
B 1
Activ
ities
(58;
59)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
97MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
GR
ADE
LESS
ON
PLA
N 5
TER
M 3
Dat
e : f
rom
....
to ..
....
Ti
me:
7 h
ours
G
rade
7
8 9
Topi
c AR
EA A
ND
PER
IMET
ER O
F 2D
SH
APES
C
APS
page
s 56
10
6 - 1
07
139
- 140
Sk
ills a
nd K
now
ledg
e Ar
ea an
d per
imete
r
Cal
cula
te th
e pe
rimet
er o
f re
gula
r and
irre
gula
r pol
ygon
s
U
se a
ppro
pria
te fo
rmul
ae to
ca
lcul
ate
perim
eter
and
are
a of
: -
squa
res
- re
ctan
gles
-
trian
gles
Ca
lculat
ions a
nd so
lving
prob
lems
• So
lve
prob
lem
s in
volv
ing
perim
eter
an
d ar
ea o
f pol
ygon
s
Area
and p
erim
eter
U
se a
ppro
pria
te fo
rmul
ae to
cal
cula
te
perim
eter
an
d ar
ea o
f: -
squa
res
- re
ctan
gles
-
trian
gles
-
circ
les
• C
alcu
late
the
area
s of
pol
ygon
s,
to a
t lea
st 2
dec
imal
pla
ces,
by
deco
mpo
sing
them
into
re
ctan
gles
and
/or t
riang
les
• U
se a
nd d
escr
ibe
the
rela
tions
hip
betw
een
the
radi
us,
diam
eter
and
circ
umfe
renc
e of
a
circ
le in
cal
cula
tions
• U
se a
nd d
escr
ibe
the
rela
tions
hip
betw
een
the
radi
us a
nd a
rea
of a
circ
le in
ca
lcul
atio
ns
Calcu
latio
ns an
d solv
ing pr
oblem
s •
Solv
e pr
oble
ms,
with
or w
ithou
t a
calc
ulat
or, i
nvol
ving
per
imet
er
and
area
of p
olyg
ons
and
circ
les
Area
and p
erim
eter
U
se a
ppro
pria
te fo
rmul
ae a
nd
conv
ersi
ons
betw
een
SI u
nits
, to
solv
e pr
oble
ms
and
calc
ulat
e pe
rimet
er a
nd a
rea
of:
- po
lygo
ns
- ci
rcle
s
Inve
stig
ate
how
dou
blin
g an
y or
all
of th
e di
men
sion
s of
a 2
D fi
gure
af
fect
s its
per
imet
er a
nd it
s ar
ea
98 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
• C
alcu
late
to a
t lea
st 1
dec
imal
pla
ce
• U
se a
nd c
onve
rt be
twee
n ap
prop
riate
SI
uni
ts,
incl
udin
g:
-
mm
2 ↔
cm
2
- cm
2 ↔
m2
• C
alcu
late
to a
t lea
st 2
dec
imal
pla
ces
• U
se a
nd d
escr
ibe
the
mea
ning
of t
he
irrat
iona
l num
ber P
i (π
) in
calc
ulat
ions
in
volv
ing
circ
les
• U
se a
nd c
onve
rt be
twee
n ap
prop
riate
SI
uni
ts,
in
clud
ing:
mm
2 ↔
cm
2 ↔
m2 ↔
km
2 Su
gges
ted
met
hodo
logy
Fo
rmul
ae le
arne
rs s
houl
d kn
ow a
nd u
se a
re:
-
perim
eter
of a
squ
are
= 4s
-
perim
eter
of a
rect
angl
e =
2(l +
b) o
r 2l +
2b
- ar
ea o
f a s
quar
e =
l2
- ar
ea o
f a re
ctan
gle
= l x
b
- ar
ea o
f a tr
iang
le =
2
(b x
h)
Solvi
ng e
quati
ons
usin
g fo
rmula
e •
The
use
of fo
rmul
ae p
rovi
des
a co
ntex
t to
prac
tice
solv
ing
equa
tions
by
insp
ectio
n.
Exam
ple
1.
If
the
perim
eter
of a
squ
are
is 3
2 cm
wha
t is
the
leng
th o
f eac
h si
de?
Lear
ners
sho
uld
writ
e th
is a
s:
4s =
32
and
solv
e by
insp
ectio
n by
ask
ing:
4 ti
mes
wha
t will
be 3
2?
2.
If
the
area
of a
rect
angl
e is
200
cm
2 , a
nd it
s le
ngth
is 5
0 cm
wha
t is
its w
idth
? Le
arne
rs s
houl
d w
rite
this
as:
50
x b
= 20
and
sol
ve b
y in
spec
tion
by a
skin
g: 5
0 tim
es w
hat w
ill be
200
? Fo
r area
s of t
riang
les:
• M
ake
sure
lear
ners
kno
w th
at th
e he
ight
of a
tria
ngle
is a
line
seg
men
t dra
wn
from
any
ver
tex
perp
endi
cula
r to
the
oppo
site
sid
e.
• Po
int o
ut th
at e
very
tria
ngle
has
3 b
ases
, eac
h w
ith a
rela
ted
heig
ht o
r alti
tude
.
99MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
• Fo
r con
vers
ions,
note:
-
if 1
cm =
10
mm
then
1 c
m2
= 10
0 m
m2
- if
1 m
= 1
00 c
m th
en 1
m2
= 10
000
cm
2 G
rade
8
Wha
t is di
ffere
nt to
Gra
de 7?
•
Area
s of
pol
ygon
s by
dec
ompo
sitio
n •
Circ
umfe
renc
e an
d ar
ea o
f a c
ircle
•
Form
ulae
lear
ners
sho
uld
know
and
use
: -
diam
eter
of a
circ
le: d
= 2
r -
circ
umfe
renc
e of
circ
le: c
= π
d or
2π
r -
area
of a
circ
le: A
= π
r2
Solvi
ng e
quati
ons
usin
g fo
rmula
e Th
e us
e of
form
ulae
pro
vide
s a
cont
ext t
o pr
actis
e so
lvin
g eq
uatio
ns b
y in
spec
tion
or u
sing
add
itive
or m
ultip
licat
ive
inve
rses
Ci
rcles
•
mak
e su
re le
arne
rs c
an id
entif
y th
e ce
ntre
, rad
ius,
dia
met
er a
nd c
ircum
fere
nce
of th
e ci
rcle
. •
Spen
d tim
e in
vest
igat
ing
the
rela
tions
hip
betw
een
radi
us, c
ircum
fere
nce
and
diam
eter
, so
that
lear
ners
dev
elop
a s
ense
of
whe
re th
e irr
atio
nal n
umbe
r Pi (π
) is
deriv
ed fr
om.
• D
evel
op a
n un
ders
tand
ing
of π
, mak
ing
sure
lear
ners
und
erst
and
that
:
- π
repr
esen
ts th
e va
lue
of th
e ci
rcum
fere
nce
divi
ded
by th
e di
amet
er, f
or a
ny c
ircle
- π
is a
n irr
atio
nal n
umbe
r and
is g
iven
as
3,14
1 59
2 65
4 co
rrect
to 9
dec
imal
pl
aces
on
the
calc
ulat
or
722
or 3
,14
are
appr
oxim
ate
ratio
nal v
alue
s of
π in
eve
ryda
y us
e.
100 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Gra
de 9
W
hat is
diffe
rent
to G
rade
8?
• Th
e ca
lcul
atio
ns a
re th
e sa
me
as in
Gra
de 8
, but
lear
ners
can
find
per
imet
ers
and
area
s of
mor
e co
mpo
site
and
com
plex
fig
ures
•
Poly
gons
can
incl
ude
trape
zium
s, p
aral
lelo
gram
s, rh
ombi
and
kite
s Fo
rmul
a fo
r lea
rner
s to
kno
w a
nd u
se.
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴
𝑜𝑜𝑜𝑜 𝐴𝐴
𝐴𝐴ℎ𝑜𝑜𝑜𝑜
𝑜𝑜𝑜𝑜𝑜𝑜
=𝑙𝑙𝐴𝐴𝑙𝑙𝑙𝑙
𝑙𝑙ℎ ×
ℎ𝐴𝐴𝑒𝑒𝑙𝑙ℎ𝑙𝑙
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴
𝑜𝑜𝑜𝑜 𝐴𝐴
𝑘𝑘𝑒𝑒𝑙𝑙𝐴𝐴
=1 2
(𝑑𝑑𝑒𝑒𝐴𝐴𝑙𝑙𝑜𝑜𝑙𝑙𝐴𝐴𝑙𝑙 1
×𝑑𝑑𝑒𝑒𝐴𝐴𝑙𝑙
𝑜𝑜𝑙𝑙𝐴𝐴𝑙𝑙
2)
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴
𝑜𝑜𝑜𝑜 𝐴𝐴
𝑝𝑝𝐴𝐴𝐴𝐴𝐴𝐴𝑙𝑙𝑙𝑙𝐴𝐴𝑙𝑙𝑜𝑜𝑙𝑙𝐴𝐴𝐴𝐴𝑜𝑜
=𝑜𝑜𝐴𝐴𝑜𝑜𝐴𝐴
×ℎ𝐴𝐴𝑒𝑒𝑙𝑙ℎ𝑙𝑙
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴
𝑜𝑜𝑜𝑜 𝐴𝐴
𝑙𝑙𝐴𝐴𝐴𝐴𝑝𝑝𝐴𝐴𝑡𝑡𝑒𝑒𝑜𝑜𝑜𝑜
=1 2
(𝑜𝑜𝑜𝑜𝑜𝑜
𝑜𝑜𝑜𝑜 𝑝𝑝𝐴𝐴
𝐴𝐴𝐴𝐴𝑙𝑙𝑙𝑙𝐴𝐴𝑙𝑙 𝑜𝑜𝑒𝑒𝑑𝑑𝐴𝐴)
×ℎ𝐴𝐴𝑒𝑒𝑙𝑙ℎ𝑙𝑙
Th
e us
e of
form
ulae
pro
vide
s a
cont
ext t
o pr
actis
e so
lvin
g eq
uatio
ns b
y in
spec
tion
or u
sing
add
itive
or m
ultip
licat
ive
inve
rses
. E
xpla
natio
n of
the
effe
ct o
f dou
blin
g of
any
dim
ensi
ons
of 2
D.
LT
SM
Res
ourc
es
Met
er s
tick,
trun
dle
whe
el, m
eter
stic
k, c
harts
, squ
are
pape
rs.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es
(52
- 55)
WB
1 Ac
tiviti
es (R
14)
WB
2 Ac
tiviti
es
( 82
- 86)
WB
1 Ac
tiviti
es
(R14
; 60-
64)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
101MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
102 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
6
TER
M 3
Dat
e : f
rom
....
to ..
....
Tim
e:
8 ho
urs
Gra
de
7 8
9 To
pic
SUR
FAC
E AR
EA A
ND
VO
LUM
E O
F 3D
C
APS
page
s 57
10
8 14
6 Sk
ills a
nd
Know
ledg
e Su
rface
area
and
volu
me
U
se a
ppro
pria
te fo
rmul
ae to
ca
lcul
ate
the
surfa
ce
area
, vol
ume
and
capa
city
of:
- cu
bes
- re
ctan
gula
r pris
ms
D
escr
ibe
the
inte
rrela
tions
hip
betw
een
surfa
ce a
rea
and
volu
me
of th
e ob
ject
s m
entio
ned
abov
e Ca
lculat
ions a
nd so
lving
prob
lems
So
lve
prob
lem
s in
volv
ing
surfa
ce a
rea,
vol
ume
and
capa
city
Surfa
ce ar
ea a
nd v
olume
Use
app
ropr
iate
form
ulae
to c
alcu
late
the
surfa
ce
area
, vol
ume
and
capa
city
of:
- cu
bes
- re
ctan
gula
r pris
ms
- tri
angu
lar p
rism
s
D
escr
ibe
the
inte
rrela
tions
hip
betw
een
surfa
ce a
rea
and
volu
me
of th
e ob
ject
s m
entio
ned
abov
e Ca
lculat
ions a
nd so
lving
prob
lems
So
lve
prob
lem
s, w
ith o
r with
out a
ca
lcul
ator
, inv
olvi
ng s
urfa
ce a
rea,
vo
lum
e an
d ca
paci
ty
Surfa
ce ar
ea a
nd v
olume
Use
app
ropr
iate
form
ulae
and
co
nver
sion
s be
twee
n SI
uni
ts to
sol
ve
prob
lem
s an
d ca
lcul
ate
the
surfa
ce
area
, vol
ume
and
capa
city
of:
- cu
bes
- re
ctan
gula
r pris
ms
- tri
angu
lar p
rism
s -
cylin
ders
Inve
stig
ate
how
dou
blin
g an
y or
all
the
dim
ensi
ons
of ri
ght p
rism
s an
d cy
linde
rs a
ffect
s th
eir v
olum
e
103MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
U
se a
nd c
onve
rt be
twee
n ap
prop
riate
SI u
nits
, in
clud
ing:
-
mm
2 ↔
cm
2 -
cm2
↔ m
2 -
mm
3 ↔
cm
3 -
cm3
↔ m
3
Use
equ
ival
ence
bet
wee
n un
its w
hen
solv
ing
prob
lem
s:
- 1
cm3
↔ 1
ml
- 1
m3
↔ 1
kl
U
se a
nd c
onve
rt be
twee
n ap
prop
riate
SI
units
, in
clud
ing:
-
mm
2 ↔
cm
2 ↔
m2
↔ k
m2
- m
m3
↔ c
m3
↔ m
3 -
ml (
cm3 )
↔ l
↔ k
l
Sugg
este
d m
etho
dolo
gy
Wha
t is di
ffere
nt to
Gra
de 6?
Fo
rmul
ae le
arne
rs s
houl
d kn
ow a
nd u
se:
- th
e vo
lum
e of
a p
rism
= th
e ar
ea o
f the
bas
e x
the
heig
ht
- th
e su
rface
are
a of
a p
rism
= th
e su
m o
f the
are
a of
all
its fa
ces
- th
e vo
lum
e of
a c
ube
= l3
- th
e vo
lum
e of
a re
ctan
gula
r pris
m =
l x
b x
h
Fo
r con
vers
ions
, not
e:
- if
1 cm
= 1
0 cm
then
1 c
m3
= 1
000
mm
3 an
d
- if
1 m
= 1
0 cm
then
1 m
3 =
1 00
0 00
0 m
m3
or 1
000
000
or 1
06 c
m3 .
