1MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

2 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

3MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

4 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

5MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Mathematics: Intermediate 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. .

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.

6 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MU

LTI-G

RA

DE

AN

NU

AL

TEA

CH

ING

PLA

N

7MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

T ER

M 1

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

5

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

O

dd a

nd e

ven

num

bers

;

Pl

ace

valu

e;

Mul

tiple

s ;

Rou

ndin

g of

f;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

O

dd a

nd e

ven

num

bers

;

Pl

ace

valu

e;

M

ultip

les;

Fact

ors;

Rou

ndin

g of

f;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

Pr

ime

num

bers

;

Pl

ace

valu

e;

M

ultip

les

;

Fact

ors;

Rou

ndin

g of

f 1

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

6 N

umbe

r sen

tenc

es

Num

ber s

ente

nces

N

umbe

r sen

tenc

es

7 Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

Addi

tion

and

subt

ract

ion

of w

hole

num

bers

Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

1 In

form

al a

sses

smen

t In

form

al a

sses

smen

t In

form

al a

sses

smen

t

3

Num

eric

and

geo

met

ric p

atte

rns

In

vest

igat

e an

d ex

tend

pat

tern

s

Inpu

t and

out

put v

alue

s

Equi

vale

nt fo

rms

Num

eric

and

geo

met

ric p

atte

rns

In

vest

igat

e an

d ex

tend

pat

tern

s

Inpu

t and

out

put v

alue

s

Equi

vale

nt fo

rms

Num

eric

and

geo

met

ric p

atte

rns

In

vest

igat

e an

d ex

tend

pat

tern

s

Inpu

t and

out

put v

alue

s

Equi

vale

nt fo

rms

9 M

ultip

licat

ion

and

divi

sion

of w

hole

nu

mbe

rs

Mul

tiplic

atio

n an

d di

visi

on o

f who

le

num

bers

M

ultip

licat

ion

and

divi

sion

of w

hole

nu

mbe

rs

6

Tim

e

M

easu

ring

inst

rum

ents

Rea

ding

tim

e

Cal

cula

tion

and

prob

lem

sol

ving

invo

lvin

g tim

e

His

tory

of t

ime

Tim

e an

d te

mpe

ratu

re

M

easu

ring

inst

rum

ents

Rea

ding

tim

e an

d te

mpe

ratu

re

C

alcu

latio

n an

d pr

oble

m s

olvi

ng in

volv

ing

time

and

tem

pera

ture

His

tory

of t

ime

Tim

e an

d te

mpe

ratu

re

M

easu

ring

inst

rum

ents

Rea

ding

tim

e an

d te

mpe

ratu

re

C

alcu

latio

n an

d pr

oble

m s

olvi

ng in

volv

ing

time

and

tem

pera

ture

His

tory

of t

ime

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

topi

cs, c

once

pts

and

skills

.

8 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

6

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng d

ata

Pr

obab

ility

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng d

ata

Pr

obab

ility

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng d

ata

Pr

obab

ility

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

2 - D

sha

pes,

R

ecog

nize

, vis

ualiz

e an

d na

me

2-D

sha

pes

in

re

gula

r and

irre

gula

r pol

ygon

s –

trian

gles

, sq

uare

s, re

ctan

gles

, oth

er q

uadr

ilate

rals

, pe

ntag

ons,

hex

agon

s an

d ci

rcle

s C

hara

cter

istic

s of

sha

pes

st

raig

ht s

ides

curv

ed s

ides

num

ber o

f sid

es

Dra

w 2

-D s

hape

s on

grid

pap

er

2 -

D s

hape

s,

Rec

ogni

ze, v

isua

lize

and

nam

e 2-

D s

hape

s in

regu

lar a

nd ir

regu

lar p

olyg

ons

– tri

angl

es,

squa

res,

rect

angl

es, o

ther

qua

drila

tera

ls,

pent

agon

s, h

exag

ons,

hep

tago

ns a

nd

circ

les

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n sq

uare

s an

d re

ctan

gles

C

hara

cter

istic

s of

sha

pes

st

raig

ht s

ides

curv

ed s

ides

num

ber o

f sid

es

an

gles

in s

hape

D

raw

2-D

sha

pes

on g

rid p

aper

In

volv

ing

angl

es

• Rec

ogni

ze a

nd d

escr

ibe

angl

es in

2-D

sh

apes

2 - D

sha

pes,

R

ecog

nize

, vis

ualiz

e an

d na

me

2-D

sha

pes

in

re

gula

r and

irre

gula

r pol

ygon

s - t

riang

les,

sq

uare

s, re

ctan

gles

, par

alle

logr

ams,

oth

er

quad

rilat

eral

s, p

enta

gons

, hex

agon

s,

hept

agon

s, o

ctag

ons

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n re

ctan

gles

and

par

alle

logr

ams

Cha

ract

eris

tics

of s

hape

s

num

ber o

f sid

es

le

ngth

s in

sid

es

si

zes

of a

ngle

s D

raw

2-D

sha

pes

on g

rid p

aper

In

volv

ing

angl

es

• Rec

ogni

ze a

nd d

escr

ibe

angl

es in

2-D

sh

apes

1

Form

al S

BA

Task

: Tes

t Fo

rmal

SB

A Ta

sk: T

est

Form

al S

BA

Task

: Tes

t 4

Rev

isio

n

Rev

isio

n

Rev

isio

n

56

IM

PORT

ANT

NOTE

: 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 to

pics

, con

cept

s an

d sk

ills.

9MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

T ER

M 2

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

3

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

O

dd a

nd e

ven

num

bers

;

Pl

ace

valu

e;

Mul

tiple

s ;

Rou

ndin

g of

f;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

O

dd a

nd e

ven

num

bers

;

Pl

ace

valu

e;

M

ultip

les;

Fact

ors;

Rou

ndin

g of

f;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

Pr

ime

num

bers

;

Pl

ace

valu

e;

M

ultip

les

;

Fact

ors;

Rou

ndin

g of

f 1

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

6 Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

Addi

tion

and

subt

ract

ion

of w

hole

num

bers

Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

3 M

ultip

licat

ion

of w

hole

num

bers

M

ultip

licat

ion

of w

hole

num

bers

M

ultip

licat

ion

of w

hole

num

bers

3

Div

isio

n of

who

le n

umbe

rs

Div

isio

n of

who

le n

umbe

rs

Div

isio

n of

who

le n

umbe

rs

1 Fo

rmal

SB

A Ta

sk: T

est

Form

al S

BA

Task

: Tes

t Fo

rmal

SB

A Ta

sk: T

est

9

Com

mon

frac

tions

Des

crib

ing

and

orde

ring

fract

ions

Cal

cula

tions

with

frac

tions

Solv

ing

prob

lem

s in

con

text

s

Equi

vale

nt fo

rms

Com

mon

frac

tions

Des

crib

ing

and

orde

ring

fract

ions

Cal

cula

tions

with

frac

tions

Solv

ing

prob

lem

s in

con

text

s

Equi

vale

nt fo

rms

Com

mon

frac

tions

Des

crib

ing

and

orde

ring

fract

ions

Cal

cula

tions

with

frac

tions

Solv

ing

prob

lem

s in

con

text

s

Perc

enta

ges

Eq

uiva

lent

form

s D

ecim

al fr

actio

ns

R

ecog

nisi

ng, o

rder

ing

and

plac

e va

lue

of d

ecim

al fr

actio

ns

C

alcu

latio

ns w

ith d

ecim

al fr

actio

ns

So

lvin

g pr

oble

ms

in c

onte

xts

Eq

uiva

lent

form

s

2 Sy

mm

etry

Rec

ogni

ze, d

raw

and

des

crib

e lin

e(s)

of

sym

met

ry in

2-D

sha

pes.

Sym

met

ry

R

ecog

nize

, dra

w a

nd d

escr

ibe

line(

s) o

f sy

mm

etry

in 2

-D s

hape

s.

Sym

met

ry

R

ecog

nize

, dra

w a

nd d

escr

ibe

line(

s) o

f sy

mm

etry

in 2

-D s

hape

s.

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

topi

cs, c

once

pts

and

skills

.

10 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

5

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

leng

th

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

leng

th

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

leng

th

5

Mas

s

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

ob

ject

s

Mea

surin

g in

stru

men

ts

U

nits

Cal

cula

tions

and

pro

blem

sol

ving

invo

lvin

g m

ass

Mas

s

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

ob

ject

s

Mea

surin

g in

stru

men

ts

U

nits

Cal

cula

tions

and

pro

blem

sol

ving

invo

lvin

g m

ass

Mas

s

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

ob

ject

s

Mea

surin

g in

stru

men

ts

U

nits

Cal

cula

tions

and

pro

blem

sol

ving

invo

lvin

g m

ass

1

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

5

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

capa

city

/vol

ume

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

capa

city

/vol

ume

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

capa

city

/vol

ume

IM

PORT

ANT

NOTE

: 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 to

pics

, con

cept

s an

d sk

ills.

11MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

5

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms,

sphe

res

cy

linde

rs

py

ram

ids

Cha

ract

eris

tics

of o

bjec

ts

sh

apes

of f

aces

flat a

nd c

urve

d su

rface

s

Furt

her a

ctiv

ities

Mak

e 3-

D m

odel

s us

ing

cut o

ut p

olyg

ons

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms

and

othe

r pris

ms

cu

bes

cy

linde

rs

co

nes

py

ram

ids

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n cu

bes

and

rect

angu

lar p

rism

s C

hara

cter

istic

s of

obj

ects

shap

e of

face

s

num

ber o

f fac

es

fla

t and

cur

ved

surfa

ces

Furt

her a

ctiv

ities

Mak

e 3-

D m

odel

s us

ing

c

ut o

ut p

olyg

ons

ne

ts

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms

cu

bes

te

trahe

dron

s

pyra

mid

s

sim

ilarit

ies

and

diffe

renc

es b

etw

een

tetra

hedr

ons

and

othe

r pyr

amid

s C

hara

cter

istic

s of

obj

ects

num

ber a

nd s

hape

of f

aces

num

ber o

f ver

tices

num

ber o

f edg

es

Furt

her a

ctiv

ities

• Mak

e 3-

D m

odel

s us

ing:

drin

king

stra

ws,

toot

hpic

ks e

tc

ne

ts

1 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

6 R

evis

ion

R

evis

ion

R

evis

ion

56

TOTA

L H

OU

RS

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

topi

cs, c

once

pts

and

skills

.

12 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

T ER

M 3

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

3

Who

le n

umbe

rs:

C

ount

ing

Ord

er, c

ompa

re a

nd re

pres

ent n

umbe

rs;

Odd

and

eve

n nu

mbe

rs;

Plac

e va

lue;

M

ultip

les

;

R

ound

ing

off;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

O

dd a

nd e

ven

num

bers

;

Pl

ace

valu

e;

M

ultip

les;

Fact

ors;

Rou

ndin

g of

f;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

Pr

ime

num

bers

;

Pl

ace

valu

e;

M

ultip

les

;

Fact

ors;

Rou

ndin

g of

f 1

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

8 Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

7 M

ultip

licat

ion

and

divi

sion

of w

hole

nu

mbe

rs

Mul

tiplic

atio

n an

d di

visi

on o

f who

le

num

bers

M

ultip

licat

ion

and

divi

sion

of w

hole

nu

mbe

rs

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

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 to

pics

, con

cept

s an

d sk

ills.