- an
obj
ect w
ith a
vol
ume
of 1
cm
3 w
ill di
spla
ce e
xact
ly 1
ml o
f wat
er; a
nd
- an
obj
ect w
ith a
vol
ume
of 1
m3
will
disp
lace
exa
ctly
1 k
l of w
ater
104 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Em
phas
ize
that
the
amou
nt o
f spa
ce in
side
a p
rism
is c
alle
d its
cap
acity
; and
the
amou
nt o
f spa
ce o
ccup
ied
by a
pris
m is
cal
led
its v
olum
e.
In
vest
igat
e th
e ne
ts o
f cub
es a
nd re
ctan
gula
r pris
ms
in o
rder
to d
educ
e fo
rmul
ae fo
r cal
cula
ting
thei
r sur
face
are
as.
Gra
de 8
Wha
t is di
ffere
nt to
Gra
de 7?
Surfa
ce a
rea
and
volu
me
of tr
iang
ular
pris
ms
Fo
rmul
ae le
arne
rs s
houl
d kn
ow a
nd u
se:
- th
e vo
lum
e of
a tr
iang
ular
pris
m =
(1b
x h)
x h
eigh
t of p
rism
Fo
r con
vers
ions
, not
e:
- if
1 cm
= 1
0 m
m th
en 1
cm
3 =
1 00
0 m
m3
and
- if
1 m
= 1
00 c
m th
en 1
m3
= 1
000
000
cm3
or 1
06 c
m3
- an
obj
ect w
ith a
vol
ume
of 1
cm
3 w
ill di
spla
ce e
xact
ly 1
ml o
f wat
er
- an
obj
ect w
ith a
vol
ume
of 1
m3
will
disp
lace
exa
ctly
1 k
l of w
ater
.
Em
phas
ize
that
the
amou
nt o
f spa
ce in
side
a p
rism
is c
alle
d its
cap
acity
; and
the
amou
nt o
f spa
ce o
ccup
ied
by a
pris
m is
cal
led
its v
olum
e.
In
vest
igat
e th
e ne
ts o
f cub
es a
nd re
ctan
gula
r pris
ms
in o
rder
to d
educ
e fo
rmul
ae fo
r cal
cula
ting
thei
r sur
face
are
as.
105MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Gra
de 9
Wha
t is di
ffere
nt to
Gra
de 8?
Surfa
ce a
rea
and
volu
me
of c
ylin
ders
Fo
rmul
ae le
arne
rs s
houl
d kn
ow a
nd u
se:
- th
e vo
lum
e of
a c
ylin
der =
(πr2
) x th
e he
ight
of t
he c
ylin
der
Fo
r con
vers
ions
, not
e:
- if
1 cm
= 1
0 m
m th
en 1
cm
3 =
1 00
0 m
m3 ;
and
- if
1 m
= 1
00 c
m th
en 1
m3
= 1
000
000
or 1
06 c
m3
7
LTSM
Res
ourc
es
Empt
y bo
xes,
geo
solid
s, m
easu
ring
jugs
,cub
es.
G
rade
7
Gra
de 8
G
rade
9
Wor
kboo
k re
fere
nce
WB
1 Ac
tiviti
es
(56
- 64)
W
B 2
Activ
ity (1
43)
WB
1 Ac
tiviti
es (R
15a&
b)
WB
2 Ac
tiviti
es
(87
- 91)
WB
1 Ac
tiviti
es (R
15a&
b)
WB
2
Activ
ities
(1
00 –
104
)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
106 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
mar
ks m
ax
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
7
TER
M 3
Dat
e : f
rom
....
to ..
....
T
ime:
4 ho
urs
G
rade
7
8 9
Topi
c G
RAP
HS
CAP
S pa
ges
65
114
145
Skills
and
Kn
owle
dge
Inte
rpre
ting
grap
hs
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
spe
cial
focu
s on
the
follo
win
g tre
nds
and
feat
ures
:
linea
r or n
on-li
near
cons
tant
, inc
reas
ing
or d
ecre
asin
g D
raw
ing
grap
hs
Dra
w g
loba
l gra
phs
from
giv
en d
escr
iptio
ns o
f a
pro
blem
situ
atio
n, id
entif
ying
feat
ures
list
ed
abov
e
Inte
rpre
ting
grap
hs
Rev
ise
the
follo
win
g do
ne in
Gra
de 7
:
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
a s
peci
al fo
cus
on th
e fo
llow
ing
trend
s an
d fe
atur
es:
♦♦
linea
r or n
on-li
near
♦
♦ co
nsta
nt, i
ncre
asin
g or
dec
reas
ing
Ex
tend
the
focu
s on
feat
ures
of g
raph
s to
incl
ude:
♦♦
max
imum
or m
inim
um
♦♦ d
iscr
ete
or c
ontin
uous
D
raw
ing
grap
hs
D
raw
glo
bal g
raph
s fro
m g
iven
de
scrip
tions
of a
pro
blem
situ
atio
n,
iden
tifyi
ng fe
atur
es li
sted
abo
ve
Inte
rpre
ting
grap
hs
Rev
ise
the
follo
win
g do
ne in
Gra
de 8
:
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
a s
peci
al fo
cus
on th
e fo
llow
ing
trend
s an
d fe
atur
es:
♦♦ li
near
or n
on-li
near
♦♦
con
stan
t, in
crea
sing
or d
ecre
asin
g
♦♦
max
imum
or m
inim
um
♦♦ d
iscr
ete
or c
ontin
uous
Exte
nd th
e ab
ove
with
spe
cial
focu
s on
th
e fo
llow
ing
feat
ures
of l
inea
r gra
phs:
♦♦
x-in
terc
ept a
nd y
-inte
rcep
t
♦
♦ gr
adie
nt
Dra
win
g gr
aphs
R
evis
e th
e fo
llow
ing
done
in G
rade
8:
dr
aw g
loba
l gra
phs
from
giv
en
desc
riptio
ns o
f a p
robl
em s
ituat
ion,
107MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
U
se ta
bles
of o
rder
ed p
airs
to p
lot
poin
ts a
nd d
raw
gra
phs
on th
e C
arte
sian
pla
ne
iden
tifyi
ng fe
atur
es li
sted
abo
ve.
u
se ta
bles
of o
rder
ed p
airs
to p
lot p
oint
s an
d dr
aw g
raph
s on
the
Car
tesi
an p
lane
Exte
nd th
e ab
ove
with
a s
peci
al fo
cus
on:
♦♦
dra
win
g lin
ear g
raph
s fro
m g
iven
eq
uatio
ns
♦♦
det
erm
ine
equa
tions
from
giv
en
linea
r gra
phs
Sugg
este
d m
etho
dolo
gy
In th
e In
term
edia
te P
hase
lear
ners
enc
ount
ered
gra
phs
in th
e fo
rm o
f dat
a ba
r gra
phs
and
pie
char
ts. T
his
mea
ns th
ey d
o ha
ve s
ome
expe
rienc
e re
adin
g an
d in
terp
retin
g gr
aphs
. How
ever
, in
the
Seni
or P
hase
, the
y ar
e in
trodu
ced
to li
ne g
raph
s th
at s
how
func
tiona
l re
latio
nshi
ps d
escr
ibed
in te
rms
of d
epen
dent
and
inde
pend
ent v
aria
bles
. In
Gra
de 7
, the
focu
s is
on
draw
ing,
ana
lysi
ng a
nd in
terp
retin
g gl
obal
gra
phs
only
. Tha
t is,
lear
ners
do
not h
ave
to p
lot p
oint
s to
dra
w
grap
hs a
nd th
ey fo
cus
on th
e fe
atur
es o
f the
glo
bal r
elat
ions
hip
show
n in
the
grap
h.
Exam
ples
of c
onte
xts
for g
loba
l gra
phs
incl
ude:
the
rela
tions
hip
betw
een
time
and
dist
ance
trav
elle
d
the
rela
tions
hip
betw
een
tem
pera
ture
and
tim
e ov
er w
hich
it is
mea
sure
d
the
rela
tions
hip
betw
een
rain
fall
and
time
over
whi
ch it
is m
easu
red,
etc
. In
Gra
de 8
:
New
feat
ures
of g
loba
l gra
phs,
nam
ely,
max
imum
and
min
imum
; dis
cret
e an
d co
ntin
uous
, are
intro
duce
d
Lear
ners
plo
t poi
nts
to d
raw
gra
phs
See
Gra
de 7
abo
ve fo
r exa
mpl
es o
f con
text
s fo
r glo
bal g
raph
s.
Exam
ples
of d
raw
ing
grap
hs b
y pl
ottin
g po
ints
a)
C
ompl
ete
the
tabl
e of
ord
ered
pai
rs b
elow
for t
he e
quat
ion
y =
x +
3
108 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Now
, plo
t the
abo
ve c
o-or
dina
te p
oint
s on
the
Car
tesi
an p
lane
. Joi
n po
ints
to fo
rm a
gra
ph.
b)
Com
plet
e th
e ta
ble
of o
rder
ed p
airs
bel
ow fo
r the
equ
atio
n y
= x
2 + 3
N
ow, p
lot t
he a
bove
co-
ordi
nate
poi
nts
on th
e C
arte
sian
pla
ne. J
oin
poin
ts to
form
a g
raph
. In
Gra
de 9
, •
x-in
terc
ept,
y-in
terc
ept a
nd g
radi
ent o
f lin
ear g
raph
s ar
e in
trodu
ced
• Le
arne
rs d
raw
line
ar g
raph
s fro
m g
iven
equ
atio
ns
• Th
ey d
eter
min
e eq
uatio
ns o
f lin
ear g
raph
s Le
arne
rs s
houl
d co
ntin
ue to
ana
lyse
and
inte
rpre
t gra
phs
of p
robl
em s
ituat
ions
. In
vest
igat
ing
linea
r gra
phs
To
ske
tch
linea
r gra
phs
from
giv
en e
quat
ions
, lea
rner
s sh
ould
firs
t dra
w u
p a
tabl
e of
ord
ered
pai
rs, t
hat i
nclu
des
the
inte
rcep
t poi
nts
(x; 0
) and
(0; y
), an
d th
en p
lotti
ng th
e po
ints
.
Lear
ners
sho
uld
inve
stig
ate
grad
ient
s by
com
parin
g 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
𝑣𝑣ℎ𝑣𝑣𝑎𝑎
𝑎𝑎𝑣𝑣ℎ𝑜𝑜
𝑣𝑣𝑣𝑣𝑜𝑜𝑜𝑜𝑎𝑎𝑣𝑣𝑣𝑣𝑣𝑣
𝑣𝑣ℎ𝑣𝑣𝑎𝑎
𝑎𝑎𝑣𝑣 b
etw
een
any
two
poin
ts o
n a
stra
ight
line
gra
ph.
Le
arne
rs s
houl
d al
so in
vest
igat
e th
e re
latio
nshi
p be
twee
n th
e va
lue
of th
e gr
adie
nt a
nd th
e co
effic
ient
of x
in th
e eq
uatio
n of
a s
traig
ht
line
grap
h.
Le
arne
rs s
houl
d co
mpa
re y
-inte
rcep
ts o
f lin
ear g
raph
s to
the
valu
e of
the
cons
tant
in th
e eq
uatio
n of
the
stra
ight
line
gra
ph.
Ex
ampl
es o
f lin
ear g
raph
s a)
Ske
tch
and
com
pare
the
grap
hs o
f: y
= 4
and
x =
4 b)
Ske
tch
and
com
pare
the
grap
hs o
f: y
= x
and
y =
–x
c) S
ketc
h an
d co
mpa
re th
e gr
aphs
of:
y =
2x; y
= 2
x +
1; y
= 2
x –
1 d)
Ske
tch
and
com
pare
the
grap
hs o
f: y
= 3x
; y =
4x;
y =
5x
e) S
ketc
h th
e gr
aphs
of:
y =
–3x
+ 2;
usi
ng th
e ta
ble
met
hod
f) D
eter
min
e th
e eq
uatio
n of
the
stra
ight
line
pas
sing
thro
ugh
the
follo
win
g po
ints
:
109MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LTSM
Res
ourc
es
G
raph
pap
er, r
uler
s, c
olou
r pen
s.
G
rade
7
Gra
de 8
G
rade
9
W
orkb
ook
refe
renc
e
WB
2 Ac
tiviti
es (8
0 - 8
5)
W
B 2
Activ
ities
(114
- 12
0)
W
B 1
Activ
ities
(88
– 99
)
Text
book
re
fere
nce
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
110 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
111MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LESS
ON P
LANS
: TER
M 4
112 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LESS
ON
PLA
NS:
TER
M 4
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
1
TER
M 4
Dat
e : f
rom
....
to ..
....
T
ime:
4 ho
urs
G
rade
7
8 9
Topi
c FU
NC
TIO
NS
AND
REL
ATIO
NSH
IPS
CAP
S pa
ges
62
88 -
89
129
Skills
and
Kn
owle
dge
Inpu
t and
out
put v
alue
s D
eter
min
e in
put v
alue
s, o
utpu
t val
ues
or ru
les
for
patte
rns
and
rela
tions
hips
usi
ng:
flo
w d
iagr
ams
ta
bles
form
ulae
Eq
uiva
lent
form
s D
eter
min
e, in
terp
ret a
nd ju
stify
equ
ival
ence
of
diffe
rent
des
crip
tions
of t
he s
ame
rela
tions
hip
or ru
le p
rese
nted
:
verb
ally
in fl
ow d
iagr
ams
in
tabl
es
by
form
ulae
by n
umbe
r sen
tenc
es
Inpu
t and
out
put v
alue
s D
eter
min
e in
put v
alue
s, o
utpu
t val
ues
or
rule
s fo
r pat
tern
s an
d re
latio
nshi
ps u
sing
:
flow
dia
gram
s
tabl
es
fo
rmul
ae
eq
uatio
ns
Equi
vale
nt fo
rms
• Det
erm
ine,
inte
rpre
t and
just
ify
equi
vale
nce
of d
iffer
ent d
escr
iptio
ns
of th
e sa
me
rela
tions
hip
or ru
le
pres
ente
d:
-- ve
rbal
ly
-- in
flow
dia
gram
s --
in ta
bles
--
by fo
rmul
ae
-- by
equ
atio
ns
Inpu
t and
out
put v
alue
s D
eter
min
e in
put v
alue
s, o
utpu
t val
ues
or
rule
s fo
r pat
tern
s an
d re
latio
nshi
ps u
sing
:
flow
dia
gram
s
tabl
es
fo
rmul
ae
eq
uatio
ns
Equi
vale
nt fo
rms
• Det
erm
ine,
inte
rpre
t and
just
ify
equi
vale
nce
of d
iffer
ent d
escr
iptio
ns o
f the
sa
me
rela
tions
hip
or ru
le
pres
ente
d:
ve
rbal
ly
in
flow
dia
gram
s
in ta
bles
by fo
rmul
ae
by
equ
atio
ns
by
gra
phs
on a
Car
tesi
an p
lane
Sugg
este
d m
etho
dolo
gy
In th
is p
hase
, it i
s us
eful
to b
egin
to s
peci
fy w
heth
er th
e in
put v
alue
s ar
e na
tura
l num
bers
, or i
nteg
ers
or ra
tiona
l num
bers
. Thi
s bu
ilds
lear
ners
’ aw
aren
ess
of th
e do
mai
n of
inpu
t val
ues.
Hen
ce, t
o fin
d ou
tput
val
ues,
lear
ners
sho
uld
be g
iven
the
rule
/form
ula
as w
ell a
s th
e do
mai
n of
the
inpu
t val
ues.
In
Gra
de 7
, the
focu
s is
on
findi
ng o
utpu
t val
ues
for g
iven
form
ulae
and
inpu
t val
ues.
Not
e, w
hen
lear
ners
find
inpu
t or o
utpu
t val
ues
for
113MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
give
n ru
les
or fo
rmul
ae, t
hey
are
actu
ally
find
ing
the
num
eric
al v
alue
of a
lgeb
raic
exp
ress
ions
usi
ng s
ubst
itutio
n.
Exam
ples
U
se th
e fo
rmul
a fo
r the
are
a of
a re
ctan
gle:
A =
l x
b to
cal
cula
te th
e fo
llow
ing:
a)
The
are
a, if
the
leng
th is
4 c
m a
nd th
e w
idth
is 2
cm
b)
The
leng
th, i
f the
are
a is
20
cm2
and
the
wid
th is
4 c
m
c) T
he w
idth
, if t
he a
rea
is 2
4 cm
2 an
d th
e le
ngth
is 8
cm
Le
arne
rs c
an w
rite
thes
e as
num
ber s
ente
nces
, and
sol
ve b
y in
spec
tion.
In
Gra
de 8
:
The
focu
s is
on
findi
ng in
put o
r out
put v
alue
s us
ing
give
n eq
uatio
ns
Th
e ru
les
and
num
ber p
atte
rns
for w
hich
lear
ners
hav
e to
find
inpu
t and
out
put v
alue
s ar
e ex
tend
ed to
incl
ude
patte
rns
with
m
ultip
licat
ion
and
divi
sion
of i
nteg
ers
and
num
bers
in e
xpon
entia
l for
m.
Flow
dia
gram
s ar
e re
pres
enta
tions
of f
unct
iona
l rel
atio
nshi
ps. H
ence
, whe
n us
ing
flow
dia
gram
s, th
e co
rresp
onde
nce
betw
een
inpu
t and
ou
tput
val
ues
shou
ld b
e cl
ear i
n its
repr
esen
tatio
nal f
orm
i.e.
the
first
inpu
t pro
duce
s th
e fir
st o
utpu
t; th
e se
cond
inpu
t pro
duce
s th
e se
cond
ou
tput
, etc
. Ex
ampl
es
a) U
se th
e gi
ven
rule
to c
alcu
late
the
valu
es o
f t fo
r eac
h va
lue
of p
, whe
re p
is a
nat
ural
num
ber.
In th
is k
ind
of fl
ow d
iagr
am, l
earn
ers
can
also
be
aske
d to
det
erm
ine
the
valu
e of
p fo
r a g
iven
t va
lue.
b)
Det
erm
ine
the
rule
for c
alcu
latin
g th
e ou
tput
val
ue fo
r eve
ry g
iven
inpu
t val
ue in
the
flow
dia
gram
bel
ow.
114 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
In
flow
dia
gram
s su
ch a
s th
ese,
mor
e th
an o
ne ru
le m
ight
be
poss
ible
to d
escr
ibe
the
rela
tions
hip
betw
een
inpu
t and
out
put v
alue
s. T
he
rule
s ar
e ac
cept
able
if th
ey m
atch
the
give
n in
put v
alue
s to
the
corre
spon
ding
out
put v
alue
s.
c) If
the
rule
for f
indi
ng y
in th
e ta
ble
belo
w is
y =
–3x
– 1
, cal
cula
te y
for t
he g
iven
x v
alue
s:
b)
Des
crib
e th
e re
latio
nshi
p be
twee
n th
e nu
mbe
rs in
the
top
row
and
bot
tom
row
in th
e ta
ble.
The
n w
rite
dow
n th
e va
lue
of m
and
n
In
tabl
es s
uch
as th
ese,
mor
e th
an o
ne ru
le m
ight
be
poss
ible
to d
escr
ibe
the
rela
tions
hip
betw
een
x an
d y
valu
es. T
he ru
les
are
acce
ptab
le if
they
mat
ch th
e gi
ven
inpu
t val
ues
to th
e co
rresp
ondi
ng o
utpu
t val
ues.
For
exa
mpl
e, th
e ru
le y
= x
–3
desc
ribes
the
rela
tions
hip
betw
een
the
x va
lues
and
giv
en y
val
ues.
To
find
m a
nd n
, lea
rner
s ha
ve to
sub
stitu
te th
e co
rresp
ondi
ng v
alue
s fo
r x o
r y in
th
e ru
le a
nd s
olve
the
equa
tion
by in
spec
tion.
In
Gra
de 9
, lea
rner
s co
nsol
idat
e w
ork
with
inpu
t and
out
put v
alue
s do
ne in
Gra
de 8
. The
y sh
ould
con
tinue
to fi
nd in
put o
r out
put v
alue
s in
flo
w d
iagr
ams,
tabl
es, f
orm
ulae
and
equ
atio
ns. L
earn
ers
shou
ld b
egin
to re
cogn
ize
equi
vale
nt re
pres
enta
tions
of t
he s
ame
rela
tions
hips
sh
own
as a
n eq
uatio
n, a
set
of o
rder
ed p
airs
in a
tabl
e or
on
a gr
aph.
115MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Exam
ples
a)
If th
e ru
le fo
r fin
ding
in th
e ta
ble
belo
w is
y =
1 2 x +
1, d
eter
min
e th
e va
lues
of y
for t
he g
iven
x v
alue
s:
b
) Des
crib
e th
e re
latio
nshi
p be
twee
n th
e nu
mbe
rs in
the
top
row
and
thos
e in
the
botto
m ro
w in
the
tabl
e. T
hen
writ
e do
wn
the
valu
e of
m
and
n
In
tabl
es s
uch
as th
ese,
mor
e th
an o
ne ru
le m
ight
be
poss
ible
to d
escr
ibe
the
rela
tions
hip
betw
een
x an
d y
valu
es. T
he ru
les
are
acce
ptab
le if
they
mat
ch th
e gi
ven
inpu
t val
ues
to th
e co
rresp
ondi
ng o
utpu
t val
ues.
For
exa
mpl
e, th
e ru
le y
= 2
x –
3 de
scrib
es th
e re
latio
nshi
p be
twee
n th
e gi
ven
valu
es fo
r x a
nd y
. To
find
m a
nd n
, lea
rner
s ha
ve to
sub
stitu
te th
e co
rresp
ondi
ng v
alue
s fo
r x o
r y in
to th
e ru
le a
nd s
olve
the
equa
tion
by in
spec
tion.
LT
SM
Res
ourc
es
C
harts
.
G
rade
7
Gra
de 8
Gra
de 9
Wor
kboo
k re
fere
nce
W
B 1
Activ
ities
(48
- 51)
W
B 2
Activ
ities
(72
– 73
)
WB
2 Ac
tiviti
es (7
2 - 7
3)
Text
book
re
fere
nce
116 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
117MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
2
TER
M 4
Dat
e : f
rom
....
to ..
....
T
ime:
4 ho
urs
G
rade
7
8 9
Topi
c G
RAP
HS
CAP
S pa
ges
65
114
145
Skills
and
Kn
owle
dge
Inte
rpre
ting
grap
hs
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
spe
cial
focu
s on
the
follo
win
g tre
nds
and
feat
ures
:
linea
r or n
on-li
near
cons
tant
, inc
reas
ing
or d
ecre
asin
g D
raw
ing
grap
hs
Dra
w g
loba
l gra
phs
from
giv
en d
escr
iptio
ns o
f a
pro
blem
situ
atio
n, id
entif
ying
feat
ures
list
ed
abov
e
Inte
rpre
ting
grap
hs
Rev
ise
the
follo
win
g do
ne in
Gra
de 7
:
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
a s
peci
al fo
cus
on th
e fo
llow
ing
trend
s an
d fe
atur
es:
♦♦
linea
r or n
on-li
near
♦
♦ co
nsta
nt, i
ncre
asin
g or
dec
reas
ing
Ex
tend
the
focu
s on
feat
ures
of g
raph
s to
incl
ude:
♦♦
max
imum
or m
inim
um
♦♦ d
iscr
ete
or c
ontin
uous
D
raw
ing
grap
hs
D
raw
glo
bal g
raph
s fro
m g
iven
de
scrip
tions
of a
pro
blem
situ
atio
n,
iden
tifyi
ng fe
atur
es li
sted
abo
ve
U
se ta
bles
of o
rder
ed p
airs
to p
lot
poin
ts a
nd d
raw
gra
phs
on th
e C
arte
sian
pla
ne
Inte
rpre
ting
grap
hs
Rev
ise
the
follo
win
g do
ne in
Gra
de 8
:
Anal
yse
and
inte
rpre
t glo
bal g
raph
s of
pr
oble
m s
ituat
ions
, with
a s
peci
al fo
cus
on th
e fo
llow
ing
trend
s an
d fe
atur
es:
♦♦ li
near
or n
on-li
near
♦♦
con
stan
t, in
crea
sing
or d
ecre
asin
g
♦♦
max
imum
or m
inim
um
♦♦ d
iscr
ete
or c
ontin
uous
Exte
nd th
e ab
ove
with
spe
cial
focu
s on
th
e fo
llow
ing
feat
ures
of l
inea
r gra
phs:
♦♦
x-in
terc
ept a
nd y
-inte
rcep
t
♦
♦ gr
adie
nt
Dra
win
g gr
aphs
R
evis
e th
e fo
llow
ing
done
in G
rade
8:
dr
aw g
loba
l gra
phs
from
giv
en
desc
riptio
ns o
f a p
robl
em s
ituat
ion,
id
entif
ying
feat
ures
list
ed a
bove
.
use
tabl
es o
f ord
ered
pai
rs to
plo
t poi
nts
and
draw
gra
phs
on th
e C
arte
sian
pla
ne
Ex
tend
the
abov
e w
ith a
spe
cial
focu
s on
:
♦♦
dra
win
g lin
ear g
raph
s fro
m g
iven
eq
uatio
ns
♦♦ d
eter
min
e eq
uatio
ns fr
om g
iven
lin
ear g
raph
s
118 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Sugg
este
d m
etho
dolo
gy
In th
e In
term
edia
te P
hase
lear
ners
enc
ount
ered
gra
phs
in th
e fo
rm o
f dat
a ba
r gra
phs
and
pie
char
ts. T
his
mea
ns th
ey d
o ha
ve s
ome
expe
rienc
e re
adin
g an
d in
terp
retin
g gr
aphs
. How
ever
, in
the
Seni
or P
hase
, the
y ar
e in
trodu
ced
to li
ne g
raph
s th
at s
how
func
tiona
l re
latio
nshi
ps d
escr
ibed
in te
rms
of d
epen
dent
and
inde
pend
ent v
aria
bles
. In
Gra
de 7
, the
focu
s is
on
draw
ing,
ana
lysi
ng a
nd in
terp
retin
g gl
obal
gra
phs
only
. Tha
t is,
lear
ners
do
not h
ave
to p
lot p
oint
s to
dra
w
grap
hs a
nd th
ey fo
cus
on th
e fe
atur
es o
f the
glo
bal r
elat
ions
hip
show
n in
the
grap
h.
Exam
ples
of c
onte
xts
for g
loba
l gra
phs
incl
ude:
the
rela
tions
hip
betw
een
time
and
dist
ance
trav
elle
d
the
rela
tions
hip
betw
een
tem
pera
ture
and
tim
e ov
er w
hich
it is
mea
sure
d
the
rela
tions
hip
betw
een
rain
fall
and
time
over
whi
ch it
is m
easu
red,
etc
. In
Gra
de 8
:
New
feat
ures
of g
loba
l gra
phs,
nam
ely,
max
imum
and
min
imum
; dis
cret
e an
d co
ntin
uous
, are
intro
duce
d
Lear
ners
plo
t poi
nts
to d
raw
gra
phs
See
Gra
de 7
abo
ve fo
r exa
mpl
es o
f con
text
s fo
r glo
bal g
raph
s.
Exam
ples
of d
raw
ing
grap
hs b
y pl
ottin
g po
ints
c)
C
ompl
ete
the
tabl
e of
ord
ered
pai
rs b
elow
for t
he e
quat
ion
y =
x +
3
N
ow, p
lot t
he a
bove
co-
ordi
nate
poi
nts
on th
e C
arte
sian
pla
ne. J
oin
poin
ts to
form
a g
raph
. d)
C
ompl
ete
the
tabl
e of
ord
ered
pai
rs b
elow
for t
he e
quat
ion
y =
x2 +
3
119MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
N
ow, p
lot t
he a
bove
co-
ordi
nate
poi
nts
on th
e C
arte
sian
pla
ne. J
oin
poin
ts to
form
a g
raph
. In
Gra
de 9
, •
x-in
terc
ept,
y-in
terc
ept a
nd g
radi
ent o
f lin
ear g
raph
s ar
e in
trodu
ced
• Le
arne
rs d
raw
line
ar g
raph
s fro
m g
iven
equ
atio
ns
• Th
ey d
eter
min
e eq
uatio
ns o
f lin
ear g
raph
s Le
arne
rs s
houl
d co
ntin
ue to
ana
lyse
and
inte
rpre
t gra
phs
of p
robl
em s
ituat
ions
. In
vest
igat
ing
linea
r gra
phs
To
ske
tch
linea
r gra
phs
from
giv
en e
quat
ions
, lea
rner
s sh
ould
firs
t dra
w u
p a
tabl
e of
ord
ered
pai
rs, t
hat i
nclu
des
the
inte
rcep
t poi
nts
(x; 0
) and
(0; y
), an
d th
en p
lotti
ng th
e po
ints
.