6

Geo

met

ric P

atte

rns

Inve

stig

ate

and

exte

nd p

atte

rns

for

rela

tions

hips

or r

ules

of p

atte

rns.

In

put a

nd o

utpu

t val

ues

Equi

vale

nt fo

rms

ve

rbal

ly

in

a fl

ow d

iagr

am

by

a n

umbe

r sen

tenc

e

Geo

met

ric P

atte

rns

Inve

stig

ate

and

exte

nd p

atte

rns

for

rela

tions

hips

or r

ules

of p

atte

rns.

In

put a

nd o

utpu

t val

ues

Equi

vale

nt fo

rms

ve

rbal

ly

in

a fl

ow d

iagr

am

by

a n

umbe

r sen

tenc

e

Geo

met

ric P

atte

rns

Inve

stig

ate

and

exte

nd p

atte

rns

for

rela

tions

hips

or r

ules

of p

atte

rns.

In

put a

nd o

utpu

t val

ues

Equi

vale

nt fo

rms

ve

rbal

ly

in

a fl

ow d

iagr

am

in

tabl

es

by

a n

umbe

r sen

tenc

e 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

.

13MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

8 2

- D s

hape

s,

Rec

ogni

ze, v

isua

lize

and

nam

e 2-

D s

hape

s in

regu

lar a

nd ir

regu

lar p

olyg

ons

– tri

angl

es,

squa

res,

rect

angl

es, o

ther

qua

drila

tera

ls,

pent

agon

s, h

exag

ons

and

circ

les

Cha

ract

eris

tics

of s

hape

s

stra

ight

sid

es

cu

rved

sid

es

nu

mbe

r of s

ides

D

raw

2-D

sha

pes

on g

rid p

aper

Tr

ansf

orm

atio

ns

Bu

ild c

ompo

site

sha

pes

Te

ssel

latio

ns

D

escr

ibe

patte

rns

Posi

tion

and

view

s Lo

catio

n an

d di

rect

ions

2 -

D s

hape

s,

Rec

ogni

ze, v

isua

lize

and

nam

e 2-

D s

hape

s in

regu

lar a

nd ir

regu

lar p

olyg

ons

– tri

angl

es,

squa

res,

rect

angl

es, o

ther

qua

drila

tera

ls,

pent

agon

s, h

exag

ons,

hep

tago

ns a

nd

circ

les

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n sq

uare

s an

d re

ctan

gles

C

hara

cter

istic

s of

sha

pes

st

raig

ht s

ides

curv

ed s

ides

num

ber o

f sid

es

an

gles

in s

hape

D

raw

2-D

sha

pes

on g

rid p

aper

In

volv

ing

angl

es

Rec

ogni

ze a

nd d

escr

ibe

angl

es in

2-D

sha

pes

Tran

sfor

mat

ions

Use

tran

sfor

mat

ions

to m

ake

com

posi

te

shap

es

U

se tr

ansf

orm

atio

ns to

mak

e te

ssel

latio

ns

D

escr

ibe

patte

rns

Posi

tion

and

view

s Lo

catio

n an

d di

rect

ions

2 - D

sha

pes,

R

ecog

nize

, vis

ualiz

e an

d na

me

2-D

sha

pes

in

re

gula

r and

irre

gula

r pol

ygon

s - t

riang

les,

sq

uare

s, re

ctan

gles

, par

alle

logr

ams,

oth

er

quad

rilat

eral

s, p

enta

gons

, hex

agon

s,

hept

agon

s, o

ctag

ons

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n re

ctan

gles

and

par

alle

logr

ams

Cha

ract

eris

tics

of s

hape

s

num

ber o

f sid

es

le

ngth

s in

sid

es

si

zes

of a

ngle

s D

raw

2-D

sha

pes

on g

rid p

aper

In

volv

ing

angl

es

Rec

ogni

ze a

nd d

escr

ibe

angl

es in

2-D

sha

pes

Tran

sfor

mat

ions

Use

tran

sfor

mat

ions

to m

ake

com

posi

te

shap

es

E

nlar

gem

ent a

nd re

duct

ions

Des

crib

e pa

ttern

s Po

sitio

n an

d vi

ews

Loca

tion

and

dire

ctio

ns

1 In

form

al a

sses

smen

t In

form

al a

sses

smen

t In

form

al a

sses

smen

t 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

.

14 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6 4

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

leng

th

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

leng

th

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

leng

th

M

ass

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts ((

milli

litre

s, li

tres)

Cal

cula

tions

and

pro

blem

sol

ving

invo

lvin

g m

ass

Mas

s

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

ob

ject

s

Mea

surin

g in

stru

men

ts

U

nits

(milli

litre

s, li

tres)

Cal

cula

tions

and

pro

blem

sol

ving

invo

lvin

g m

ass

Mas

s

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

ob

ject

s

Mea

surin

g in

stru

men

ts

U

nits

(milli

litre

s, li

tres,

kilo

litre

s)

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

mas

s (in

clud

e fra

ctio

n an

d de

cim

al fo

rms

to

2 de

cim

al p

lace

s)

C

apac

ity/V

olum

e

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

ob

ject

s

Mea

surin

g in

stru

men

ts

U

nits

Cal

cula

tions

and

pro

blem

sol

ving

invo

lvin

g ca

paci

ty/v

olum

e

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

capa

city

/vol

ume

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D

obje

cts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng in

volv

ing

capa

city

/vol

ume

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

.

15MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

12

Com

mon

frac

tions

D

escr

ibin

g an

d or

derin

g fr

actio

ns:

• C

ompa

re a

nd o

rder

(hal

ves;

third

s,

quar

ters

; fift

hs; s

ixth

s; s

even

ths;

eig

hths

)

• Com

mon

frac

tions

in d

iagr

am fo

rm

Cal

cula

tions

with

frac

tions

:

Addi

tion

of c

omm

on fr

actio

ns w

ith th

e sa

me

deno

min

ator

s

Equi

vale

nce

of d

ivis

ion

and

fract

ions

So

lvin

g pr

oble

ms

So

lve

prob

lem

s in

con

text

s in

volv

ing

fract

ions

, inc

ludi

ng g

roup

ing

and

equa

l sh

arin

g Eq

uiva

lent

form

s:

Com

mon

frac

tions

D

escr

ibin

g an

d or

derin

g fr

actio

ns:

C

ount

forw

ards

and

bac

kwar

ds

C

ompa

re a

nd o

rder

com

mon

frac

tions

to a

t le

ast t

wel

fths

Cal

cula

tions

with

frac

tions

:

Addi

tion

and

subt

ract

ion

of c

omm

on

fract

ions

with

the

sam

e de

nom

inat

ors

Ad

ditio

n an

d su

btra

ctio

n of

mix

ed n

umbe

rs

Fr

actio

ns o

f who

le n

umbe

rs

Eq

uiva

lenc

e of

div

isio

n an

d fra

ctio

ns

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, i

nclu

ding

gro

upin

g an

d sh

arin

g Eq

uiva

lent

form

s:

Com

mon

frac

tions

, dec

imal

frac

tions

and

pe

rcen

tage

s

Com

pare

and

ord

er c

omm

on fr

actio

ns,

incl

udin

g te

nths

and

hun

dred

ths

Cal

cula

tions

with

frac

tions

:

Addi

tion

and

subt

ract

ion

of c

omm

on

fract

ions

in

w

hich

one

den

omin

ator

is a

mul

tiple

of

anot

her

Ad

ditio

n an

d su

btra

ctio

n of

mix

ed n

umbe

rs

Fr

actio

ns o

f who

le n

umbe

rs

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, i

nclu

ding

gro

upin

g an

d sh

arin

g Eq

uiva

lent

form

s:

Perc

enta

ges

Fi

nd p

erce

ntag

es o

f who

le n

umbe

rs

1 Fo

rmal

SB

A Ta

sk: T

est

Form

al S

BA

Task

: Tes

t Fo

rmal

SB

A Ta

sk: T

est

4 R

evis

ion

Rev

isio

n R

evis

ion

56

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

.

16 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

T ER

M 4

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

3

Who

le n

umbe

rs:

C

ount

ing

Ord

er, c

ompa

re a

nd re

pres

ent n

umbe

rs;

Odd

and

eve

n nu

mbe

rs;

Plac

e va

lue;

M

ultip

les

;

R

ound

ing

off;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

O

dd a

nd e

ven

num

bers

;

Pl

ace

valu

e;

M

ultip

les;

Fact

ors;

Rou

ndin

g of

f;

Who

le n

umbe

rs:

Cou

ntin

g

O

rder

, com

pare

and

repr

esen

t num

bers

;

Pr

ime

num

bers

;

Pl

ace

valu

e;

M

ultip

les

;

Fact

ors;

Rou

ndin

g of

f 1

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

6 Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

Addi

tion

and

subt

ract

ion

of w

hole

num

bers

Ad

ditio

n an

d su

btra

ctio

n of

who

le n

umbe

rs

6 M

ultip

licat

ion

and

divi

sion

of w

hole

nu

mbe

rs

Mul

tiplic

atio

n an

d di

visi

on o

f who

le

num

bers

M

ultip

licat

ion

and

divi

sion

of w

hole

nu

mbe

rs

6

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms,

sphe

res

cy

linde

rs

py

ram

ids

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms

and

othe

r pris

ms

cu

bes

cy

linde

rs

co

nes

py

ram

ids

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n cu

bes

and

rect

angu

lar p

rism

s

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms

cu

bes

te

trahe

dron

s

pyra

mid

s

sim

ilarit

ies

and

diffe

renc

es b

etw

een

tetra

hedr

ons

and

othe

r pyr

amid

s

1

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

Info

rmal

ass

essm

ent

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 to

pics

, con

cept

s an

d sk

ills.

17MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

char

acte

ristic

s of

obj

ects

shap

es o

f fac

es

fla

t and

cur

ved

surfa

ces

Furt

her a

ctiv

ities

Mak

e 3-

D m

odel

s us

ing

cut o

ut p

olyg

ons

char

acte

ristic

s of

obj

ects

shap

e of

face

s

num

ber o

f fac

es

fla

t and

cur

ved

surfa

ces

Fu

rthe

r act

iviti

es

M

ake

3-D

mod

els

usin

g

cu

t out

pol

ygon

s

nets

char

acte

ristic

s of

obj

ects

num

ber a

nd s

hape

of f

aces

num

ber o

f ver

tices

num

ber o

f edg

es

Furt

her a

ctiv

ities

Mak

e 3-

D m

odel

s us

ing:

drin

king

stra

ws,

toot

hpic

ks e

tc

ne

ts

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

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 to

pics

, con

cept

s an

d sk

ills.