Lear
ners
sho
uld
inve
stig
ate
grad
ient
s by
com
parin
g 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
𝑣𝑣ℎ𝑣𝑣𝑎𝑎
𝑎𝑎𝑣𝑣ℎ𝑜𝑜
𝑣𝑣𝑣𝑣𝑜𝑜𝑜𝑜𝑎𝑎𝑣𝑣𝑣𝑣𝑣𝑣
𝑣𝑣ℎ𝑣𝑣𝑎𝑎
𝑎𝑎𝑣𝑣 b
etw
een
any
two
poin
ts o
n a
stra
ight
line
gra
ph.
Le
arne
rs s
houl
d al
so in
vest
igat
e th
e re
latio
nshi
p be
twee
n th
e va
lue
of th
e gr
adie
nt a
nd th
e co
effic
ient
of x
in th
e eq
uatio
n of
a s
traig
ht
line
grap
h.
Le
arne
rs s
houl
d co
mpa
re y
-inte
rcep
ts o
f lin
ear g
raph
s to
the
valu
e of
the
cons
tant
in th
e eq
uatio
n of
the
stra
ight
line
gra
ph.
Ex
ampl
es o
f lin
ear g
raph
s a)
Ske
tch
and
com
pare
the
grap
hs o
f: y
= 4
and
x =
4 b)
Ske
tch
and
com
pare
the
grap
hs o
f: y
= x
and
y =
–x
c) S
ketc
h an
d co
mpa
re th
e gr
aphs
of:
y =
2x; y
= 2
x +
1; y
= 2
x –
1 d)
Ske
tch
and
com
pare
the
grap
hs o
f: y
= 3x
; y =
4x;
y =
5x
e) S
ketc
h th
e gr
aphs
of:
y =
–3x
+ 2;
usi
ng th
e ta
ble
met
hod
f) D
eter
min
e th
e eq
uatio
n of
the
stra
ight
line
pas
sing
thro
ugh
the
follo
win
g po
ints
:
120 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LTSM
Res
ourc
es
G
raph
pap
ers,
rule
rs, c
olou
r pen
s.
G
rade
7
Gra
de 8
G
rade
9
W
orkb
ook
refe
renc
e
WB
2 Ac
tiviti
es (8
0 - 8
5)
W
B 2
Activ
ities
(114
- 12
0)
W
B 1
Activ
ities
(88
– 99
)
Text
book
re
fere
nce
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
121MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
3
TER
M 4
Dat
e: fr
om ..
......
to ..
......
Tim
e:
5
hour
s
Gra
de
7 8
9 To
pic
CO
NST
RU
CTI
ON
OF
GEO
MET
RIC
FIG
UR
ES
CAP
S pa
ges
45
95
134
Skills
and
Kno
wle
dge
Mea
surin
g an
gles
Ac
cura
tely
use
a p
rotra
ctor
to
mea
sure
and
cla
ssify
ang
les:
< 90
° (a
cute
ang
les)
Rig
ht-a
ngle
s
> 90
° (o
btus
e an
gles
)
Stra
ight
ang
les
>
180°
(ref
lex
angl
es)
Con
stru
ctio
ns
Accu
rate
ly c
onst
ruct
geo
met
ric
figur
es a
ppro
pria
tely
usi
ng c
ompa
ss,
rule
r and
pro
tract
or, i
nclu
ding
:
angl
es, t
o on
e de
gree
of
accu
racy
circ
les
pa
ralle
l lin
es
pe
rpen
dicu
lar l
ines
Con
stru
ctio
ns
Accu
rate
ly c
onst
ruct
geo
met
ric fi
gure
s ap
prop
riate
ly u
sing
a c
ompa
ss, r
uler
and
pr
otra
ctor
, inc
ludi
ng:
Bi
sect
ing
lines
and
ang
les
Pe
rpen
dicu
lar l
ines
at a
giv
en p
oint
or
from
a g
iven
poi
nt
Tr
iang
les
Q
uadr
ilate
rals
Angl
es o
f 30°
, 45°
and
60°
and
thei
r m
ultip
les
with
out u
sing
a p
rotra
ctor
In
vest
igat
ing
prop
ertie
s of
geo
met
ric
figur
es
Con
stru
ctio
ns
Ac
cura
tely
con
stru
ct g
eom
etric
figu
res
appr
opria
tely
usi
ng a
com
pass
, rul
er a
nd
prot
ract
or, i
nclu
ding
bis
ectin
g an
gles
of
a tri
angl
e
C
onst
ruct
ang
les
of 4
5°, 3
0°, 6
0°an
d th
eir m
ultip
les
with
out u
sing
a p
rotra
ctor
In
vest
igat
ing
prop
ertie
s of
geo
met
ric
figur
es
122 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
By c
onst
ruct
ion,
inve
stig
ate
the
angl
es in
a
trian
gle,
focu
sing
on:
the
sum
of t
he in
terio
r ang
les
of tr
iang
les
th
e si
ze o
f ang
les
in a
n eq
uila
tera
l tri
angl
e
the
side
s an
d ba
se a
ngle
s of
an
isos
cele
s tri
angl
e By
con
stru
ctio
n, in
vest
igat
e si
des
and
angl
es in
qua
drila
tera
ls, f
ocus
ing
on:
th
e su
m o
f the
inte
rior a
ngle
s of
qu
adril
ater
als
th
e si
des
and
oppo
site
ang
les
of
para
llelo
gram
s
By
con
stru
ctio
n, in
vest
igat
e th
e an
gles
in
a tri
angl
e, fo
cusi
ng o
n th
e re
latio
nshi
p be
twee
n th
e ex
terio
r ang
le o
f a tr
iang
le
and
its in
terio
r ang
les
By
con
stru
ctio
n, e
xplo
re th
e m
inim
um
cond
ition
s fo
r tw
o tri
angl
es to
be
cong
ruen
t
By c
onst
ruct
ion,
inve
stig
ate
side
s,
angl
es a
nd d
iago
nals
in q
uadr
ilate
rals
, fo
cusi
ng o
n th
e di
agon
als
of re
ctan
gles
, sq
uare
s, p
aral
lelo
gram
s, rh
ombi
and
ki
tes
By
con
stru
ctio
n ex
plor
e th
e su
m o
f the
in
terio
r ang
les
of p
olyg
ons
Sugg
este
d m
etho
dolo
gy
In G
rade
7, l
earn
ers
M
easu
re a
ngle
s w
ith a
pro
tract
or
D
o ge
omet
ric c
onst
ruct
ions
usi
ng a
com
pass
, rul
er a
nd p
rotra
ctor
M
easu
ring
angl
es
Le
arne
rs h
ave
to b
e sh
own
how
to p
lace
the
prot
ract
or o
n th
e ar
m o
f the
ang
le to
be
mea
sure
d.
Le
arne
rs a
lso
have
to le
arn
how
to re
ad th
e si
ze o
f ang
les
on a
pro
tract
or.
Con
stru
ctio
ns
C
onst
ruct
ions
pro
vide
a u
sefu
l con
text
to e
xplo
re o
r con
solid
ate
know
ledg
e of
ang
les
and
shap
es.
Le
arne
rs h
ave
to b
e sh
own
how
to u
se a
com
pass
to d
raw
circ
les,
alth
ough
they
mig
ht h
ave
done
this
in G
rade
6.
Le
arne
rs s
houl
d be
aw
are
that
the
cent
re o
f the
circ
le is
at t
he fi
xed
poin
t of t
he c
ompa
ss a
nd th
e ra
dius
of t
he c
ircle
is
depe
nden
t on
how
wid
e th
e co
mpa
ss is
ope
ned
up.
M
ake
sure
lear
ners
und
erst
and
that
arc
s ar
e pa
rts o
f the
circ
les
of a
par
ticul
ar ra
dius
.
Initi
ally
, lea
rner
s ha
ve to
be
give
n ca
refu
l ins
truct
ions
abo
ut h
ow to
do
the
cons
truct
ions
of t
he v
ario
us s
hape
s
O
nce
they
are
com
forta
ble
with
the
appa
ratu
s an
d ca
n do
the
cons
truct
ions
, the
y ca
n pr
actis
e by
dra
win
g pa
ttern
s, fo
r ex
ampl
e of
circ
les
or p
aral
lel l
ines
. In
Gra
de 8
, all
the
cons
truct
ions
are
new
. In
Gra
de 7
, lea
rner
s on
ly c
onst
ruct
ed a
ngle
s, p
erpe
ndic
ular
and
par
alle
l lin
es. G
rade
8
lear
ners
als
o us
e co
nstru
ctio
ns to
exp
lore
pro
perti
es o
f tria
ngle
s an
d qu
adril
ater
als
123MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Con
stru
ctio
ns
C
onst
ruct
ions
pro
vide
a u
sefu
l con
text
to e
xplo
re o
r con
solid
ate
know
ledg
e of
ang
les
and
shap
es
M
ake
sure
lear
ners
are
com
pete
nt a
nd c
omfo
rtabl
e w
ith th
e us
e of
a c
ompa
ss a
nd k
now
how
to m
easu
re a
nd re
ad a
ngle
si
zes
on a
pro
tract
or
R
evis
e th
e co
nstru
ctio
ns o
f ang
les
if ne
cess
ary
befo
re p
roce
edin
g w
ith th
e ne
w c
onst
ruct
ions
Star
t with
the
cons
truct
ions
of l
ines
, so
that
lear
ners
can
firs
t exp
lore
ang
le re
latio
nshi
ps o
n st
raig
ht li
nes.
Whe
n co
nstru
ctin
g tri
angl
es le
arne
rs s
houl
d dr
aw o
n kn
own
prop
ertie
s an
d co
nstru
ctio
n of
circ
les.
Con
stru
ctio
n of
spe
cial
ang
les
with
out p
rotra
ctor
s ar
e do
ne b
y:
o bi
sect
ing
a rig
ht a
ngle
to g
et 4
5°
o dr
awin
g an
equ
ilate
ral t
riang
le to
get
60°
o
bise
ctin
g th
e an
gles
of a
n eq
uila
tera
l tria
ngle
to g
et 3
0°
In G
rade
9, l
earn
ers:
Bise
ct a
ngle
s in
a tr
iang
le
C
onst
ruct
30°
with
out a
pro
tract
or
In
vest
igat
ion
of n
ew p
rope
rties
of t
riang
les,
qua
drila
tera
ls a
nd p
olyg
ons
Con
stru
ctio
ns
Sam
e as
for G
rade
8.
LTSM
Res
ourc
es
Mat
hem
atic
al in
stru
men
ts, g
eom
etric
set
s, c
olou
r pen
s, c
harts
.
G
rade
7
Gra
de 8
Gra
de 9
Wor
kboo
k re
fere
nce
W
B-1
Act
iviti
es (2
5)
W
B 2
Activ
ities
(132
- 13
3)
WB
1 Ac
tiviti
es (3
9 –
46)
WB
2 Ac
tiviti
es (
121
– 12
2)
Text
book
refe
renc
e
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
124 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
4
TER
M 4
Dat
e : f
rom
....
to ..
....
T
ime:
6 ho
urs
G
rade
7
8 (6
hou
rs)
9 To
pic
TRAN
SFO
RM
ATIO
N G
EOM
ETR
Y C
APS
page
s 65
11
5 12
9 Sk
ills a
nd
Know
ledg
e Tr
ansf
orm
atio
ns
R
ecog
nise
, des
crib
e an
d pe
rform
tra
nsla
tions
, ref
lect
ions
and
rota
tions
with
ge
omet
ric fi
gure
s an
d sh
apes
on
squa
red
pape
r.
Id
entif
y an
d dr
aw li
nes
of s
ymm
etry
in
geom
etric
figu
res
Enla
rgem
ents
and
redu
ctio
ns
D
raw
enl
arge
men
ts a
nd re
duct
ions
of
geom
etric
figu
res
on s
quar
ed p
aper
and
co
mpa
re th
em in
term
s of
sha
pe a
nd s
ize
Tran
sfor
mat
ions
Rec
ogni
se, d
escr
ibe
and
perfo
rm
trans
form
atio
ns w
ith p
oint
s on
a c
o-or
dina
te p
lane
, foc
usin
g on
: -
Ref
lect
ing
a po
int i
n th
e X
-axi
s or
Y-
axis
-
Tran
slat
ing
a po
int w
ithin
and
ac
ross
qua
dran
ts
R
ecog
nise
, des
crib
e an
d pe
rform
tra
nsfo
rmat
ions
with
tria
ngle
s on
a c
o-or
dina
te p
lane
, foc
usin
g on
the
co-
ordi
nate
s of
the
verti
ces
whe
n:
- R
efle
ctin
g a
trian
gle
in th
e X
-axi
s or
Y-a
xis
- Tr
ansl
atin
g a
poin
t with
in a
nd
acro
ss q
uadr
ants
-
Rot
atin
g a
trian
gle
arou
nd th
e or
igin
En
larg
emen
ts a
nd re
duct
ions
Use
pro
porti
on to
des
crib
e th
e ef
fect
of
enla
rgem
ent o
r red
uctio
n on
are
a an
d
Tran
sfor
mat
ions
Rec
ogni
se, d
escr
ibe
and
perfo
rm
trans
form
atio
ns w
ith p
oint
s, li
ne
segm
ents
and
sim
ple
geom
etric
figu
res
on
a c
o-or
dina
te p
lane
, foc
usin
g on
: -
Ref
lect
ing
a po
int i
n th
e X
-axi
s or
Y-
axis
-
Tran
slat
ing
a po
int w
ithin
and
ac
ross
qua
dran
ts
- R
efle
ctin
g in
the
line
y =
x
Iden
tify
wha
t the
tran
sfor
mat
ion
of a
po
int i
s, if
giv
en th
e co
-ord
inat
es o
f its
im
age.
En
larg
emen
ts a
nd re
duct
ions
Use
pro
porti
on to
des
crib
e th
e ef
fect
of
enla
rgem
ent o
r red
uctio
n on
are
a an
d pe
rimet
er o
f geo
met
ric fi
gure
s.
125MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
perim
eter
of g
eom
etric
figu
res.
Inve
stig
ate
the
co-o
rdin
ates
of t
he
verti
ces
of fi
gure
s th
at h
ave
been
en
larg
ed o
r red
uced
by
a gi
ven
scal
e fa
ctor
.
Sugg
este
d m
etho
dolo
gy
Wha
t is
diffe
rent
to G
rade
6
Unl
ike
in G
rade
6,
Le
arne
rs in
Gra
de 7
hav
e to
do
trans
form
atio
ns o
n sq
uare
d pa
per f
or a
ccur
acy
and
for e
nabl
ing
them
to c
ompa
re th
e sh
ape
and
size
of
figu
res.
Al
l Gra
des
Le
arne
rs s
houl
d re
cogn
ize
that
tran
slat
ions
, ref
lect
ions
and
rota
tions
onl
y ch
ange
the
posi
tion
of th
e fig
ure
and
not t
he s
hape
or s
ize.
They
sho
uld
reco
gniz
e th
at th
e ab
ove
trans
form
atio
ns p
rodu
ce c
ongr
uent
figu
res.
They
sho
uld
reco
gniz
e th
at e
nlar
gem
ents
and
redu
ctio
ns c
hang
e si
ze o
f fig
ures
by
incr
easi
ng o
r dec
reas
ing
the
leng
th o
f sid
es, b
ut
keep
ing
angl
es th
e sa
me,
pro
duci
ng s
imila
r rat
her t
han
cong
ruen
t fig
ures
.
They
sho
uld
also
be
able
to w
ork
out t
he fa
ctor
of e
nlar
gem
ent o
r red
uctio
n.
G
rade
8 a
nd 9
•
Doi
ng tr
ansf
orm
atio
ns o
n a
co-o
rdin
ate
plan
e fo
cuse
s at
tent
ion
on th
e co
ordi
nate
s of
poi
nts
and
verti
ces
of s
hape
s.
• Le
arne
rs d
o no
t hav
e to
lear
n ge
nera
l rul
es fo
r the
tran
sfor
mat
ions
at t
his
stag
e, b
ut s
houl
d ex
plor
e th
e w
ay th
e co
-ord
inat
es o
f poi
nts
chan
ge w
hen
perfo
rmin
g di
ffere
nt tr
ansf
orm
atio
ns w
ith li
nes
or s
hape
s.
Exam
ples
of t
rans
form
atio
n pr
oble
ms
• Pl
ot p
oint
A(4
;3) a
nd A
', its
imag
e, a
fter r
efle
ctio
n in
: a)
th
e x-
axis
b)
th
e y-
axis
. •
Writ
e do
wn
the
co-o
rdin
ates
of T
' if T
(–2;
3) is
tran
slat
ed 4
uni
ts d
ownw
ards
. •
The
perim
eter
of s
quar
e ab
cd =
48c
m
a)
Writ
e do
wn
the
perim
eter
of t
he s
quar
e if
the
leng
th o
f eac
h si
de is
dou
bled
. b)
W
ill th
e ar
ea o
f the
enl
arge
d sq
uare
be
twic
e or
four
tim
es th
at o
f the
orig
inal
squ
are?
G
rade
9
Wha
t is
diffe
rent
to G
rade
8?
126 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
• R
efle
ctio
n in
the
line
y =
x •
Iden
tify
trans
form
atio
ns fr
om c
o-or
dina
te p
oint
s of
the
imag
e •
Co-
ordi
nate
s of
ver
tices
C
o-or
dina
te p
lane
•
Doi
ng tr
ansf
orm
atio
ns o
n th
e co
-ord
inat
e pl
ane
is a
n op
portu
nity
to p
ract
ise
read
ing
and
plot
ting
poin
ts w
ith o
rder
ed p
airs
, and
link
s to
dr
awin
g al
gebr
aic
grap
hs.
• M
ake
sure
lear
ners
kno
w h
ow to
plo
t poi
nts
on th
e co
-ord
inat
e pl
ane
and
can
read
the
co-o
rdin
ates
of p
oint
s of
f the
x-a
xis
and
y-ax
is.
• M
ake
sure
lear
ners
kno
w th
e co
nven
tion
for w
ritin
g or
dere
d pa
irs (x
; y)
• Po
int o
ut th
e di
ffere
nces
bet
wee
n th
e ax
es in
the
four
qua
dran
ts.
Focu
s of
tran
sfor
mat
ions
•
Doi
ng tr
ansf
orm
atio
ns o
n a
co-o
rdin
ate
plan
e fo
cuse
s at
tent
ion
on th
e co
ordi
nate
s of
poi
nts
and
verti
ces
of s
hape
s.
• Le
arne
rs s
houl
d be
gin
to s
ee p
atte
rns
in te
rms
of th
e co
-ord
inat
e po
ints
, for
the
diffe
rent
tran
sfor
mat
ions
, suc
h as
: --
for t
rans
latio
ns to
the
right
or l
eft,
the
x–va
lue
chan
ges
and
y–va
lue
stay
s th
e sa
me
-- fo
r tra
nsla
tions
up
or d
own,
the
y–va
lue
chan
ges
and
the
x–va
lue
stay
s th
e sa
me
-- fo
r ref
lect
ions
in th
e y–
axis
, the
x–v
alue
cha
nges
sig
n an
d th
e y–
valu
e st
ays
the
sam
e --
for r
efle
ctio
ns in
the
x–ax
is, t
he y
–val
ue c
hang
es s
ign
and
the
x–va
lue
stay
s th
e sa
me
-- fo
r ref
lect
ions
in th
e lin
e y
= x,
the
x–va
lue
and
y–va
lue
are
inte
rcha
nged
. •
Lear
ners
sho
uld
also
be
able
-- re
flect
ion
in th
e lin
e y
= x
• Id
entif
y w
hat t
he tr
ansf
orm
atio
n of
a p
oint
is, i
f giv
en th
e co
-ord
inat
es o
f its
imag
e En
larg
emen
ts a
nd re
duct
ions
•
Use
pro
porti
on to
des
crib
e th
e ef
fect
of e
nlar
gem
ent o
r red
uctio
n on
are
a an
d pe
rimet
er o
f geo
met
ric fi
gure
s •
Inve
stig
ate
the
co-o
rdin
ates
of t
he fi
gure
s th
at h
ave
been
enl
arge
d or
redu
ced
by a
giv
en s
cale
fact
or L
earn
ers
shou
ld a
lso
be a
ble
to
wor
k ou
t the
fact
or o
f enl
arge
men
t or r
educ
tion
of a
figu
re.
LT
SM
Res
ourc
es
R
uler
s, s
quar
e pa
per,
dotte
d pa
per.
127MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
G
rade
7
Gra
de 8
Gra
de 9
Wor
kboo
k re
fere
nce
W
B 2
Act
iviti
es (8
6 - 9
4)
W
B 1
Activ
ities
(R12
) W
B 2
Activ
ities
(121
- 12
6)
WB
1 Ac
tiviti
es (R
12)
WB
2 Ac
tiviti
es (1
05 –
113
)
Text
book
re
fere
nce
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t –
Test
: mar
ks
max
128 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
5
TER
M 4
Dat
e: fr
om ..
.. to
.....
.
Tim
e:
9
hour
s
Gra
de
7 8
9 To
pic
GEO
MET
RY
OF
3-D
OB
JEC
TS
CAP
S pa
ges
66
116
148
Skills
and
Kn
owle
dge
Cla
ssify
ing
3-D
obj
ects
Des
crib
e, s
ort a
nd c
ompa
re p
olyh
edra
in
term
s of
o
shap
e an
d nu
mbe
r of f
aces
o
num
ber o
f ver
tices
o
num
ber o
f edg
es
Bui
ldin
g 3-
D m
odel
Rev
ise
usin
g ne
ts to
cre
ate
mod
els
of
geom
etric
sol
ids,
incl
udin
g:
o cu
bes
o pr
ism
s
Cla
ssify
ing
3-D
obj
ects
Des
crib
e, n
ame
and
com
pare
the
5 Pl
aton
ic s
olid
s in
term
s of
the
shap
e an
d nu
mbe
r of f
aces
, the
num
ber o
f ve
rtice
s an
d th
e nu
mbe
r of e
dges
Bui
ldin
g 3-
D m
odel
s
Rev
ise
usin
g ne
ts to
mak
e m
odel
s of
ge
omet
ric s
olid
s, in
clud
ing:
o
cube
s o
pris
ms
o py
ram
ids
Cla
ssify
ing
3-D
obj
ects
Rev
ise
prop
ertie
s an
d de
finiti
ons
of th
e 5
Plat
onic
sol
ids
in te
rms
of th
e sh
ape
and
num
ber o
f fac
es, t
he n
umbe
r of
verti
ces
and
the
num
ber o
f edg
es
R
ecog
nize
and
des
crib
e th
e pr
oper
ties
of: o
sphe
res
o cy
linde
rs
B
uild
ing
3-D
mod
els
U
se n
ets
to c
reat
e m
odel
s of
geo
met
ric
solid
s, in
clud
ing:
o
cube
s o
pris
ms
o py
ram
ids
o cy
linde
rs
129MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Sugg
este
d m
etho
dolo
gy
GR
ADE
7 W
hat i
s di
ffere
nt to
Gra
de 6
?
Mos
t of t
his
wor
k co
nsol
idat
es w
hat h
as b
een
done
in G
rade
6.
Poly
hedr
a Ex
ampl
es o
f sor
ting
or g
roup
ing
cate
gorie
s:
cu
bes
(onl
y sq
uare
face
s)
re
ctan
gula
r pris
ms
(onl
y re
ctan
gula
r fac
es)
tri
angu
lar p
rism
s (o
nly
trian
gula
r and
rect
angu
lar f
aces
)
pyra
mid
s (s
quar
e an
d tri
angu
lar f
aces
)
cylin
ders
(circ
ular
and
rect
angu
lar f
aces
). U
sing
and
con
stru
ctin
g ne
ts
U
sing
and
con
stru
ctin
g ne
ts a
re u
sefu
l con
text
s fo
r exp
lorin
g or
con
solid
atin
g pr
oper
ties
of p
olyh
edra
.
Lear
ners
sho
uld
reco
gniz
e th
e ne
ts o
f diff
eren
t sol
ids.
Lear
ners
sho
uld
draw
ske
tche
s of
the
nets
usi
ng th
eir k
now
ledg
e of
sha
pe a
nd n
umbe
r of f
aces
of t
he s
olid
s, b
efor
e dr
awin
g an
d cu
tting
out
the
nets
to b
uild
mod
els.
The
cons
truct
ion
of n
ets
is b
ased
on
the
num
ber a
nd s
hape
of f
aces
of t
he s
olid
s, a
nd d
o no
t req
uire
mea
surin
g of
inte
rnal
ang
les
of
poly
gons
.
Lear
ners
hav
e to
wor
k ou
t the
rela
tive
posi
tion
of th
e fa
ces
of th
e ne
ts, a
nd u
se tr
ial a
nd e
rror t
o m
atch
up
the
edge
s an
d ve
rtice
s, in
or
der t
o bu
ild th
e 3D
obj
ect.
GR
ADE
8 W
hat i
s di
ffere
nt to
Gra
de 7
? •
Nam
ing
and
com
parin
g Pl
aton
ic s
olid
s •
Net
s of
pyr
amid
s
Plat
onic
sol
ids
• Pl
aton
ic s
olid
s ar
e a
spec
ial g
roup
of p
olyh
edra
that
hav
e fa
ces
that
are
con
grue
nt re
gula
r pol
ygon
s.
• Th
ere
are
only
5 P
lato
nic
solid
s:
o Te
trahe
dron
o
Hex
ahed
ron
(cub
e)
o O
ctah
edro
n o
Dod
ecah
edro
n o
Icos
ahed
rons
•
The
nam
e of
eac
h Pl
aton
ic s
olid
is d
eriv
ed fr
om it
s nu
mbe
r of f
aces
. •
Plat
onic
sol
ids
prov
ide
an in
tere
stin
g co
ntex
t in
whi
ch to
inve
stig
ate
the
rela
tions
hip
betw
een
the
num
ber o
f fac
es, v
ertic
es a
nd e
dges
.
130 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
By li
stin
g th
ese
prop
ertie
s fo
r all
the
Plat
onic
sol
ids,
lear
ners
can
inve
stig
ate
the
patte
rn th
at e
mer
ges,
to c
ome
up w
ith th
e ge
nera
l ru
le: V
– e
+ f
= 2,
whe
re V
= n
umbe
r of v
ertic
es; e
= n
umbe
r of e
dges
; f =
num
ber o
f fac
es.
U
sing
and
con
stru
ctin
g ne
ts
• U
sing
and
con
stru
ctin
g ne
ts a
re u
sefu
l con
text
s fo
r exp
lorin
g or
con
solid
atin
g pr
oper
ties
of p
olyh
edra
. •
Lear
ners
sho
uld
reco
gniz
e th
e ne
ts o
f diff
eren
t sol
ids.
•
Lear
ners
sho
uld
mak
e sk
etch
es o
f the
net
s us
ing
thei
r kno
wle
dge
of th
e sh
ape
and
num
ber o
f fac
es o
f the
sol
ids,
bef
ore
draw
ing
and
cutti
ng o
ut th
e ne
ts to
bui
ld m
odel
s.
• Si
nce
lear
ners
hav
e m
ore
know
ledg
e ab
out t
he s
ize
of a
ngle
s in
equ
ilate
ral t
riang
les,
and
can
mea
sure
ang
les,
thei
r con
stru
ctio
ns o
f ne
ts s
houl
d be
mor
e ac
cura
te.