8

Perim

eter

M

easu

re p

erim

eter

usi

ng ru

lers

or m

easu

ring

tape

s M

easu

rem

ent o

f are

a

Find

are

as o

f reg

ular

and

irre

gula

r sha

pes

by c

ount

ing

squa

res

on g

rids

in o

rder

to

deve

lop

an u

nder

stan

ding

of s

quar

e un

its

Mea

sure

men

t of v

olum

e

Find

vol

ume/

capa

city

of o

bjec

ts b

y pa

ckin

g or

fillin

g th

em in

ord

er to

dev

elop

an

unde

rsta

ndin

g of

cub

ic u

nits

Perim

eter

M

easu

re p

erim

eter

usi

ng ru

lers

or m

easu

ring

tape

s M

easu

rem

ent o

f are

a

Find

are

as o

f reg

ular

and

irre

gula

r sha

pes

by c

ount

ing

squa

res

on g

rids

in o

rder

to

deve

lop

an u

nder

stan

ding

of s

quar

e un

its

Mea

sure

men

t of v

olum

e

Find

vol

ume/

capa

city

of o

bjec

ts b

y pa

ckin

g or

fillin

g th

em in

ord

er to

dev

elop

an

unde

rsta

ndin

g of

cub

ic u

nits

Perim

eter

M

easu

re p

erim

eter

usi

ng ru

lers

or m

easu

ring

tape

s M

easu

rem

ent o

f are

a

Con

tinue

to fi

nd a

reas

of r

egul

ar a

nd

irreg

ular

sha

pes

by c

ount

ing

squa

res

on

grid

s

Dev

elop

rule

s fo

r cal

cula

ting

the

area

s of

sq

uare

s an

d re

ctan

gles

M

easu

rem

ent o

f vol

ume

C

ontin

ue to

find

vol

ume/

capa

city

of o

bjec

ts

by p

acki

ng o

r filli

ng th

em

D

evel

op a

n un

ders

tand

ing

of w

hy th

e vo

lum

e of

rect

angu

lar p

rism

s is

giv

en b

y le

ngth

mul

tiplie

d by

wid

th m

ultip

lied

by

heig

ht

Inve

stig

ate

R

elat

ions

hip

betw

een

perim

eter

and

are

a of

re

ctan

gles

and

squ

ares

.

Rel

atio

nshi

p be

twee

n su

rface

are

a an

d vo

lum

e of

rect

angu

lar p

rism

s

18 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

6

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng d

ata

Pr

obab

ility

Toss

ing

a co

in &

rollin

g a

die

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng d

ata

U

ngro

uped

dat

a

Prob

abilit

y

To

ssin

g a

coin

& ro

lling

a di

e

spin

ning

a s

pinn

er

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng d

ata

in

clud

ing

doub

le b

ar g

raph

s

cent

ral t

ende

ncie

s –

(mod

e an

d m

edia

n)

U

ngro

uped

dat

a

Prob

abilit

y

To

ssin

g a

coin

& ro

lling

a di

e

spin

ning

a s

pinn

er

4

Com

mon

frac

tions

D

escr

ibin

g an

d or

derin

g fr

actio

ns:

C

ompa

re a

nd o

rder

(hal

ves;

third

s, q

uarte

rs;

fifth

s; s

ixth

s; s

even

ths;

eig

hths

)

Com

mon

frac

tions

in d

iagr

am fo

rm

Cal

cula

tions

with

frac

tions

:

Addi

tion

of c

omm

on fr

actio

ns w

ith th

e sa

me

deno

min

ator

s

Equi

vale

nce

of d

ivis

ion

and

fract

ions

So

lvin

g pr

oble

ms

So

lve

prob

lem

s in

con

text

s in

volv

ing

fract

ions

, inc

ludi

ng g

roup

ing

and

equa

l sh

arin

g Eq

uiva

lent

form

s:

Com

mon

frac

tions

D

escr

ibin

g an

d or

derin

g fr

actio

ns:

C

ount

forw

ards

and

bac

kwar

ds

C

ompa

re a

nd o

rder

com

mon

frac

tions

to a

t le

ast t

wel

fths

Cal

cula

tions

with

frac

tions

:

Addi

tion

and

subt

ract

ion

of c

omm

on

fract

ions

with

the

sam

e de

nom

inat

ors

Ad

ditio

n an

d su

btra

ctio

n of

mix

ed n

umbe

rs

Fr

actio

ns o

f who

le n

umbe

rs

Eq

uiva

lenc

e of

div

isio

n an

d fra

ctio

ns

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, i

nclu

ding

gro

upin

g an

d sh

arin

g Eq

uiva

lent

form

s:

Com

mon

frac

tions

, dec

imal

frac

tions

and

pe

rcen

tage

s C

ompa

re a

nd o

rder

com

mon

frac

tions

, in

clud

ing

tent

hs a

nd h

undr

edth

s C

alcu

latio

ns w

ith fr

actio

ns:

Ad

ditio

n an

d su

btra

ctio

n of

com

mon

fra

ctio

ns in

whi

ch o

ne d

enom

inat

or is

a

mul

tiple

of a

noth

er

Ad

ditio

n an

d su

btra

ctio

n of

mix

ed n

umbe

rs

Fr

actio

ns o

f who

le n

umbe

rs

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, i

nclu

ding

gro

upin

g an

d sh

arin

g Eq

uiva

lent

form

s:

Perc

enta

ges

Fi

nd p

erce

ntag

es o

f who

le n

umbe

rs

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 to

pics

, con

cept

s an

d sk

ills.

19MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TIM

E

(HO

UR

S)

GR

ADE

4 G

RAD

E 5

GR

ADE

6

3

Num

ber s

ente

nces

nu

mbe

r sen

tenc

es d

escr

ibin

g pr

oble

m

situ

atio

ns

So

lve

and

com

plet

e nu

mbe

r sen

tenc

es

by in

spec

tion,

-tria

l and

impr

ovem

ent,

chec

k so

lutio

n by

sub

stitu

tion

Num

ber s

ente

nces

nu

mbe

r sen

tenc

es d

escr

ibin

g pr

oble

m

situ

atio

ns

so

lve

and

com

plet

e nu

mbe

r sen

tenc

es

by in

spec

tion,

tria

l and

impr

ovem

ent,

chec

k so

lutio

n by

sub

stitu

tion

Num

ber s

ente

nces

nu

mbe

r sen

tenc

es d

escr

ibin

g pr

oble

m

situ

atio

ns

So

lve

and

com

plet

e nu

mbe

r sen

tenc

es

by in

spec

tion,

tria

l and

impr

ovem

ent,

chec

k so

lutio

n by

sub

stitu

tion

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

End

of y

ear e

xam

En

d of

yea

r exa

m

End

of y

ear e

xam

6 R

evis

ion

Rev

isio

n R

evis

ion

52

TOTA

L H

OU

RS

IMPO

RTA

NT

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

.

20 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LE

SSO

N P

LAN

S: T

ERM

1

21MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

1

TER

M 1

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 5

HO

UR

S G

RAD

E 4

CAP

S pp

. 37

- 38

5 C

APS

pp. 1

25 -1

26

6 C

APS

pp. 2

15 –

216

TO

PIC

W

hole

num

bers

C

once

pts,

Ski

lls a

nd

know

ledg

e

Cou

nt fo

rwar

ds a

nd b

ackw

ards

in 2

s,

3s, 5

s, 1

0s, 2

5s, 5

0s, 1

00s

betw

een

0 an

d at

leas

t 1 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

0 -

999.

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 3

-dig

it nu

mbe

rs

Rou

nd o

ff to

the

near

est 1

0, 1

00 a

nd

1 00

0

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

4-d

igit

num

bers

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

.

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 6

dig

it nu

mbe

rs.

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00

and

1 00

0

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

9-d

igit

num

bers

Rep

rese

nt p

rime

num

bers

to a

t lea

st

100

R

ecog

nizi

ng th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 9

-dig

it nu

mbe

rs

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00,

1 00

0, 1

00 0

00, a

nd 1

000

000

22 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Expl

ain

from

kno

wn

(low

num

ber r

ange

); w

ork

with

con

cret

e LT

SM

Cou

ntin

g, o

rder

ing,

com

parin

g, re

pres

entin

g of

dig

its

St

art w

ith 2

-dig

it nu

mbe

rs a

nd b

uild

up

to 3

-dig

it nu

mbe

rs

Not

e: A

ccor

ding

to C

APS

num

ber r

ange

per

gra

de p

er te

rm a

s in

con

tent

cla

rific

atio

n

R

ead,

say

and

writ

e up

to a

t lea

st 4

-dig

it nu

mbe

rs, (

Lear

ners

can

writ

e nu

mbe

rs w

ith fi

nger

s in

the

air w

hile

they

are

read

ing

/ say

ing

the

wor

ds.)

e.g.

Tw

enty

-nin

e

C

onve

rt fro

m w

ords

to n

umbe

rs a

nd n

umbe

rs to

wor

ds

W

rite

the

num

ber t

hat i

s eq

uiva

lent

to:

Ar

rang

e th

e nu

mbe

rs b

elow

from

the

smal

lest

to th

e bi

gges

t: 11

1; 1

101

; 1 1

10; 1

011

M

ake

the

bigg

est/s

mal

lest

num

ber y

ou c

an w

ith th

ese

digi

ts: 3

, 2, 5

, 4, 0

.

Odd

and

eve

n nu

mbe

rs:

Ex

plor

e pr

actic

ally

by

wor

king

with

num

bers

to b

e ab

le to

des

crib

e th

e pa

ttern

e.g

. all

odd

num

bers

end

in 1

, 3, 5

, 7 o

r 9

ther

efor

e 20

1, 2

03, 2

05, 2

07, 2

09 a

re a

ll od

d nu

mbe

rs

Prim

e nu

mbe

rs in

trod

uced

to G

rade

6

U

se n

umbe

r cha

rt to

isol

ate

prim

e nu

mbe

rs fr

om c

ompo

site

num

bers

Mul

tiple

s:

If

you

coun

t in

mul

tiple

s of

3, u

p to

100

.

o W

ill yo

u co

unt t

he n

umbe

r 13?

Why

?

o W

ill yo

u co

unt t

he n

umbe

r 42?

Why

?

23MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Plac

e va

lue:

D

istin

guis

h be

twee

n nu

mer

ic v

alue

and

pla

ce v

alue

U

se p

lace

val

ue c

ards

to b

uild

up

and

brea

k do

wn

num

bers

Wha

t is

diffe

rent

to G

rade

3?

R

ound

ing

off t

o th

e ne

ares

t

Rou

ndin

g of

f:

R

ound

off

to th

e ne

ares

t te

n, h

undr

ed a

nd th

ousa

nd (u

se b

ase

ten

bloc

ks, n

umbe

r lin

es, p

lace

val

ue)

Pl

ace

valu

e Ta

ble

U

se th

e co

ncep

t of r

ound

ing

off

in p

robl

em s

olvi

ng c

onte

xt

Gra

de 4

and

5:

In T

erm

1, l

earn

ers

shou

ld r

evis

e an

d co

nsol

idat

e w

ork

done

in G

rade

3. R

ecom

men

ded

spec

ifica

tions

are

pro

vide

d be

low.

Cou

ntin

g

Cou

nt fo

rwar

ds a

nd b

ackw

ards

in 2

s, 3

s, 5

s, 1

0s, 2

5s, 5

0s, 1

00s

betw

een

0 an

d at

leas

t 1 0

00

C

ount

ing

shou

ld n

ot o

nly

be th

ough

t of a

s ve

rbal

cou

ntin

g. L

earn

ers

shou

ld c

ount

usi

ng a

ppar

atus

suc

h as

o C

ount

ers

o co

untin

g be

ads

o nu

mbe

r grid

s

o st

ruct

ured

, sem

i-stru

ctur

ed a

nd e

mpt

y nu

mbe

r lin

es

o pi

ctur

es o

f obj

ects

, esp

ecia

lly p

ictu

res

of la

rge

num

bers

of o

bjec

ts th

at a

re p

rese

nted

in a

gro

uped

or s

truct

ured

way

. An

exam

ple

of a

pic

ture

of o

bjec

ts s

uita

ble

for c

ount

ing

is p

rovi

ded

at th

e en

d of

the

Gra

de 4

sec

tion

of

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

.

o ar

rays

or d

iagr

ams

of a

rrays

e.g

.