• Le
arne
rs h
ave
to w
ork
out t
he re
lativ
e po
sitio
n of
face
s of
the
nets
in o
rder
to b
uild
the
3D m
odel
. G
RAD
E 9
Wha
t is
diffe
rent
to G
rade
8?
• Pr
oper
ties
of s
pher
es a
nd c
ylin
ders
•
Net
s of
cyl
inde
rs
Pl
aton
ic s
olid
s (S
ame
as G
rade
8)
U
sing
and
con
stru
ctin
g ne
ts (S
ame
as G
rade
9)
LT
SM
Res
ourc
es
So
lids,
3D
obj
ects
.
G
rade
7
Gra
de 8
Gra
de 9
Wor
kboo
k re
fere
nce
W
B 2
Act
iviti
es (9
6 - 1
04)
W
B 2
Activ
ities
(127
- 13
2)
WB
2 Ac
tiviti
es (1
14; 1
22
Text
book
re
fere
nce
HO
MEW
OR
K
131MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
132 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MAT
HEM
ATIC
S SE
NIO
R P
HAS
E
MU
LTI G
RAD
E LE
SSO
N P
LAN
6
TER
M 4
Dat
e: fr
om ..
......
to ..
......
Tim
e:
4.
5 ho
urs
G
rade
7
8 9
Topi
c PR
OB
ABIL
ITY
CAP
S pa
ges
73
117
152
Skills
and
Kn
owle
dge
Prob
abili
ty
P
erfo
rm s
impl
e ex
perim
ents
whe
re th
e po
ssib
le o
utco
mes
are
equ
ally
like
ly
and:
o
list t
he p
ossi
ble
outc
omes
bas
ed
on th
e co
nditi
ons
of th
e ac
tivity
o
dete
rmin
e th
e pr
obab
ility
of e
ach
poss
ible
out
com
e us
ing
the
defin
ition
of p
roba
bilit
y
Prob
abili
ty
C
onsi
der a
sim
ple
situ
atio
n (w
ith e
qual
ly
likel
y ou
tcom
es) t
hat c
an b
e de
scrib
ed
usin
g pr
obab
ility
and:
o
list a
ll th
e po
ssib
le o
utco
mes
o
dete
rmin
e th
e pr
obab
ility
of e
ach
poss
ible
o
outc
ome
usin
g th
e de
finiti
on o
f pr
obab
ility
o pr
edic
t with
reas
ons
the
rela
tive
frequ
ency
o
of th
e po
ssib
le o
utco
mes
for a
se
ries
of tr
ials
o
base
d on
pro
babi
lity
o co
mpa
re re
lativ
e fre
quen
cy w
ith
prob
abilit
y an
d ex
plai
ns p
ossi
ble
diffe
renc
es
Prob
abili
ty
C
onsi
der s
ituat
ions
with
equ
ally
pr
obab
le o
utco
mes
, and
: o
dete
rmin
e pr
obab
ilitie
s fo
r co
mpo
und
even
ts u
sing
two-
way
ta
bles
and
tree
dia
gram
s o
dete
rmin
e th
e pr
obab
ilitie
s fo
r ou
tcom
es o
f eve
nts
and
pred
ict t
heir
rela
tive
frequ
ency
in s
impl
e ex
perim
ents
o
com
pare
rela
tive
frequ
ency
with
pr
obab
ility
and
expl
ains
pos
sibl
e di
ffere
nces
Sugg
este
d m
etho
dolo
gy
GR
ADE
7 Pr
obab
ility
exp
erim
ents
In
the
Inte
rmed
iate
Pha
se le
arne
rs d
id e
xper
imen
ts w
ith c
oins
, dic
e an
d sp
inne
rs. I
n th
is g
rade
exp
erim
ents
can
be
done
with
oth
er
obje
cts,
like
, diff
eren
t col
oure
d bu
ttons
in a
bag
; cho
osin
g sp
ecifi
c ca
rds
from
a d
eck
of c
ards
, etc
. Ex
ampl
e If
you
toss
a c
oin
ther
e ar
e tw
o po
ssib
le o
utco
mes
(hea
d or
tail)
. The
pro
babi
lity
of a
hea
d is
1 2
whi
ch is
equ
ival
ent t
o 50
% (s
ince
it is
one
ou
t of t
wo
poss
ible
.
133MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
GR
ADE
8 Pr
obab
ility
exp
erim
ents
In
the
Inte
rmed
iate
Pha
se a
nd G
rade
7 le
arne
rs d
id p
roba
bilit
y ex
perim
ents
with
coi
ns, d
ice
and
spin
ners
. In
Gra
de 8
doi
ng a
ctua
l tria
ls o
f ex
perim
ents
bec
ome
less
impo
rtant
, and
lear
ners
sho
uld
cons
ider
pro
babi
lity
for h
ypot
hetic
al e
vent
s e.
g. th
e pr
obab
ility
of w
hite
as
a su
cces
sful
out
com
e on
a ro
ulet
te ta
ble,
or t
he p
roba
bilit
y of
get
ting
a C
oca
Col
a at
the
shop
if y
ou k
now
wha
t the
tota
l num
ber o
f drin
ks is
th
at th
ey s
tock
and
how
man
y ca
ns o
f Coc
a C
ola
they
hav
e.
Com
parin
g re
lativ
e fr
eque
ncy
and
prob
abili
ty
Th
e re
lativ
e fre
quen
cy is
the
obse
rved
num
ber o
f suc
cess
ful o
utco
mes
for a
fini
te s
ampl
e of
tria
ls.
Fo
r exa
mpl
e, if
you
toss
a c
oin
50 ti
mes
, the
resu
lts a
re 2
7 he
ads
and
23 ta
ils. D
efin
e a
head
as
a su
cces
sful
out
com
e. T
he
rela
tive
frequ
ency
of h
eads
is: 2
7 50
= 5
4%
Th
e pr
obab
ility
of a
hea
d is
50%
(one
of t
wo
likel
y ou
tcom
es).
The
diffe
renc
e be
twee
n th
e re
lativ
e fre
quen
cy o
f 54%
and
the
prob
abilit
y of
50%
is d
ue to
sm
all s
ampl
e si
ze.
Th
e m
ore
trial
s yo
u do
, the
clo
ser t
he re
lativ
e fre
quen
cy g
ets
to th
e pr
obab
ility.
Thi
s ca
n be
com
pare
d in
cla
ss b
y co
mbi
ning
re
sults
from
tria
ls d
one
in g
roup
s or
pai
rs.
GR
ADE
9 Pr
obab
ility
exp
erim
ents
In
Gra
des
8 an
d 9
prob
abilit
y ex
perim
ents
are
less
impo
rtant
, and
lear
ners
sho
uld
cons
ider
pro
babi
lity
for h
ypot
hetic
al e
vent
s. In
Gra
de 9
, le
arne
rs h
ave
to c
onsi
der o
utco
mes
of c
ompo
und
even
ts a
nd u
se tw
o-w
ay ta
bles
and
tree
dia
gram
s to
wor
k ou
t the
pro
babi
lity
of a
n ou
tcom
e.
Prob
abili
ty o
f com
poun
d ev
ents
Fo
r exa
mpl
e, w
hat i
s th
e pr
obab
ility
of a
wom
an g
ivin
g bi
rth to
two
boys
afte
r eac
h ot
her?
A tw
o-w
ay ta
ble
can
be u
sed:
Firs
t Birt
h Se
cond
Birt
h Bo
y G
irl
Boy
BB
BG
Girl
G
B G
G
Two
boys
afte
r eac
h ot
her (
BB) i
s 1
of 4
pos
sibl
e ou
tcom
es, s
o th
e pr
obab
ility
of tw
o bo
ys is
1 o
ut o
f 4, o
r 25%
. H
ow d
oes
this
pro
babi
lity
chan
ge if
you
ask
wha
t is
the
prob
abilit
y of
giv
ing
birth
to th
ree
boys
afte
r eac
h ot
her?
A
tree
diag
ram
can
then
be
used
. See
an
exam
ple
of a
tree
dia
gram
on
CAP
S p.
152
134 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
LTSM
Res
ourc
es
D
ice,
spi
nner
s, m
arbl
es, b
ag.
G
rade
7
Gra
de 8
Gra
de 9
Wor
kboo
k re
fere
nce
W
B 1
Activ
ities
(R15
) W
B 2
Activ
ities
(140
)
WB
2 Ac
tiviti
es (1
35 -
138)
W
B 2
Activ
ities
(138
- 14
30
Text
book
re
fere
nce
HO
MEW
OR
K
ASSE
SSM
ENT
E.g.
Info
rmal
as
sess
men
t – T
est:
mar
ks m
ax
135MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
TASKS & MEMORANDA
136 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
INVESTIGATION EXEMPLAR: GRADE 7 TERM 2
ACTIVITY: INTERSECTING LINES Intersecting lines are any lines that cut each other at a point, and that point where they cut each other
is called the point of intersection.
Example
a) Describe the above diagram (diagram in the example) to a friend who does not see it.
_____________________________________________________________________________
___________________________________________________________________
(2)
b) Draw at least three diagrams with two intersecting lines and label the lines and the diagrams
(Diagram 1, Diagram 2, etc.). Label the points of intersection too. The lines in all the sets should
not be perpendicular to each other
(3)
c) Measure ALL angles that are formed at the points of intersection. Record your measurements in
the table like the one drawn below.
DIAGRAM ANGLE NAME SIZE
(3 x 4)
d) Explain what you notice about these angles
_____________________________________________________________________________
___________________________________________________________________
(1)
e) Which angles are equal in each diagram?
_____________________________________________________________________________
___________________________________________________________________
________________________________________________________________________
(1)
137MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
f) Describe the position of equal angles in relation to each other. [Hint: Are thy next to each other OR on opposite sides of the common vertex?] ________________________________________________________________________
(1)
g) How many sets of equal angles are there in each diagram?
________________________________________________________________________
(1)
h) The equal pairs of angles in the diagram are called VERTICALLY OPPOSITE ANGLES. In your
own words, define what vertically opposite angles are.
_____________________________________________________________________________
___________________________________________________________________
(2)
i) From the observations made during the activity, what conclusion can one make about vertically
opposite angles formed by intersecting lines?
_____________________________________________________________________________
___________________________________________________________________
(2)
TOTAL 25
138 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
INVESTIGATION EXEMPLAR: GRADE 8 TERM 2
Activity: Corresponding angles
A transversal is a line that is drawn such that it intersects other lines. Angles on the same side of the
transversal, and also on the same side of the lines cut by the transversal are called corresponding
angles.
Example
In the diagram, AD is a transversal. Angles FBC and GCD are corresponding angles.
Activity
Use the diagrams below for the activity. Parallel lines and corresponding angles are indicated.
Diagram 1 Diagram 2
CB
F
E
G
H
A
D
KHG
I
J
C
D
A B
E
F
139MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Diagram 3 j) Measure ALL corresponding angles and then complete the table that follows.
DIAGRAM ANGLE NAME SIZE
1
2
3
(9)
k) Specify equal pairs of corresponding angles in each diagram.
_____________________________________________________________________________
___________________________________________________________________
________________________________________________________________________
(3)
l) Which lines form equal corresponding angles if cut by a transversal? ________________________________________________________________________
(1)
m) Construct and label your own parallel lines that are cut by a transversal. (2)
________________________________________________________________________
E
D
C
F
G
A
BH
I
J
K
140 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
n) Measure and compare corresponding angles
______________________________________________________________________
______________________________________________________________________
(3)
o) From the observations made during the activity, what conclusion can one make about
corresponding angles formed if parallel lines are cut by a transversal?
_____________________________________________________________________________
___________________________________________________________________
(2)
TOTAL 25
141MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
INVESTIGATION EXEMPLAR: GRADE 9
Relating angles in a triangle
In the diagram 1 below, the marked angle ACD is an exterior angle of triangle ABE. An exterior angle is formed by producing one of the sides. The marked angles BAC and ABC are interior, opposite angles to the exterior angle ACD.
Diagram 1 Diagram 2
Diagram 3
(a) Measure the marked angles and complete the table below. Diagram Interior angle 1 Interior angle 2 Exterior angle Sum of interior angles
1 2 3
(12)
S
Q
P
R
P
NO
M
142 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
(b) How does the exterior angle compare to the sum of the interior opposite angles? _________________________________________________________________ (1)
(c) Draw your own triangle with one side produced to form an exterior angle. (2)
(d) Measure and record the size of the exterior angle and the interior opposite angles. __________________________________________________________________ (3)
(e) Is the exterior angle equal to the sum of the interior angle? _________________________________________________________________ (1)
(f) What conclusion can be drawn from the activities? ____________________________________________________________________________________________________________________________________________________ (1)
TOTAL 25
143MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
MULTIGRADE FORMAL ASSESSMENT
MATHEMATICS GRADE 7, 8, 9: FORMAL ASSESSMENT TASK: TERM 1 TEST
Examiner: ________________________ Time: 1 Hour
Moderator: ________________________ Date: __________
Total Grade 7 40
Total Grade 8 50
Total Grade 9 75
INSTRUCTIONS:
Answer all the questions underneath each other. Leave a line open between every sum. Show all your calculations. Write legibly. Round off to 2 decimal digits. Answer the questions relevant to your grade
QUESTION 1
1.1Calculate:
GRADE No Marks
7,8,9 1.1.1 −10 + (18)
(1)
8,9 1.1.2 (-2)2(-3)3 (3)
8,9 1.1.3
−4 × 5 + (−5)2
(3)
8,9 1.1.4 √−643 +(−2)3
(3)
7,8,9 1.1.5 (37)2
(2)
8,9 1.1.6 √2 7
9
(3)
9 1.1.7
−42 + (−4)3
(3)
9 1.1.8
√169 − √−0,0643
(3)
9 1.1.9
√1 916 × 1 2
3 ÷ 9, 3̇
(4)
144 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
7 1.1.10 7 836 524 + 4 853 477 (2)
7 1.1.11 42 746 ÷ 274 (2)
7,8 1.2 Arrange the following numbers in ascending order (2)
20; −18; −22; 2; −8; 6
7,8,9 1.3 Write 54 as the product of its prime factors (3)
Grade 7 [7]
Grade 8 [15]
Grade 9 [25]
9 1.4 Use prime factors and determine the HCF of 112 and 210 (3)
7,8,9 1.5.1 Write the following ratios in their simplest form
20 : 24 : 44
(1) 7,8,9 1.5.2 Divide R2500 in the ratio of 2 : 3
(3) 7,8,9 1.6.1 You cover a distance of 18km in 2h30 min. What is your
average speed in km/h?
(3) 7,8,9 1.6.2 John buys and sells T-shirts. He buys the shirts at a cost of
R50. He sells them at a profit of 70%. Calculate his selling price.
(2) Grade 7 [9]
Grade 8 [9]
Grade 9 [12]
QUESTION 2
7,8,9 7,8,9 7,8,9
2.1 2.1.1 2.1.2 2.1.3
Use the number sequence below to answer the questions that follow:
1; 4 ; 9; 16; …. Write down the next term in the number sequence. Determine the 7th term in the number sequence. State the rule in words of the number sequence.
(1)
(2)
(1)
7,8,9 2.2 Calculate the value of the following expression if : a = -2 ; b = 3 and c = -1
Expression: a + b – (2c)2
(2)
145MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
7,8,9 2.3 Write the following relationship as a symbolic formula: To calculate the output number (y), 7 is subtracted from the input number(x) and the answer is multiplied by 5.
(2)
7,8,9
2.4
Write 620 000 000 000 in scientific notation.
(2)
Grade 7 [10]
Grade 8 [10]
Grade 9 [10]
QUESTION 3
3.1 Consider the following expression and answer the questions that follow.
2x4 + y − 3x + 5 + 2y
7,8,9 3.1.1 How many terms does the expression have? (1)
7,8,9 3.1.2 Write down two terms which are alike. (1)
7,8,9 3.1.3 What is the exponent of x in the first term? (1)
7,8,9 3.1.4 Which term is a constant? (1)
8,9 3.1.5 What is the coefficient of the third term? (1)
Grade 7 [4]
Grade 8 [5]
Grade 9 [5]
QUESTION 4
4.1 Consider the following pattern that is built with matches and answer the questions.
figure 1 figure 2 figure 3
7,8,9 4.1.1 Draw the next figure (Fig 4) in the pattern.
(1)
7,8,9 4.1.2 Consider the following table and write down the values of a and b. (2)
Figure (n) 1 2 3 4 10 Number of matches (Tn) 8 15 22 a b
8,9 4.1.3 Write down the formula for the nth term. (2)
7,8,9 4.1.4 Which figure will consist of 50 matches? (2)
146 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
7,8,9 4.1.5 How many matches will be needed to make the 25th figure? (1)
Grade 7 [6]
Grade 8 [8]
Grade 9 [8]
QUESTION 5
5.1 You receive R2500 from your father on your birthday. You decide to invest the money in the bank for you studies four years from now. The bank gives you the following two options. OPTION A: Invest with 8% compound interest per year for four years. OPTION B: Invest at 7,5% simple interest rate for four years.
9 5.1.1 Decide which one will generate more money in the given time period.
(7)
9 5.1.2 Calculate the percentage increase if the price of diesel increases from R11,21 per litre to R12,30 per litre. Round your answer off to 2 decimal places.
(4)
9 5.1.3 An aircraft travels at a speed of 0,9 x 103 km/h for 24 hours. How far has the aircraft travelled? Give the answer in scientific notation.
(3)
Grade 9 [14]
QUESTION 6
6.1 Study the weather temperature graph provided and answer the questions that follow
14
12
10
8
6
4
2
- 2
Temperature in °C
18:0017:0016:0015:0014:0013:0012:0011:0010:0009:0008:0007:00
147MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
7,8 6.1.1 What was the temperature at 7 am?
(1)
7,8 6.1.2 What was the maximum day temperature?
(1)
7,8 6.1.3 What is the difference in temperatures at 18:00 and at 8:00? (1)
7,9 6.1.4 During which period did the temperature increase? (1)
Grade 7 [4]
Grade 8 [3]
Grade 9 [1]
TOTAL Grade 7 [40]
TOTAL Grade 8 [50]
TOTAL Grade 9 [75]
148 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
TEST
TERM 1
Gr 7,8,9
M Method S Simplification KA / CA Constant accuracy A Answer
QUESTION 1 1.1.1 -10 + (18)
= 8 A A= 1 mark
1.1.2 (-2)2(-3)3 4A × -27A = -108 CA
A=2 marks CA = 1 mark
1.1.3 -4 × 5 + (-5)2
= -20 S+ 25S = 5CA/KA
S = -20 1 mark S = 25 1 mark CA/KA = 5 1 mark
1.1.4 -4 S + (-8) S
= -12CA/KA S = -4 1 mark S = -8 1 mark CA/KA = -12 1 mark
1.1.5
499AA
A = 9 1 mark A = 49 1 mark (Answer must be a fraction)
1.1.6 259
53
SA A
S = 25/9 1 mark A = 5 1 mark A = 3 1 mark
1.1.7 -42 + (-4)3 = -16-64 M = -80 KA/CA
-16: 1 mark -64: 1 mark answer: 1 mark
1.1.8 3 064,0169 = 13 - (-0,4) M = 13 + 0,4 = 13,4 KA/CA
13: 1 mark -0,4: 1 mark/answer: 1 mark
1.1.9
319
321
1691 M
= 328
35
1625
M
= 283
35
54
M
= 11225
KA/CA
319 : 1 mark
1625
: 1 mark
reciprocal: 1 mark answer: 1 mark
149MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
1.1.10 7 836 542 + 4 853 477 12 690 019
019 - 1 mark 12 690 – 1 mark
1.1.11 42 746 ÷ 274 = 156 r 2
156 – 1 mark Remainder 2 – 1 mark
1.2 -22 ; -18 ; -8 ; 2 ; 6 ; 20 A A
A = -22 ; -18 ; -8 1 mark A = 2 ; 6 ; 20 1 mark
1.3 2 54 3 27 3 9 3 3 1 M M 2 × 3× 3 × 3 = 54 CA
M = correctly divide l 2 (1 mark)
M = correctly divide l 3
(1 mark
CA/KA = Write as a product (1 mark)
1.4 2 112 2 210 2 56 3 105 2 48 5 35 2 24 7 7 7 14 1 7 7 1
S S 2 × 7 = 14 S
S = 112 correctly divide by
prime numbers (1 mark)
S = 210 correctly divide by
prime numbers (1 mark)
A = GGF / HCF = 14
1.5.1 5 : 6 : 11 A
A = 4 : 6 : 11 (1 A punt/mark)
1.5.2 2 2500 1000
53 2500 15005
M A
A
M = 2 25005 (1 mark)
A = R1000 (1 mark)
A = R1500 (1 mark)
1.6.1 Average speed = 18 km2,5 h
= 7,2 km/h M ; C ; CA/KA
M = formula (1 mark) C = convert to 2,5 h (1 mark)
CA/KA = 7,2 km(1 mark)
1.6.2
10070
R50 = R35 S
= R50 + R35 = R85 OF /OR 1,7 S × 50 = R85 CA/KA
S = R35/1,7 (1 mark) CA/KA = R85 (1 mark)
QUESTION 2 2.1.1 25 A A = 1 mark 2.1.2 T7 = 49 A A = 2 marks 2.1.3 Each term number is squared. A A = 1 mark 2.2 = -2 + 3 – (2. -1)2S
= -2 + 3 – 4 = -3CA / KA
S = 1 mark CA/KA = 1 mark
2.3 y = 5( x – 7) AA A = 2 marks 2.4 A 6,2 × 1011A A =2 marks
150 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 3 3.1 3.1.1 5 A A = 1 mark 3.1.2 y + 2y A A = 1 mark 3.1.3 4 A A = 1 mark 3.1.4 4th term A A = 1 mark 3.1.5 -3 A A = 1 mark
QUESTION 4 4.1.1
correct pattern (1)
4.1.2 a = 29 b = 71
value of a (1) value of b (1)
4.1.3 Tn = 7n + 1
7n (1) 1 (1)
4.1.4 Tn = 7n + 1 50 = 7n + 1 n = 7
substitution of 50 (1) (CA) answer (1)
4.1.5 T25 = 7(25) + 1 = 176
correct answer (1)
QUESTION 5 5.1.1 Option A:
A = P(1 + i)n M = 2 500(1 + 0,08)4 M = R3 401,22 KA/CA Option B: A = P(1 + in) M = 2 500(1 + 0,75 x 4) M = R3 250 KA/CA ∴ Option A is better KA/CA
formula: 1 mark substitution: 1 mark
answer: 1 mark formula: 1 mark
substitution: 1 mark answer: 1 mark
Option A: 1 mark
5.1.2
%72,9
100x21,11
21,1130,12
100xOP
OPNPincrease/verhoging%
12.30 – 11,21: 1 mark 11,21/divide by 11,21: 1 mark
X 100: 1 mark answer: 1 mark
If answer not rounded off to 2 dec places: only ¾
5.1.3 distance = speed x time M = 0,9 x 103 x 24 M = 2,16 x 104 km KA/CA
Formula/: 1 mark 0,9 x 103 x 24: 1 mark
answer: 1 mark
151MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
EXEMPLAR TEST GRADE 8 TERM 1 Marks : 60 Instructions to learners: 1. Read all the instructions carefully. 2. Answer Questions 1 – 8 in the spaces or frames provided. 3. Show all working on the question paper. 4. The test duration is 90 minutes.
QUESTION 1 Calculate each of the following: 1.1 235 292 + 782 354
_________________________________________________ _________________________________________________ _________________________________________________
(2)
1.2 9 634 567 – 6 546 321 _________________________________________________ _________________________________________________ _________________________________________________
(2)
152 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 2
Given the following numbers: 144 and 150 2.1 Write down the first 5 multiples of 144 and 150
_________________________________________________ _________________________________________________ _________________________________________________
(2)
1.3 12 421 × 25 _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________
(3)
1.3 12 421 × 25 _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________
(3)
1.4 10 625 ÷ 25 _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________
(3)
1.5
90 + 18 ÷ 2 × 8 - 13 _________________________________________________ _________________________________________________ _________________________________________________
(3)
_________________________________________________ _________________________________________________ _________________________________________________
153MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 5
5.1 Arrange the following sets of numbers in an ascending order. 63 ; − 49 ; − 56 ; 0 ; − 28 _____________________________________________________
(1)
5.2 Fill in <; > , =
2.2 Write 144 and 150 as product of their prime factors. (2) 2.3 Write down the HCF of 120 and 144 (2) QUESTION 3 3.1 Lucy earned R4500 and gave 10% of it to her mother. How much did she give to her? (2) 3.2 Tshepo invested R1 500 into his savings account. The investment grew/increased by 10%
simple interest per year. How much was the investment worth at the end of the third year? (1)
3.3 Thobeka wants to order a book that costs $56,67. The rand-dollar exchange rate is R11 to a dollar. What is the price of the book in rands?
(3)
3.4 Calculate a discount of 6% on each of the following marked prices of article R3 600. (3) QUESTION 4 4.1 Write down the ratio 15 : 27 as a fraction and then simplify if possible.
_________________________________________________ _________________________________________________ _________________________________________________
(2)
4.2 Tom has R7 and Lucy has R9; what is the ratio of the money Tom has to Lucy?
____________________________________________
(1)
4.3 The ratio of boys to girls in a class is 7 : 5. If there are 36 learners in a class, how many are boys and how many are girls? _________________________________________________ _________________________________________________ _________________________________________________
(4)
4.4 Divide 24 in the ratio 1 : 2 : 5 ______________________________________________________ ______________________________________________________ ______________________________________________________
(3)
4.5 If 5kg of nuts cost R40 how much will 7kg cost? ______________________________________________________ ______________________________________________________ ______________________________________________________
(2)
154 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
5.2.1 0 ____ − 999
(1)
52.2 − 5,5 _____ 0,5
(1)
5.2.3 67 ____ 60 + 7
(1)
5.3 Fill in the missing numbers:
3; 12, 48; -------; -------
(2)
5.4 Calculate each of the following.
5.4.1 4 + (−1) – (−9) ________________________________________________
(1)
5.4.2 200 ÷(+4) × (−2) _________________________________________________
(1)
[8]
QUESTION 6
6.1 (− 6)2 ________________________________________________________________________
(1)
6.2 √1253 _____________________________________________________
(1)
[2]
QUESTION 7
7.1 Determine the rule and solve m and n.
x 1 2 3 m 15 y 1 4 9 64 n
__________________________________________________ _________________________________________________ __________________________________________________
(3)
155MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 8
Study the weather temperature graph provided and answer the questions that follow
8.1 What was the temperature at 7 am? _________________________________________________
(1)
8.2 What was the maximum day temperature? _________________________________________________
(1)
8.3 What is the difference in temperatures at 18:00 and at 8:00? __________________________________________________
(1)
8.4
During which period did the temperature increase? __________________________________________________
(1)
8.5 Do you think the graph represents a summer or winter day in South Africa? Give a reason for your answer.
(2)
14
12
10
8
6
4
2
- 2
Temperature in °C
18:0017:0016:0015:0014:0013:0012:0011:0010:0009:0008:0007:00
156 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
EXEMPLAR TEST MEMO GRADE 8 TERM 1 Marks : 60
QUESTION 1
1.1 235 292 + 782 354 1 017 646 √ √
For the correct answer: full marks.
(2) 1.2 9 634 567
– 6 546 321 3 088 246 √ √
For the correct answer: full marks.
(2) 1.3 12 421
× 25 62 105 + 248 420 √ 310 525 √ √
For the correct answer: full marks.
(3) 1.4 10 625 ÷ 25
425
000000012500125005000621001062525
√√√ QUESTION 2
For the correct answer: full marks.
(3)
1.5 90 + 18 ÷ 2 × 8 − 13 = 90 + 9 x 8 – 13 = 90+72-13√ = 149 √
For the correct answer: full marks.