24 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

ot

her d

iagr

ams

for c

ount

ing

e.g.

Cou

ntin

g sh

ould

not

alw

ays

star

t on

the

first

mul

tiple

, nor

sho

uld

it al

way

s st

art o

n an

y ot

her m

ultip

le e

.g. c

ount

ing

in 2

s

can

star

t fro

m 5

or 2

7 or

348

.

Plac

e va

lue

(num

ber r

ange

0 to

999

)

Le

arne

rs s

houl

d be

abl

e to

bre

ak u

p nu

mbe

rs in

to h

undr

eds,

tens

and

uni

ts u

sing

o th

e nu

mbe

r nam

es (n

umbe

r wor

ds)

o pl

ace

valu

e or

flas

h ca

rds

o ex

pand

ed n

otat

ion

R

ecom

men

ded

appa

ratu

s: p

lace

val

ue/fl

ash

card

s; D

iene

s bl

ocks

Com

pare

and

ord

er (n

umbe

r ran

ge 0

to 9

99)

Le

arne

rs s

houl

d be

giv

en a

rang

e of

exe

rcis

es s

uch

as:

o Ar

rang

e th

e gi

ven

num

bers

bel

ow fr

om th

e sm

alle

st to

the

bigg

est o

r big

gest

to s

mal

lest

o Fi

ll in

mis

sing

num

bers

in

◊ a

sequ

ence

◊ on

a n

umbe

r grid

25MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

o Sh

ow a

giv

en n

umbe

r on

a st

ruct

ured

or s

emi-s

truct

ured

num

ber l

ine,

e.g

. sho

w w

hich

num

ber i

s ha

lfway

bet

wee

n

340

and

350

on a

num

ber l

ine

o In

dica

te w

hich

of t

wo

num

bers

is g

reat

er o

r sm

alle

r e.g

. 5 4

31 o

r 5 4

13

o R

epla

ce *

with

<,=

or >

Exa

mpl

e: 8

9 * 9

8, 1

09 *

190

A

ll w

ork

deve

lope

d he

re c

an b

e pr

actis

ed th

roug

hout

the

year

in th

e M

enta

l Mat

hem

atic

s pr

ogra

mm

e.

Gra

de 6

: Pl

ace

valu

e (n

umbe

r ran

ge 0

to 9

99 9

99)

Le

arne

rs s

houl

d be

abl

e to

bre

ak u

p nu

mbe

rs in

to h

undr

eds,

tens

and

uni

ts u

sing

o th

e nu

mbe

r nam

es (n

umbe

r wor

ds)

o pl

ace

valu

e or

flas

h ca

rds

o ex

pand

ed n

otat

ion

R

ecom

men

ded

appa

ratu

s: p

lace

val

ue, f

lash

car

ds, D

iene

s bl

ocks

Com

pare

and

ord

er

Her

e le

arne

rs s

houl

d be

giv

en a

rang

e of

exe

rcis

es

Ar

rang

e th

e gi

ven

num

bers

bel

ow f

rom

the

smal

lest

to th

e bi

gges

t: or

big

gest

to s

mal

lest

Fi

ll in

mis

sing

num

bers

in

o a

sequ

ence

o on

a n

umbe

r grid

◊ Sh

ow a

giv

en n

umbe

r on

a nu

mbe

r lin

e –

stru

ctur

ed o

r sem

i-stru

ctur

ed e

.g. s

how

on

a nu

mbe

r lin

e w

hich

num

ber i

s ha

lfway

bet

wee

n 47

1 34

0 an

d 47

1 35

0.

In

dica

te w

hich

of t

wo

num

bers

is g

reat

er o

r sm

alle

r: 39

5431

or 3

9541

3?

Fi

ll in

<, =

or >

a)

247

889

* 2

47 8

98

26 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

b)

784

109

* 7

85 1

90

All w

ork

deve

lope

d he

re c

an b

e pr

actis

ed th

roug

hout

the

year

in th

e M

enta

l Mat

hem

atic

s pr

ogra

mm

e.

LTSM

Res

ourc

es

Num

ber l

ines

: 0 –

120

, Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, F

lash

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, mul

tiplic

atio

n ta

ble,

aba

cus

, bea

ds

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

1: p

p. 2

– 7

A

ctiv

ities

: 1a&

b; 2

W

B 1

: pp.

2

Activ

ities

: 1a&

b W

B 1

: pp.

2 –

4

Activ

ities

: 1a&

b, 2

, 3

DB

E Te

xtbo

ok re

fere

nce

Term

1: U

nit 1

: Who

le N

umbe

rs p

. 3

Term

1: U

nit 1

: Who

le N

umbe

rs p

. 3

Term

1: U

nit 1

: Who

le N

umbe

rs p

. 3

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

27MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

2

TER

M 1

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to

……

…

Tim

e: 6

HO

UR

S G

RAD

E 4

CAP

S pp

. 39

– 42

5

CAP

S pp

. 127

-131

6

CAP

S pp

. 217

– 2

22

TOPI

C

Num

ber s

ente

nces

C

once

pts,

Ski

lls a

nd

know

ledg

e

• W

rite

num

ber s

ente

nces

to

desc

ribe

prob

lem

situ

atio

ns

•

Solv

e an

d co

mpl

ete

num

ber

sent

ence

s by

- in

spec

tion

-

trial

and

impr

ovem

ent

• C

heck

sol

utio

n by

sub

stitu

tion

• W

rite

num

ber s

ente

nces

to

desc

ribe

prob

lem

situ

atio

ns

• So

lve

and

com

plet

e nu

mbe

r se

nten

ces

by

-

insp

ectio

n

- tri

al a

nd im

prov

emen

t •

Che

ck s

olut

ion

by s

ubst

itutio

n

• W

rite

num

ber s

ente

nces

to

desc

ribe

prob

lem

situ

atio

ns

• So

lve

and

com

plet

e nu

mbe

r se

nten

ces

by

-

insp

ectio

n

- tri

al a

nd im

prov

emen

t •

Che

ck s

olut

ion

by s

ubst

itutio

n

SUG

GES

TED

M

ETH

OD

OLO

GY

U

se n

umbe

r sen

tenc

es to

des

crib

e pr

oble

m s

ituat

ions

. Ex

ampl

e: P

robl

em s

ituat

ion:

I bo

ught

750

g of

sw

eets

and

div

ided

them

am

ong

25 c

hild

ren.

Num

ber s

ente

nce:

750

÷25

U

se n

umbe

r sen

tenc

es a

s eq

uiva

lent

form

of e

xpre

ssio

n to

sec

tions

of f

low

dia

gram

or t

able

s.

Exam

ple:

Tab

le

1 2

3 4

×3

28 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Flow

dia

gram

E

quiv

alen

t rep

rese

ntat

ion

of th

e ta

ble

and

the

flow

dia

gram

: ×

3

• U

sing

num

ber

sent

ence

s to

hel

p le

arne

rs u

nder

stan

d an

d us

e th

e fa

ct t

hat

addi

tion

and

subt

ract

ion

are

inve

rse

oper

atio

ns

Exa

mpl

e: 3

7 –

4 +

4 =

∆ •

Use

num

ber s

ente

nces

hel

ps le

arne

rs d

evel

op a

dditi

on a

nd s

ubtra

ctio

n te

chni

ques

Exa

mpl

e: 3

6+13

= th

eref

ore

49 –

13

=

•

Com

mut

ativ

e pr

oper

ty o

f add

ition

Num

bers

can

be

adde

d in

any

ord

er.

Exam

ple:

29

+ 19

= 1

9 +

26

T

here

fore

13

+ 49

= o

r 49

+ 1

3 =

•

Ass

ocia

tive

prop

erty

of a

dditi

on

Th

e as

soci

ativ

e pr

oper

ty a

llow

s nu

mbe

rs t

o be

gro

uped

in

diffe

rent

way

s w

hen

addi

ng m

ore

than

wo

num

bers

, with

out i

t af

fect

ing

the

answ

er.

Exa

mpl

e: (3

1 +

26) +

19=

i

s th

e sa

me

as 3

1 +(

26+

19) =

• U

se n

umbe

r sen

tenc

es to

hel

p le

arne

rs s

ee a

nd u

se p

atte

rns

in a

dditi

on a

nd s

ubtra

ctio

n nu

mbe

r bon

ds fo

r:

- 10

- m

ultip

les

of 1

0

- m

ultip

les

of 1

00

29MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

E

xam

ples

:

- Te

n

3 +

7=

4

+ 6

= 2

+ 8

=

5 +

5 =

- M

ultip

les

of 1

0

1

3 +

7 =

14

+ 6

=

12

+ 8=

15 +

5 =

- M

ultip

les

of 1

00

Sim

ilar e

xam

ples

can

be

give

n fo

r mul

tiple

s of

100

suc

h as

200

; 300

; 400

; 500

; 600

; 700

; 800

; 900

[S

ee C

APS

at th

e in

dica

ted

page

s fo

r mor

e ex

ampl

e / d

etai

ls]

LTSM

R

esou

rces

N

umbe

r lin

es, S

truct

ured

-, s

emi-

stru

ctur

ed -

, uns

truct

ured

num

ber l

ines

; Cou

nter

s, A

bacu

s, P

ictu

res,

arra

ys/ d

iagr

ams,

flow

di

agra

ms,

mat

hs v

ideo

clip

s, E

lear

ning

reso

urce

s on

line,

Mat

hs g

ames

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

1: p

p. 1

4 - 1

6

Activ

ities

: 6 a

& b

; 29

WB

1: p

p. 1

0 –

12

Activ

ities

: 4 a

nd 5

W

B 1

: pp.

73

-74

Ac

tivity

: 4

DB

E Te

xtbo

ok

refe

renc

e Te

rm 1

: Uni

t 2: N

umbe

r Sen

tenc

es p

. 18

Term

1: U

nit 2

: Num

ber s

ente

nces

p.

13

Term

1: U

nit 2

: Num

ber s

ente

nces

p.

18

HO

MEW

OR

K

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

30 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

3

TER

M 1

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to

……

…

Tim

e: 7

HO

UR

S G

RAD

E 4

CAP

S pp

. 43

– 45

5

CAP

S pp

. 132

-135

6

CAP

S pp

. 222

– 2

25

TOPI

C

Addi

tion

and

subt

ract

ion

of w

hole

num

bers

C

once

pts,

Ski

lls a

nd

know

ledg

e

Addi

tion

and

subt

ract

ion

of w

hole

nu

mbe

rs o

f at l

east

4 d

igits

C

alcu

latio

n te

chni

ques

U

se a

rang

e of

tech

niqu

es to

per

form

an

d ch

eck

writ

ten

and

men

tal

calc

ulat

ions

of w

hole

num

bers

in

clud

ing:

•

estim

atio

n

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

• us

ing

a nu

mbe

r lin

e •

usin

g ad

ditio

n an

d su

btra

ctio

n as

inve

rse

oper

atio

ns

Addi

tion

and

subt

ract

ion

of w

hole

nu

mbe

rs o

f at l

east

5 d

igits

C

alcu

latio

n te

chni

ques

U

sing

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal

calc

ulat

ions

with

who

le n

umbe

rs

incl

udin

g:

• es

timat

ion

• ad

ding

and

sub

tract

ing

in c

olum

ns

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

• us

ing

a nu

mbe

r lin

e •

usin

g ad

ditio

n an

d su

btra

ctio

n as

inve

rse

oper

atio

ns

• A

dditi

on a

nd s

ubtra

ctio

n of

who

le

num

bers

with

at l

east

6-d

igit

num

ber

• M

ultip

le o

pera

tions

on

who

le

num

bers

with

or w

ithou

t br

acke

ts

Cal

cula

tion

tech

niqu

es

Usi

ng a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l cal

cula

tions

with

who

le

num

bers

incl

udin

g:

• es

timat

ion

• ad

ding

, sub

tract

ing

in c

olum

ns

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g

• us

ing

addi

tion

and

subt

ract

ion

as in

vers

e op

erat

ions

•

usin

g a

calc

ulat

or

31MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Prop

ertie

s of

who

le n

umbe

rs

Rec

ogni

ze a

nd u

se th

e co

mm

utat

ive

and

asso

ciat

ive

prop

ertie

s of

who

le n

umbe

rs

Sol

ving

pro

blem

s So

lve

prob

lem

s in

con

text

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

fina

ncia

l co

ntex

ts

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e pr

oper

ties

with

w

hole

num

bers

•

0 in

term

s of

its

addi

tive

prop

erty

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

finan

cial

con

text

s

mea

sure

men

t con

text

s

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

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

So

lvin

g pr

oble

ms

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs a

nd d

ecim

al

fract

ions

, inc

ludi

ng

fin

anci

al c

onte

xts

m

easu

rem

ent c

onte

xts

• So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

SUG

GES

TED

M

ETH

OD

OLO

GY

• R

evis

e an

d co

nsol

idat

e w

ork

done

in G

rade

3.