(3) 2.1 Write down the first 5 multiples of 144 and 150
Multiples of 144 = {144 ; 288 ; 432 ; 576 ; 702} √ Multiples of 150 = {150 ; 300 ; 450 ; 600 ; 750} √
(2)
2.2 120 = 23 x 3 x 5
144 = 24x32
(2) 2.3 HCF of 120 and 144 = 24 √
(1) [5]
157MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 3
3.1 10% of R4 500 = 10% × 4 500 √ = 450 She gave her mother R450. √
(2)
3.2 SI = 𝑃𝑃𝑃𝑃𝑃𝑃100 = 1 500 ×3 ×10
100 √ = R450 √ At the end of the third year the investment was worth R1 950.√
(3)
3.3 11 x 56,67 = R623,37 √√√ (3) 3.4
216
3600100
6
R
[8]
QUESTION 4
4.1 Write down the ratio 15 : 27 as a fraction and then simplify if possible. 1527 = 59 √√
(2)
4.2 7 : 5 (1)
4.3 712 × 36 √ = 21 boys √ 512 × 36 √ = 15 girls. √
(4)
4.4 1 8 × 24 = 3 √ 28 × 24 = 6 √ 58 × 24 15 √
(3)
4.5 If 5kg of nuts cost R40 how much will 7kg cost? 5kg costs R40 1kg cost R8 ∴ 7kg will cost 7 × R8 = R56 √ √
(2)
158 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 5
5.1 − 56 ; − 49 ; ; − 28 ; 0 ; 63 (1) 5.2 5.2.1 0 > − 999
(1) 52.2 − 5,5 < 0,5
(1) 5.2.3 67 = 60 + 7
(1) 5.3 3; 12, 48; 192; 768 √ √
(2)
5.4 5.4.1 4 + (-1)-(-9) = 12
(1) 5.4.2 (+6) – (−2) = + 8
(1) 5.4.3 (+4) + (−1) – (−9)
= 4 – 1 + 9 = 12 √
(1) 5.4.4 200 ÷4 × (−2)
= − 100 √
(1)
[10]
QUESTION 6
6.1 (− 6)2 = 36 √
(1)
6.2 √1253 = 5 √
(1)
[2]
QUESTION 7
x 1 2 3 m 15 y 1 4 9 64 n
Rule y = x2 √ m = 8 √ n = 225 √
[3]
QUESTION 8
8.1 2C (1) 8.2 12C (1) 8.3 6C 0C = 6C (1) 8.4 From 7:00 to 12:00 (1) 8.5 The graph represents a winter day in South Africa.
The maximum temperature of 12C is very cold.
(2) [6]
159MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
GRADE 9 EXEMPLAR TEST - TERM 1 Marks : 60 Instructions to learners: 1. Read all the instructions carefully. 2. Answer Questions 1 – 8 in the spaces or frames provided. 3. Show all working on the question paper. 4. The test duration is 90 minutes
QUESTION 1
Calculate each of the following.
1.1 81 892 + 321 456 ________________________________________________ _________________________________________________ _________________________________________________
(2)
1.2 1 234 567 – 654 321 _________________________________________________ _________________________________________________ _________________________________________________
(2)
1.3 26 342 × 32 _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________
(3)
1.4 7550 ÷ 25 _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________
(3)
1.5
60 + 10 ÷ 2 × 5 - 1 _________________________________________________ _________________________________________________ _________________________________________________
(2)
[12]
160 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 2
Given the following numbers: 120 and 144
2.1 Write 120 and 144 as products of their prime factors. _________________________________________________ _________________________________________________ _________________________________________________
(2)
2.2 Write down the LCM and HCF in 120 and 144 _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________
(2)
[4]
QUESTION 3 [calculators may be used]
3.1 Lucy earned R4500 and gave 15% of it to her mother, how much did she give to her?
_________________________________________________ _________________________________________________ _________________________________________________
(2) 3.2 Tshepo invested R1 500into his savings account. The investment grew /increased by
10 % compound interest per year. How much was the investment worth at the end of the third year? _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________
(3)
3.3 The marked prices of an article is R850. A discount of 15% is offered to customers who pay cash. Calculate how much a customer who pays cash will actually pay.
(3)
3.4 Sara buys a flat screen television on hire purchase. The cash price is R4 199. She
has to pay a deposit of R950 and 12 monthly instalments of R360. Calculate the total hire purchase price.
(4)
3.5 Tim bought £650 at the foreign exchange desk at Gatwick Airport in the UK at a rate (4)
161MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
of R15,66 per £1. The desk also charged 2,5% commission on the transaction. How much did Tim spend to buy the pounds?
(16)
QUESTION 4
4.1 Write down the ratio 45 : 60 as a fraction and then simplify if possible. _________________________________________________ _________________________________________________ _________________________________________________
(2)
4.2 To make biscuits of a certain kind, 5 parts of flour has to be mixed with 2 parts of
oatmeal, and 1 part of cocoa powder. How much oatmeal and how much cocoa powder must be used if 500 g of flour is used? _________________________________________________ _________________________________________________ _________________________________________________
(4)
4.4 Divide 60 in the ratio 3 : 2 : 1
______________________________________________________ ______________________________________________________ ______________________________________________________
(3)
4.5 If 7kg of nuts cost R50 how much will 16kg cost? ______________________________________________________ ______________________________________________________ ______________________________________________________
(2)
[11]
QUESTION 5
5.1 Calculate :−15 + (−14) − 9 ____________________________________________________
(2) 5.2 Calculate:
(–30) × (–10) – (–30) × (–8) (2)
5.3 Calculate each of the following:
4p - 6q + (-5p) - (4q)
(2)
[6]
162 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 6 Determine the values of each of the following.
6.1 (− 6)2
________________________________________________________________________
(1)
6.2
_________________________________________________
(1)
[2]
QUESTION 7
7.1 Study the table below.
x 1 2 3 m 15 y 2 5 10 65 n
7.1.1 Determine the value of m and n __________________________________________________ 7.1.2 Write down the rule you use to calculate m and n. ________________________________________________ __________________________________________________
(2)
(1)
[3]
163MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 8
Study the weather temperature graph provided and answer the questions that follow
8.1 What was the temperature at 7 am?
_________________________________________________
(1)
8.2 What was the maximum day temperature? _________________________________________________
(1)
8.3 What is the difference in temperatures at 18:00 and at 8:00? __________________________________________________
(1)
8.4 During which period did the temperature increase? __________________________________________________
(1)
8.5 Do you think the graph represents a summer or winter day in South Africa? Give a reason for your answer. ________________________________________________________ ________________________________________________________
(2)
[6]
164 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8
%
Req %
12 6 11 12 8 2 3 6 60 100 R 12 6 5 3 26 43 45 CP 11 7 18 30 20 K 6 2 6 14 23 25 PS 2 2 3 10
165MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
GRADE 9 EXEMPLAR TEST MEMO - TERM 1 Marks : 60
QUESTION 1
1.1 81 892
+ 321 456 403 348 √ √
For the correct answer: full marks.
(2)
1.2 1 234 567 – 654 321 580 246 √ √
For the correct answer: full marks.
(2)
1.3 26 342 × 32 52 684 +790 260 √ 842 944 √ √
For the correct answer: full marks.
(3)
1.4 7550 ÷ 25 √√√
For the correct answer: full marks.
(3)
1.5 60 + 10 ÷ 2 × 5 − 1
= 60 + 25 – 1 √ = 84 √
For the correct answer: full marks.
(2)
[12]
QUESTION 2
2.1 120 = 2 × 2 × 2 × 3 × 5 √ 144 = 2 × 2 × 2 × 2 × 3 × 3 √
(2)
2.2 LCM of 120 and 144 = 720 √ HCF of 120 and 144 = 24 √
(2)
[4]
QUESTION 3
3.1 15% of R4 500 = 15% × 4 500 √ = 675 She gave her mother R675. √
(2)
166 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
3.2 CI = P(1+i)n = 1500(1+0,1)3 √ = = R1996,50 √ At the end of the third year the investment was worth R1996,50.√
(3)
3.3 R850 x 15 ÷ 100 = R 127,50√
= R850 - R127,50√
= R722,50√
(3)
3.4 R950 + (R360 x 12) √√
= R950 + R4320√
= R5270√
(4)
3.5 R15,66 x £650 = R 10 179 x 2,5 ÷ 100
= R 254,48√√
= R254,48 + R 10 179 = R10 433, 48√√
(4)
[16]
QUESTION 4
4.1 Write down the ratio 15 : 27 as a fraction and then simplify if possible. = 15 √ 27 = 5 √ 9
(2)
4.2 200 g of oatmeal,√√ and 100g of cocoa powder. √√ (4)
4.4 60 x 3 ÷ 6 = 30 √
60 x 2 ÷ 6 = 20√ 60 × 1 ÷ 6 = 10 √
(3)
4.5 If 5kg of nuts cost R40 how much will 7kg cost? 5kg costs R40 1kg cost R8 7kg will cost 7 × R8 = R56 √ √
(2)
[11]
167MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 5
5.1 −15 + (−14) − 9 √ = -15 -14 - 9 = - 38 √
(2)
5.2 (–30) × (–10) – (–30) × (–8)
= 300 - 240 √ = 60√
(2)
5.3 5.3.1 4p - 6q + (-5p) - (4q)
= 4p - 5p - 6q - 4q = - p - 10q
(2)
[6]
QUESTION 6
6.1 (− 6)2 = 36 √
(1)
6.2 =
=
= 8
(1)
[2]
QUESTION 7
x 1 2 3 m 15 y 2 5 10 65 n
7.1.1 m = 8 √ and n = 226 √ 7.1.2 square the value of "x" and add 1 or y = x2 + 1 √
(2)
(1)
[3]
QUESTION 8
8.1 - 2oC (1) 8.2 12oC (1) 8.3 6oC - 0oC = 6C (1) 8.4 From 7:00 to 12:00 (1) 8.5 The graph represents a winter day in South Africa.
The minimum temperatue is - 2oC maximum temperature of 12oC is very cold. (2)
[6]
168 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
GRADE 9 EXEMPLAR TEST MEMO - TERM 1 Marks : 60
QUESTION 1
1.1 81 892
+ 321 456 403 348 √ √
For the correct answer: full marks.
(2)
1.2 1 234 567 – 654 321 580 246 √ √
For the correct answer: full marks.
(2)
1.3 26 342 × 32 52 684 +790 260 √ 842 944 √ √
For the correct answer: full marks.
(3)
1.4 7550 ÷ 25 √√√
For the correct answer: full marks.
(3)
1.5 60 + 10 ÷ 2 × 5 − 1 = 60 + 25 – 1 √ = 84 √
For the correct answer: full marks.
(2)
[12]
QUESTION 2
2.1 120 = 2 × 2 × 2 × 3 × 5 √ 144 = 2 × 2 × 2 × 2 × 3 × 3 √
(2)
2.2 LCM of 120 and 144 = 720 √ HCF of 120 and 144 = 24 √
(2)
[4]
QUESTION 3
3.1 15% of R4 500 = 15% × 4 500 √ = 675 She gave her mother R675. √
(2)
3.2 CI = P(1+i)n
169MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
= 1500(1+0,1)3 √ = = R1996,50 √ At the end of the third year the investment was worth R1996,50.√
(3)
3.3 R850 x 15 ÷ 100 = R 127,50√
= R850 - R127,50√
= R722,50√
(3)
3.4 R950 + (R360 x 12) √√
= R950 + R4320√
= R5270√
(4)
3.5 R15,66 x £650 = R 10 179 x 2,5 ÷ 100
= R 254,48√√
= R254,48 + R 10 179 = R10 433, 48√√
(4)
[16]
QUESTION 4
4.1 Write down the ratio 15 : 27 as a fraction and then simplify if possible. = 15 √ 27 = 5 √ 9
(2)
4.2 200 g of oatmeal,√√ and 100g of cocoa powder. √√ (4)
4.4 60 x 3 ÷ 6 = 30 √
60 x 2 ÷ 6 = 20√ 60 × 1 ÷ 6 = 10 √
(3)
4.5 If 5kg of nuts cost R40 how much will 7kg cost? 5kg costs R40 1kg cost R8 7kg will cost 7 × R8 = R56 √ √
(2)
[11]
QUESTION 5
5.1 −15 + (−14) − 9 √ = -15 -14 - 9 = - 38 √
(2)
170 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
5.2 (–30) × (–10) – (–30) × (–8)
= 300 - 240 √ = 60√
(2)
(2) 5.3 5.3.1 4p - 6q + (-5p) - (4q)
= 4p - 5p - 6q - 4q = - p - 10q
(2) [6]
QUESTION 6
6.1 (− 6)2 = 36 √
(1)
6.2 =
=
= 8
(1)
[2]
QUESTION 7
x 1 2 3 m 15 y 2 5 10 65 n
7.1.1 m = 8 √ and n = 226 √ 7.1.2 square the value of "x" and add 1 or y = x2 + 1 √
(2)
(1)
[3]
QUESTION 8
8.1 - 2oC (1) 8.2 12oC (1) 8.3 6oC - 0oC = 6C (1) 8.4 From 7:00 to 12:00 (1) 8.5 The graph represents a winter day in South Africa.
The minimum temperatue is - 2oC maximum temperature of 12oC is very cold. (2)
[6]
171MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
SENIOR PHASE EXEMPLAR PROJECT QUESTION 1
Study the nets provided. [Grade 8 and 9 only]
1. 2. 3.
4. 5 Which net would you link to each of the polyhedra listed below? Write the number of the net next to the name of the correct polyhedron. Explain each answer
a) A tetrahedron - _____________________________________________________
b) A hexahedron/Cube - _____________________________________________________
c) An octahedron - _____________________________________________________
d) A dodecahedron - _____________________________________________________
e) An icosahedron - _____________________________________________________
(10)
f) What is the collective name for the group of polyhedra above?
_______________________________________________________________ (1)
g) What is common in the group of polyhedra provided above? Name any TWO things.
__________________________________________________________________
__________________________________________________________________ (4)
(15)
172 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
QUESTION 2 [All Grades]
Cut out each of the nets provided and build each polyhedron. You will need a pair of scissors and glue to complete this section. You may need to use cardboard to make your polyhedra rigid. Marks will be awarded for:
Cutting correctly leaving flaps for folding. Accuracy Neatness
QUESTION 3 [All Grades]
a) Complete the table below using the model of objects you made. Polyhedron Number of faces (F) Number of vertices (V) Number of Edges (E) Tetrahedron Cube Octahedron Dodecahedron Icosahedron
b) From your results in Question 3(a), write down an equation which relates the values F, V and E. ________________________________________________________________________________________________________________________________________________________________
173MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
174 MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE
175MULTIGRADE TOOLKIT MATHEMATICS SENIOR PHASE