• Le

arne

rs s

houl

d so

lve

prob

lem

s in

con

text

s an

d do

con

text

free

cal

cula

tions

. •

Lear

ners

sho

uld

be tr

aine

d to

judg

e th

e re

ason

able

ness

of s

olut

ions

. •

Lear

ners

sho

uld

know

that

they

can

: o

chec

k an

add

ition

cal

cula

tion

by s

ubtra

ctio

n.

Exam

ple:

If 9

6 +4

8 =1

44, t

hen

144

– 48

= 9

6 o

chec

k a

subt

ract

ion

calc

ulat

ion

by a

ddin

g.

Exa

mpl

e:14

4 –

48 =

96,

then

96

+ 48

= 1

44

32 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Cal

cula

tion

tech

niqu

es

• B

reak

ing

dow

n al

l num

bers

acc

ordi

ng to

pla

ce v

alue

par

ts to

add

.

Exam

ple:

Cal

cula

te 3

62 +

486

36

2 +

486

= 30

0 +

60 +

2 +

400

+ 8

0 +

6

2

+ 6

=

8

=

300

+ 40

0 +

60 +

80

+ 2

+ 6

or

and

60

+ 8

0

= 1

4

= 70

0 +

140

+ 8

an

d

30

0 +

400

= 70

0

= 84

8

mea

ns 3

62 +

486

= 8

48

• A

ddin

g on

(by

brea

king

dow

n th

e nu

mbe

r to

be a

dded

) Ex

ampl

e: C

alcu

late

362

+ 4

86

362

+ 40

0 →

762

+ 8

0 →

842

+ 6

→ 8

48

• Fi

lling

up

tens

(by

brea

king

dow

n th

e nu

mbe

r to

be a

dded

). Th

is c

an a

lso

be c

alle

d ro

undi

ng o

ff an

d co

mpe

nsat

ing.

Exa

mpl

e: C

alcu

late

96

+ 48

9

6 +

48 =

96

+ 4

– 4

+ 48

= 1

00 +

48

– 4

= 10

0 +

44 =

144

• B

reak

ing

dow

n bo

th n

umbe

rs a

ccor

ding

to p

lace

val

ue p

arts

to s

ubtr

act.

Ex

ampl

e: C

alcu

late

687

– 1

43

687

– 14

3 =

600

+ 80

+ 7

– 1

00 –

40

– 3

7 –

3

= 4

= 6

00 –

100

+ 8

0 –

40 +

7 –

3

or

and

80 –

40

=

40

= 5

00 +

40

+ 4

and

6

00 –

100

= 5

00

= 5

44

mea

ns 6

87 –

143

= 5

44

• B

reak

ing

dow

n al

l num

bers

acc

ordi

ng to

pla

ce v

alue

par

ts to

add

usi

ng c

ompe

nsat

ion

(cou

nter

bala

nce)

Le

arne

rs c

anno

t sub

tract

4 fr

om 3

or 8

0 fro

m 4

0. In

stea

d of

bre

akin

g do

wn

743

into

700

+ 4

0 +

3 th

ey w

ill br

eak

dow

n 74

3 in

to 6

00 +

130

+ 1

3. T

hen

they

can

sub

tract

4 fr

om 1

3 an

d 80

from

130

. Ex

ampl

e: C

alcu

late

: 743

– 6

84

743

– 68

4 =

700

+ 40

+ 3

– 6

00 –

80

– 4

= 60

0 +

130

+ 13

– 6

00 –

80

– 4

(Bre

ak u

p 74

3 in

to 6

00 +

130

+ 1

3)

= 60

0 - 6

00 +

130

– 8

0 +

13 –

4

= 5

0 +

9 =

59

33MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• Su

btra

ctin

g by

bre

akin

g do

wn

the

num

ber t

o be

sub

trac

ted

exam

ple:

C

alcu

late

687

– 1

43

687

– 10

0 →

587

– 4

0 →

547

– 3

= 5

44

or

687

– 14

0 –

3 =

547

– 3

= 54

4 •

Col

umn

met

hod

for a

ddin

g By

Gra

de 6

lear

ners

sho

uld

have

had

eno

ugh

expe

rienc

e w

ith b

reak

ing

up n

umbe

rs to

add

and

sub

tract

them

. The

ho

rizon

tal m

etho

d of

exp

andi

ng n

umbe

rs b

efor

e ad

ding

them

can

get

unw

ield

y w

hen

mor

e th

an tw

o 5-

digi

t num

bers

are

ad

ded.

Ter

m 1

lear

ners

can

revi

sit t

he e

xpan

ded

verti

cal m

etho

d, a

nd th

en m

ove

on to

the

tradi

tiona

l col

umn

met

hod

• Ex

pand

ed v

ertic

al c

olum

n m

etho

d to

add

5

6 42

3 =

5

0 00

0 +

6 00

0 +

400

+

20

+

3

+

7 58

1 =

7

000

+

5

00

+

80

+

1

+21

479

=

20

000

+

1

000

+

4

00

+

70

+

9

Tot

al

=

70 0

00

+

14 0

00

+

1 3

00

+

170

+

13

This

can

be

writ

ten

as 7

0 00

0 +

10 0

00 +

5 0

00 +

400

+ 8

0 +

3 =

85 4

83

• Th

e ve

rtic

al c

olum

n m

etho

d to

add

.

1 1

1 1

56 4

23

+ 2

1 47

9 +

7 5

81

85 48

3 •

Expa

nded

ver

tical

col

umn

met

hod

to s

ubtr

act e

xam

ple:

Cal

cula

te: 9

8 74

3 –

45 6

84

600

130

13

98

743

=

90

000

+

8

000

+

7

00

+

40

+

3

– 45

684

=

40 0

00

+

5 0

00

+

600

+

80

+

4

T

otal

=

5

0 00

0 +

3 00

0 +

0

+

50

+

9

Th

eref

ore

50 0

00 +

3 0

00 +

0 +

50

+ 9

= 53

059

34 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• T

he v

ertic

al c

olum

n m

etho

d to

sub

trac

t

6 1

3 13

98

743

– 4

5 684

53 05

9

Wor

king

with

cal

cula

tors

•

The

men

tal m

athe

mat

ics

prog

ram

me

cont

ains

wor

k on

num

ber c

once

pt, n

umbe

r fac

ts a

nd m

enta

l cal

cula

tion

tech

niqu

es.

Dai

ly w

ork

on m

enta

l Mat

hem

atic

s co

mbi

ned

with

dai

ly w

ritte

n ca

lcul

atio

ns w

ill pr

even

t lea

rner

s fro

m b

ecom

ing

depe

nden

t on

cal

cula

tors

and

not

kno

win

g ho

w to

cal

cula

te w

ithou

t the

m.

• C

alcu

lato

rs a

re a

use

ful w

ay fo

r lea

rner

s to

exp

lore

num

ber p

atte

rns

and

whe

n w

orki

ng w

ith v

ery

larg

e nu

mbe

rs.

• Le

arne

rs s

houl

d be

taug

ht h

ow to

use

cal

cula

tors

incl

udin

g ho

w to

cle

ar a

n in

corre

ctly

ent

ered

num

ber.

Lear

ners

sho

uld

alw

ays

estim

ate

answ

ers

befo

re d

oing

a c

alcu

latio

n on

a c

alcu

lato

r. Le

arne

rs s

houl

d es

timat

e w

heth

er th

eir a

nsw

ers

will

be

in te

ns, h

undr

eds,

thou

sand

s, te

n th

ousa

nds,

hun

dred

thou

sand

s or

milli

ons.

For

exa

mpl

e w

hen

addi

ng 1

2 34

5 an

d 87

654

th

ey s

houl

d es

timat

e th

at th

e an

swer

will

be b

etw

een

90 a

nd 1

00 th

ousa

nd.

Kin

ds o

f pro

blem

s •

Sum

mat

ion,

incr

ease

and

dec

reas

e, c

ompa

rison

by

diffe

renc

e •

See

the

desc

riptio

n of

pro

blem

type

s at

the

end

of th

e gr

ade

note

s LT

SM

Res

ourc

es

Num

ber l

ines

: 0 –

120

, Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, F

lash

ca

rds,

Bas

e 10

blo

cks,

100

cha

rts, m

aths

vid

eo c

lips,

E le

arni

ng re

sour

ces

onlin

e, M

aths

gam

es

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

1: p

p. 1

8 –

38

Activ

ities

: 7 a

&b –

11a&

b; 1

2

WB

1: p

p. 1

4 -2

8 Ac

tiviti

es: 6

a&b

; 7a&

b; 8

a&b;

9a&

b p

98

Activ

ity: 3

1

WB

1: p

p. 1

4 - 2

4 Ac

tiviti

es: 6

a&b;

7a&

b

DB

E Te

xtbo

ok re

fere

nce

Term

1: U

nit 3

: Add

ition

and

su

btra

ctio

n, p

. 26

Term

1: U

nit 3

: Add

ition

and

su

btra

ctio

n, p

. 21

Term

1: U

nit 3

: Add

ition

and

su

btra

ctio

n, p

. 25

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

35MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

4

TER

M 1

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

……

T

ime:

3 H

OU

RS

GR

ADE

4 C

APS

pp. 4

6 –

51

5 C

APS

pp. 1

36 -1

39

6 C

APS

pp. 2

35 –

238

TO

PIC

N

umer

ic a

nd g

eom

etric

pat

tern

s C

once

pts,

Ski

lls a

nd

know

ledg

e

Inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps

or ru

les

of p

atte

rns:

-

sequ

ence

s in

volv

ing

a co

nsta

nt d

iffer

ence

or r

atio

-

of le

arne

r’s o

wn

crea

tion

• D

escr

ibe

obse

rved

rela

tions

hips

or

rule

s in

lear

ner’s

ow

n w

ords

In

put a

nd o

utpu

t val

ues

Det

erm

ine

inpu

t val

ues,

out

put v

alue

s an

d ru

les

for p

atte

rns

and

rela

tions

hips

usi

ng fl

ow d

iagr

ams

Equi

vale

nt fo

rms

Det

erm

ine

equi

vale

nce

of d

iffer

ent

desc

riptio

ns o

f the

sam

e re

latio

nshi

p or

rule

pre

sent

ed:

• ve

rbal

ly

• in

a fl

ow d

iagr

am

• by

a n

umbe

r sen

tenc

e

Inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps

or ru

les

of p

atte

rns

- se

quen

ces

inv

olvi

ng

a

cons

tant

diff

eren

ce o

r rat

io

- of

lear

ner’s

ow

n cr

eatio

n •

Des

crib

e ob

serv

ed re

latio

nshi

ps o

r ru

les

in le

arne

r’s o

wn

wor

ds

Inpu

t and

out

put v

alue

s D

eter

min

e in

put v

alue

s, o

utpu

t val

ues

and

rule

s fo

r pat

tern

s an

d re

latio

nshi

ps

usin

g flo

w d

iagr

ams

Equi

vale

nt fo

rms

Det

erm

ine

equi

vale

nce

of d

iffer

ent

desc

riptio

ns o

f the

sam

e re

latio

nshi

p or

rule

pre

sent

ed

• ve

rbal

ly

• in

a fl

ow d

iagr

am

• by

a n

umbe

r sen

tenc

e

Inve

stig

ate

and

exte

nd p

atte

rns

• In

vest

igat

e an

d ex

tend

num

eric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps o

r ru

les

of p

atte

rns

- se

quen

ces

invo

lvin

g a

co

nsta

nt d

iffer

ence

or r

atio

-

of le

arne

r’s o

wn

crea

tion

• D

escr

ibe

obse

rved

rela

tions

hips

or

rule

s in

lear

ner’s

ow

n w

ords

In

put a

nd o

utpu

t val

ues

Det

erm

ine

inpu

t val

ues,

out

put v

alue

s an

d ru

les

for p

atte

rns

and

rela

tions

hips

us

ing

flow

dia

gram

s Eq

uiva

lent

form

s D

eter

min

e eq

uiva

lenc

e of

diff

eren

t de

scrip

tions

of t

he s

ame

rela

tions

hip

or

rule

pre

sent

ed

• ve

rbal

ly

• in

a fl

ow d

iagr

am

• by

a n

umbe

r sen

tenc

e

36 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Num

eric

pat

tern

s:

All

grad

es:

Rem

embe

r tha

t pro

gres

sion

is ta

king

pla

ce in

the

resp

ectiv

e gr

ades

as

indi

cate

d in

the

CAP

S do

cum

ent.

Sequ

ence

s of

num

bers

: Ex

ampl

es o

f the

abo

ve a

re il

lust

rate

d in

Ter

m 3

. For

Ter

m 1

the

reco

mm

enda

tion

is to

focu

s on

usi

ng in

put-o

utpu

t dia

gram

s,

with

a fo

cus

on d

evel

opin

g m

ultip

licat

ion

tabl

es a

nd th

e pr

oper

ties

of o

pera

tions

.

Patte

rns

give

n in

inpu

t-out

put d

iagr

ams

Inpu

t-out

put d

iagr

ams

are

som

etim

es c

alle

d fu

nctio

n di

agra

ms,

func

tion

mac

hine

s or

flow

dia

gram

s be

caus

e th

ey a

re a

way

of in

trodu

cing

lear

ners

to fu

nctio

nal r

elat

ions

hips

dia

gram

mat

ical

ly. F

unct

iona

l rel

atio

nshi

ps b

ecom

e ve

ry im

porta

nt in

the

Seni

or P

hase

and

FET

Mat

hem

atic

s.

The

form

s of

inpu

t-out

put d

iagr

ams

that

lear

ners

use

in th

e In

term

edia

te P

hase

mos

t ofte

n ar

e flo

w d

iagr

ams

or s

pide

rgra

ms.

Whe

n us

ing

flow

dia

gram

s, th

e co

rresp

onde

nce

betw

een

inpu

t and

out

put v

alue

s sh

ould

be

clea

r in

its re

pres

enta

tiona

l for

m

i.e. t

he fi

rst i

nput

pro

duce

s th

e fir

st o

utpu

t, th

e se

cond

inpu

t pro

duce

s th

e se

cond

out

put,

etc.

Exa

mp

le

37MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

An in

put-o

utpu

t dia

gram

can

allo

w le

arne

rs to

see

or w

ork

out t

he

• in

put v

alue

s, if

the

rule

and

a c

orre

spon

ding

out

put v

alue

are

giv

en

• ou

tput

val

ues,

if th

e ru

le a

nd a

cor

resp

ondi

ng in

put v

alue

s ar

e gi

ven

• ru

le, i

f the

rule

wor

ks fo

r eve

ry g

iven

inpu

t val

ue a

nd it

s co

rresp

ondi

ng o

utpu

t val

ue

Tabl

es a

re a

use

ful w

ay to

reco

rd p

atte

rns

in G

rade

s 4

& 5.

In G

rade

4 it

is u

sefu

l to

som

etim

es in

clud

e th

e ru

le in

a ta

ble.

Exa

mpl

e:

In T

erm

1 it

is re

com

men

ded

that

num

ber p

atte

rns

be u

sed

to d

evel

op c

once

pts

and

skills

that

will

be u

sed

in m

ultip

licat

ion

and

divi

sion

. The

focu

s ca

n be

on

inpu

t-out

put f

low

dia

gram

s th

at h

elp

lear

ners

to u

nder

stan

d an

d le

arn

abou

t

• th

e in

vers

e op

erat

ion

betw

een

mul

tiplic

atio

n an

d di

visi

on

• th

e m

ultip

licat

ion

of u

nits

by

mul

tiple

s of

ten

• th

e as

soci

ativ

e pr

oper

ty w

ith w

hole

num

bers

and

how

we

can

use

this

pro

perty

whe

n w

e m

ultip

ly b

y m

ultip

les

of 1

0

Usi

ng fl

ow d

iagr

ams

help

lear

ners

to u

nder

stan

d an

d us

e th

e fa

ct th

at m

ultip

licat

ion

and

divi

sion

are

inve

rse

oper

atio

ns

Lear

ners

are

not

exp

ecte

d to

use

the

expr

essi

on “i

nver

se o

pera

tions

”. Th

ey a

re e

xpec

ted

to k

now

that

• th

ey c

an u

se m

ultip

licat

ion

to c

heck

div

isio

n ca

lcul

atio

ns

• th

ey c

an u

se d

ivis

ion

to c

heck

mul

tiplic

atio

n ca

lcul

atio

ns

38 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Afte

r com

plet

ing

a nu

mbe

r of s

imila

r exa

mpl

es, l

earn

ers

can

be a

sked

to e

xpla

in w

hat t

hey

notic

e in

thei

r ow

n w

ords

. If l

earn

ers

writ

e pa

irs o

f mat

chin

g nu

mbe

r sen

tenc

es b

ased

on

the

inpu

t and

out

put v

alue

s in

the

flow

dia

gram

s, th

ey c

an d

iscu

ss u

sing

mul

tiplic

atio

n to

che

ck d

ivis

ion

and

usin

g di

visi

on to

che

ck m

ultip

licat

ion.

Furt

her e

xam

ple

Lear

ners

can

use

the

abov

e kn

owle

dge

to in

dica

te h

ow th

ey c

ould

com

plet

e th

e m

issi

ng in

put n

umbe

rs in

a fl

ow d

iagr

am

O

nce

lear

ners

hav

e co

mpl

eted

the

flow

dia

gram

, the

y ca

n di

scus

s ho

w th

ey fo

und

the

mis

sing

inpu

t val

ues

from

the

corre

spon

ding

out

put v

alue

s an

d ru

le.

39MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Usi

ng fl

ow d

iagr

ams

to h

elp

lear

ners

dev

elop

mul

tiplic

atio

n an

d di

visi

on te

chni

ques

Ass

ocia

tive

prop

erty

Num

bers

can

be

mul

tiplie

d in

any

ord

er.

Exam

ple:

11

x (3

x 2

) = (1

1 x

3) x

2

Lear

ners

can

dis

cuss

wha

t the

y no

tice

whe

n th

ey c

ompa

re th

e ex

ampl

es. L

earn

ers

are

not r

equi

red

to k

now

the

nam

es o

f

the

prop

ertie

s. T

hey

are

only

exp

ecte

d to

use

them

to m

ake

calc

ulat

ions

eas

ier o

r use

equ

ival

ent n

umbe

r sen

tenc

es.

Usi

ng fl

ow d

iagr

ams

to h

elp

lear

ners

thin

k ab

out a

nd u

se te

chni

ques

for m

ultip

lyin

g by

10

Lear

ners

com

plet

e a

flow

dia

gram

like

the

one

belo

w. T

hey

then

exp

lain

usi

ng th

eir o

wn

wor

ds w

hat t

hey

notic

e ab

out t

he in

put

and

outp

ut v

alue

40 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Lear

ners

can

then

be

aske

d: “W

hat i

s an

othe

r way

to m

ultip

ly b

y 6?

”

Lear

ners

can

dev

elop

fast

men

tal a

nd w

ritte

n te

chni

ques

bas

ed o

n th

is.

All c

once

pts

and

calc

ulat

ing

tech

niqu

es d

evel

oped

her

e ca

n be

pra

ctic

ed th

roug

hout

the

year

in th

e m

enta

l Mat

hem

atic

s

prog

ram

me.

Gra

de 6

: C

omm

utat

ive

prop

erty

Num

bers

can

be

mul

tiplie

d in

any

ord

er.

Exam

ple:

13

x 5

x 2

= 13

x 2

x 5

..

41MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Lear

ners

can

dis

cuss

wha

t the

y no

tice

whe

n th

ey c

ompa

re th

e

exam

ples

. Lea

rner

s ar

e no

t req

uire

d to

kno

w th

e na

me

of th

e

com

mut

ativ

e pr

oper

ty. T

hey

are

only

exp

ecte

d to

be

able

to u

se

it to

sim

plify

cal

cula

tions

or t

o us

e eq

uiva

lent

sta

tem

ents

.

Geo

met

ric p

atte

rns

LTSM

Res

ourc

es

Num

ber l

ines

: 0 –

120

, Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, F

lash

ca

rds,

Bas

e 10

blo

cks,

100

cha

rts, m

ultip

licat

ion

tabl

e, m

aths

vid

eo c

lips,

E le

arni

ng o

nlin

e re

sour

ces,

Mat

hs g

ames

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

ook

refe

renc

e W

B 1

: pp

. 40

– 42

Ac

tiviti

es: 1

3 &

14

pp. 1

36 –

138

Ac

tiviti

es: 5

1, 5

2

WB

1:

pp

. 34

- 40

Activ

ities

11;

12;

13

& 14

pp

. 152

– 1

54

Activ

ities

55,

56

WB

1:

pp. 7

2 –

74 ;

98 -

102

Activ

ities

: 24

a&b)

; 35

- 37

DB

E Te

xtbo

ok re

fere

nce

Te

rm 1

: Uni

t 4: N

umer

ic P

atte

rns,

p. 5

1 Te

rm 2

: Uni

t 7: G

eom

etric

Pat

tern

s, p

. 18

1

Term

1: U

nit 4

: Num

eric

Pat

tern

s, p

. 4

Term

2: U

nit 7

: Geo

met

ric P

atte

rns,

p.

176

Term

1: U

nit 8

: Num

eric

Pat

tern

s, p

. 10

8 Te

rm 2

: Uni

t 4: G

eom

etric

Pat

tern

s,

p. 1

52

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent –

Te

st

42 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

5

TER

M 1

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to

……

…

Tim

e: 5

HO

UR

S G

RAD

E 4

CAP

S pp

. 52

– 55

5

CAP

S pp

. 140

-143

6

CAP

S pp

. 241

– 2

43 &

pp.

250

- 25

1 TO

PIC

M

ultip

licat

ion

and

divi

sion

of w

hole

num

bers

C

once

pts,

Ski

lls a

nd

know

ledg

e

Mul

tiplic

atio

n of

at l

east

who

le

2-di

git b

y 2-

digi

t num

bers

Div

isio

n of

at l

east

who

le 3

-dig

it by

1-

digi

t num

bers

C

alcu

latio

n te

chni

ques

U

se a

rang

e of

tech

niqu

es to

per

form

an

d ch

eck

writ

ten

and

men

tal

calc

ulat

ions

of w

hole

num

bers

incl

udin

g:

es

timat

ion

bu

ildin

g up

and

bre

akin

g do

wn

num

bers

roun

ding

off

and

com

pens

atin

g

doub

ling

and

halv

ing

us

ing

a nu

mbe

r lin

e

usin

g ad

ditio

n an

d su

btra

ctio

n as

in

vers

e op

erat

ions

usin

g m

ultip

licat

ion

and

divi

sion

as

inve

rse

oper

atio

ns

M

ultip

licat

ion

of a

t lea

st w

hole

3

-di

git b

y 2-

digi

t num

bers

Div

isio

n of

at l

east

who

le 3

-dig

it by

2-

digi

t num

bers

C

alcu

latio

n te

chni

ques

U

sing

a ra

nge

of te

chni

ques

to p

erfo

rm

and

chec

k w

ritte

n an

d m

enta

l ca

lcul

atio

ns o

f who

le n

umbe

rs in

clud

ing:

estim

atio

n

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

us

ing

a nu

mbe

r lin

e

roun

ding

off

and

com

pens

atin

g

doub

ling

and

halv

ing

us

ing

addi

tion

and

subt

ract

ion

as

inve

rse

oper

atio

ns

us

ing

mul

tiplic

atio

n an

d di

visi

on a

s in

vers

e op

erat

ions

M

ultip

licat

ion

of a

t lea

st w

hole

4-

digi

t by

3-di

git n

umbe

rs

D

ivis

ion

of a

t lea

st w

hole

4-d

igit

by

3-di

git n

umbe

rs

M

ultip

le o

pera

tions

on

who

le

num

bers

with

or w

ithou

t bra

cket

s Cal

cula

tion

tech

niqu

es in

clud

e U

sing

a ra

nge

of te

chni

ques

to p

erfo

rm

and

chec

k w

ritte

n an

d m

enta

l ca

lcul

atio

ns o

f who

le n

umbe

rs in

clud

ing:

estim

atio

n

mul

tiply

ing

in c

olum

ns

bu

ildin

g up

and

bre

akin

g do

wn

num

bers

roun

ding

off

and

com

pens

atin

g

usin

g a

calc

ulat

or

us

ing

the

reci

proc

al re

latio

nshi

p be

twee

n m

ultip

licat

ion

and

divi

sion

long

div

isio

n

43MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s M

ultip

les

of 1

-dig

it nu

mbe

rs to

at l

east

10

0 Pr

oper

ties

of w

hole

num

bers

R

ecog

nize

and

use

the

com

mut

ativ

e;

asso

ciat

ive;

and

dis

tribu

tive

prop

ertie

s of

who

le n

umbe

rs

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g w

hole

num

bers

: -

finan

cial

con

text

s -

mea

sure

men

t con

text

s

So

lve

prob

lem

s in

volv

ing

who

le

num

bers

: -

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

- co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

- gr

oupi

ng a

nd e

qual

sha

ring

with

rem

aind

ers

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s

Mul

tiple

s of

2-d

igit

who

le n

umbe

rs

to a

t lea

st 1

00

Fa

ctor

s of

2-d

igit

who

le n

umbe

rs to

at

leas

t 100

Prop

ertie

s of

who

le n

umbe

rs

R

ecog

nize

and

use

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

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

-

finan

cial

con

text

s -

mea

sure

men

t con

text

s

So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng:

- co

mpa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd

(ratio

) -

com

parin

g tw

o qu

antit

ies

of

diffe

rent

kin

ds (r

ate)

-

grou

ping

and

equ

al s

harin

g w

ith re

mai

nder

s

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s

Mul

tiple

s of

2-d

igit

and

3-di

git

num

bers

Fact

ors

of 2

-dig

it an

d 3-

digi

t who

le

num

bers

Prim

e fa

ctor

s of

num

bers

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

R

ecog

nize

and

use

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

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs a

nd d

ecim

al fr

actio

ns,

incl

udin

g -

finan

cial

con

text

s -

mea

sure

men

t con

text

s

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

-

com

parin

g tw

o or

mor

e qu

antit

ies

of

the

sam

e ki

nd

(ratio

) -

com

parin

g tw

o qu

antit

ies

of

diffe

rent

kin

ds (r

ate)

-

grou

ping

and

equ

al s

harin

g w

ith re

mai

nder

s

44 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Le

arne

rs s

houl

d do

roun

ding

off

to th

e ne

ares

t 10,

to e

stim

ate

answ

ers.

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

Judg

ing

reas

onab

lene

ss o

f sol

utio

ns

Lear

ners

sho

uld

estim

ate

thei

r ans

wer

s be

fore

cal

cula

ting.

The

y co

uld

roun

d of

f the

num

bers

invo

lved

in th

e ca

lcul

atio

ns.

Le

arne

rs c

an ro

und

off t

o th

e ne

ares

t 10

whe

n m

ultip

lyin

g or

div

idin

g w

ith 2

-dig

it nu

mbe

rs

Che

ckin

g so

lutio

ns

Le

arne

rs s

houl

d kn

ow th

at th

ey c

an c

heck

a d

ivis

ion

calc

ulat

ion

by m

ultip

lyin

g Ex

ampl

e: If

69

÷ 3

= 23

; the

n 23

x 3

= 6

9

Lear

ners

sho

uld

know

that

to c

heck

a d

ivis

ion

calc

ulat

ion

with

a re

mai

nder

, the

y sh

ould

firs

t mul

tiply

div

iden

d by

the

divi

sor

and

then

add

the

rem

aind

er

Exam

ple:

If 7

0 ÷

3 =

23 re

mai

nder

1; t

hen

23 x

3 =

69

ther

efor

e 69

+ 1

=70

Usi

ng th

e in

vers

e op

erat

ion

to c

heck

sol

utio

ns is

one

reas

on fo

r tea

chin

g m

ultip

licat

ion

and

divi

sion

toge

ther

. Ano

ther

reas

on

for l

ooki

ng a

t mul

tiplic

atio

n an

d di

visi

on to

geth

er is

that

we

alm

ost a

lway

s us

e m

ultip

licat

ion

to s

olve

div

isio

n.

In G

rade

4 le

arne

rs b

reak

up

num

bers

to m

ultip

ly. T

here

are

diff

eren

t way

s of

doi

ng th

is.

Som

etim

es th

e nu

mbe

rs in

volv

ed

in th

e ca

lcul

atio

n m

ake

diffe

rent

met

hods

eas

ier

or m

ore

diffi

cult.

Lea

rner

s ha

ve a

lread

y se

en h

ow to

use

the

asso

ciat

ive

and

com

mut

ativ

e pr

oper

ties

to m

ake

mul

tiplic

atio

n ea

sier

. Mul

tiplic

atio

n an

d th

e di

strib

utiv

e pr

oper

ty o

f mul

tiplic

atio

n ov

er a

dditi

on/s

ubtr

actio

n O

ne w

ay fo

r lea

rner

s to

und

erst

and

how

and

why

the

dist

ribut

ive

prop

erty

wor

ks, i

s to

bre

ak u

p ar

rays

and

writ

e nu

mbe

r se

nten

ces

to d

escr

ibe

the

arra

ys.

9

x 5

=

5 x

5

+

4

x 5

45MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

The

dist

ribut

ive

law

allo

ws

lear

ners

to s

plit

the

num

ber a

nd th

en m

ultip

ly e

ach

part

sepa

rate

ly.

Usi

ng fa

ctor

s to

mul

tiply

Ex

ampl

e:

Cal

cula

te 4

7 x

6 47

x 6

= 4

7 x

2 x

3 =

94 x

3

= (9

0 +

4) x

3

= 90

x 3

+ 4

x 3

=

270

+ 12

=

282

Usi

ng th

e di

strib

utiv

e pr

oper

ty to

mul

tiply

Ex

ampl

e:

47 x

5 =

40

x 5

+ 7

x 5

(

usi

ng th

e di

strib

utiv

e pr

oper

ty)

= 4

x 10

x 5

+ 3

5 =

4 x

5 x

10 +

35

= 20

0 +

35

=

235

or

47 x

5 =

(50

– 3)

x 5

(usi

ng th

e di

strib

utiv

e pr

oper

ty)

= (5

0 x

5) –

(3 x

5)

= 5

x 5

x 10

– 1

5 =

250

– 1

5 =

235

Div

idin

g Le

arne

rs u

se w

hat t

hey

know

abo

ut m

ultip

licat

ion

to d

o di

visi

on.

Exam

ple

75 ÷

4

46 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Lear

ners

can

writ

e ou

t a “

clue

boa

rd”

of w

hat t

hey

know

abo

ut m

ultip

lyin

g by

4

Exam

ple:

4

x 10

= 4

0 4

x 20

= 8

0 (d

oubl

ing

the

first

sta

tem

ent)

4 x

5 =

20 (h

alvi

ng th

e fir

st s

tate

men

t) 4

x 4

= 16

4

x 3

= 12

75

cou

ld b

e w

ritte

n as

40

+ 20

+ 1

2 +

3 =

4×

10+

4×

5+

4×3

+3

The

refo

re, 7

5 ÷

4 =

40÷

4+20

÷4+

12÷

4+𝑡𝑡ℎ𝑒𝑒 𝑟𝑟

𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟

𝑒𝑒𝑟𝑟 𝑜𝑜𝑜𝑜

3 =

10

+ 5

+ 3

+ re

mai

nder

3 =

18

rem

aind

er 3

Le

arne

rs s

houl

d ch

eck

thei

r cal

cula

tions

by

mul

tiply

ing

and

addi

ng th

e re

mai

nder

: 18

x 4

= 72

and

72

+ 3

= 75

. K

inds

of p

robl

ems

Shar

ing,

gro

upin

g, tr

eatin

g gr

oups

as

units

, rat

e,

See

the

desc

riptio

n of

pro

blem

typ

es a

t th

e en

d of

the

gra

de

note

s LT

SM

Res

ourc

es

Num

ber l

ines

: 0 –

120

, Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, F

lash

ca

rds,

Bas

e 10

blo

cks,

100

cha

rts, m

ultip

licat

ion

tabl

e, m

aths

vid

eo c

lips,

E le

arni

ng o

nlin

e re

sour

ces,

Mat

hs g

ames

, cal

cula

tors

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

ook

refe

renc

e W

B 1

: pp

. 118

– 1

24

Activ

ities

: 44a

&b; 4

5a&b

pp

. 158

– 1

60

Activ

ities

: 62,

63

WB

1:

pp. 4

2 –

48

Activ

ities

: 15a

&b, 1

6 a&

b pp

. 128

Ac

tiviti

es: 4

4a&b

WB

1:

pp. 8

4 –

86

Activ

ities

28;

29

pp

. 122

- 12

4

Activ

ities

45,

46

DB

E Te

xtbo

ok

refe

renc

e Te

rm 1

: Uni

t 5: M

ultip

licat

ion

and

divi

sion

of

who

le n

umbe

rs, p

. 60

Term

1: U

nit 9

: Who

le N

umbe

r: M

ultip

licat

ion

and

divi

sion

, p. 1

04

Term

1: U

nit 5

: Mul

tiplic

atio

n an

d di

visi

on o

f who

le n

umbe

rs, p

. 55

Term

2: U

nit 2

: Mul

tiplic

atio

n of

who

le

num

bers

, p. 1

25

Term

2: U

nit 6

: Div

isio

n of

who

le

num

bers

, p. 1

65

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

47MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTIG

RAD

E LE

SSO

N P

LAN

6

TER

M 1

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

……

Tim

e: 6

HO

UR

S G

RAD

E 4

CAP

S p.

55

5 C

APS

pp. 1

44 &

186

6 C

APS

pp. 2

28 &

266

TO

PIC

Ti

me

Con

cept

s, S

kills

and

kn

owle

dge

Rea

ding

tim

e an

d tim

e in

stru

men

ts

Rea

d, te

ll an

d w

rite

time

in 1

2-ho

ur a

nd

24-h

our f

orm

ats

on b

oth

anal

ogue

and

di

gita

l ins

trum

ents

in:

• ho

urs

• m

inut

es

• se

cond

s In

stru

men

ts in

clud

e cl

ocks

and

wat

ches

R

eadi

ng c

alen

dars

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng

with

tim

e in

clud

e •

Cal

cula

tion

of

the

num

ber

of

days

be

twee

n an

y tw

o da

tes

with

in t

he s

ame

or c

onse

cutiv

e ye

ars

•

Cal

cula

tion

of ti

me

inte

rval

s w

here

tim

e is

giv

en in

min

utes

or h

ours

on

ly

Rea

ding

tim

e an

d tim

e in

stru

men

ts

Rea

d, te

ll an

d w

rite

time

in 1

2-ho

ur

and

24-h

our f

orm

ats

on b

oth

anal

ogue

an

d di

gita

l ins

trum

ents

in

• ho

urs

• m

inut

es

• se

cond

s In

stru

men

ts in

clud

e cl

ocks

, wat

ches

an

d st

opw

atch

es

Rea

ding

cal

enda

rs

Cal

cula

tions

and

pro

blem

sol

ving

re

late

d to

tim

e in

clud

e C

alcu

latio

n of

tim

e in

terv

als

whe

re

time

is g

iven

in

• se

cond

s an

d/or

min

utes

Rea

ding

tim

e an

d tim

e in

stru

men

ts

Rea

d, te

ll an

d w

rite

time

in 1

2-ho

ur a

nd

24-h

our f

orm

ats

on b

oth

anal

ogue

and

di

gita

l ins

trum

ents

in

• ho

urs

• m

inut

es

• se

cond

s In

stru

men

ts in

clud

e cl

ocks

, wat

ches

an

d st

opw

atch

es

Rea

ding

cal

enda

rs

Cal

cula

tions

and

pro

blem

-sol

ving

re

late

d to

tim

e •

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g tim

e •

Rea

d tim

e zo

ne m

aps

and

calc

ulat

ing

time

diffe

renc

es

base

d on

tim

e zo

nes

• C

alcu

latio

n of

tim

e in

terv

als

whe

re

time

is g

iven

in

• se

cond

s an

d/or

min

utes

;

48 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

His

tory

of t

ime

Know

s ho

w ti

me

was

mea

sure

d an

d re

pres

ente

d in

anc

ient

tim

es

• m

inut

es a

nd/o

r hou

rs

• ho

urs

and/

or d

ays

• da

ys a

nd/o

r wee

ks a

nd/o

r mon

ths

• m

onth

s an

d/or

yea

rs

• ye

ars

and/

or d

ecad

es

His

tory

of t

ime

Know

how

tim

e w

as m

easu

red

and

expr

esse

d in

anc

ient

tim

es.

Prac

tical

mea

surin

g of

tem

pera

ture

by

es

timat

ing,

mea

surin

g, re

cord

ing,

co

mpa

ring

and

orde

ring

Mea

surin

g in

stru

men

ts:

ther

mom

eter

s U

nits

de

gree

s C

elsi

us (o

C)

Cal

cula

tions

and

pro

blem

-so

lvin

g re

late

d to

tem

pera

ture

•

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g to

tem

pera

ture

•

Cal

cula

te te

mpe

ratu

re d

iffer

ence

s lim

ited

to p

ositi

ve w

hole

num

bers

• m

inut

es a

nd/o

r hou

rs

• ho

urs

and

/or d

ays

• da

ys a

nd/o

r wee

ks a

nd/o

r mon

ths

• ye

ars

and/

or d

ecad

es

• ce

ntur

ies,

dec

ades

and

yea

rs

His

tory

of t

ime

Know

som

e w

ays

in w

hich

tim

e w

as

mea

sure

d an

d re

pres

ente

d in

the

past

.

Prac

tical

mea

surin

g of

tem

pera

ture

by

es

timat

ing,

mea

surin

g, re

cord

ing,

co

mpa

ring

and

orde

ring

M

easu

ring

inst

rum

ents

ther

mom

eter

s (a

nalo

gue

and

digi

tal)

Uni

ts

degr

ees

Cel

sius

(oC

) C

alcu

latio

ns a

nd p

robl

em-

solv

ing

rela

ted

tem

pera

ture

So

lvin

g pr

oble

ms

in c

onte

xts

rela

ted

to

tem

pera

ture

s

SUG

GES

TED

M

ETH

OD

OLO

GY

Tim

e:

Lear

ners

sho

uld

cont

inue

to re

ad c

lock

s an

d te

ll th

e tim

e at

freq

uent

inte

rval

s du

ring

the

entir

e ye

ar. T

his

can

be d

one

durin

g th

e M

enta

l Mat

hem

atic

s tim

e or

just

bef

ore

or a

fter b

reak

tim

e or

bef

ore

lear

ners

go

hom

e, o

r whe

n th

ey c

ome

in fr

om a

cla

ss in

an

othe

r ven

ue.

All g

rade

s In G

rade

3 le

arne

rs w

ork

with

ana

logu

e an

d di

gita

l clo

cks

usin

g 12

-hou

r for

mat

. In

Gra

de 4

lear

ners

mov

e on

to d

igita

l 24

-hou

r for

mat

.

49MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Onc

e le

arne

rs h

ave

been

lear

nt to

tell

the

time,

furth

er p

ract

ise

can

take

pla

ce d

urin

g m

enta

l mat

hem

atic

s tim

e.

Lear

ners

con

tinue

to re

ad c

alen

dars

. C

alcu

latio

ns a

nd p

robl

em-s

olvi

ng w

ith ti

me

incl

ude

• ca

lcul

atio

n of

the

num

ber o

f day

s be

twee

n an

y tw

o da

tes

with

in th

e sa

me

or c

onse

cutiv

e ye

ars

• ca

lcul

atio

n of

tim

e in

terv

als

whe

re ti

me

is g

iven

in m

inut

es a

nd/o

r hou

rs o

nly

• ca

lcul

atio

ns s

houl

d be

lim

ited

to w

hole

num

bers

and

com

mon

frac

tions

Le

arne

rs s

houl

d co

ntin

ue to

read

clo

cks

and

tell

the

time

at fr

eque

nt in

terv

als

durin

g th

e en

tire

year

. Thi

s ca

n be

don

e du

ring

the

men

tal M

athe

mat

ics

time

or ju

st b

efor

e or

afte

r bre

ak ti

me

or b

efor

e le

arne

rs g

o ho

me,

or w

hen

they

com

e in

from

a c

lass

in

anot

her v

enue

. G

rade

s 5

and

6:

Stop

wat

ches

are

intro

duce

d.

Lear

ners

can

eith

er u

se s

topw

atch

es th

at o

ccur

as

sing

le in

stru

men

ts, o

r sto

pwat

ches

on

cell

phon

es o

r wris

t wat

ches

. Le

arne

rs c

ontin

ue to

read

, rec

ord

and

calc

ulat

e tim

e in

12-h

our a

nd 2

4-ho

ur fo

rmat

s an

d to

wor

k w

ith a

nalo

gue

and

digi

tal

inst

rum

ents

. Th

is is

pra

ctis

ed re

gula

rly. O

nce

lear

ners

hav

e be

en ta

ught

to te

ll th

e tim

e, it

can

be

prac

tised

dur

ing

the

men

tal m

athe

mat

ics

sect

ion

of th

e le

sson

, and

freq

uent

ly a

t oth

er ti

mes

dur

ing

the

day.

Le

arne

rs c

ontin

ue to

read

cal

enda

rs

Cal

cula

tions

and

pro

blem

-sol

ving

rela

ted

to ti

me

Dec

ades

are

intro

duce

d.

Cal

cula

tions

sho

uld

be li

mite

d to

who

le n

umbe

rs a

nd fr

actio

ns.

Gra

de 6

: W

hat i

s di

ffere

nt to

Gra

de 5

? • T

ime

zone

s ar

e in

trodu

ced.

50 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• C

entu

ries

are

intro

duce

d O

nce

lear

ners

hav

e be

en ta

ught

to te

ll th

e tim

e, th

is c

an b

e pr

actis

ed d

urin

g th

e m

enta

l Mat

hem

atic

s se

ctio

n of

the

less

on.

Lear

ners

con

tinue

to re

ad c

alen

dars

, and

do

calc

ulat

ions

bas

ed o

n da

tes.

C

alcu

latio

ns a

nd p

robl

em-s

olvi

ng re

late

d to

tim

e in

clud

e C

alcu

latio

ns w

ith a

nd c

onve

rsio

ns b

etw

een

all t

he u

nits

men

tione

d in

the

colu

mn

on th

e le

ft.

Tim

e zo

nes

Lear

ners

sho

uld

be a

ble

to:

• rea

d tim

e zo

ne m

aps

and

do c

alcu

latio

ns u

sing

zon

ed m

aps.

Hel

p le

arne

rs to

und

erst

and

why

ther

e ar

e tim

e zo

ne

diffe

renc

es b

etw

een

diffe

rent

pla

ces

in th

e w

orld

• c

alcu

late

tim

e di

ffere

nces

whe

n gi

ven

cloc

k fa

ces

show

ing

the

times

in d

iffer

ent p

lace

s.

Tem

pera

ture

: Intro

duce

dec

ades

in G

rade

5. C

alcu

latio

ns s

houl

d be

lim

ited

to w

hole

num

bers

and

frac

tions

. G

rade

5 a

nd 6

lear

ners

sho

uld

also

use

com

mon

tem

pera

ture

refe

rent

s e.

g.

• th

e fre

ezin

g po

int o

f pur

e w

ater

is 0

o C

• th

e bo

iling

poin

t of p

ure

wat

er is

100

o C

• th

e av

erag

e no

rmal

hum

an b

ody

tem

pera

ture

is 3

7oC

•

the

daily

env

ironm

enta

l tem

pera

ture

s.

Lear

ners

sho

uld

read

tem

pera

ture

s on

pic

ture

s of

ther

mom

eter

s. W

here

pos

sibl

e le

arne

rs s

houl

d re

ad te

mpe

ratu

res

on re

al

ther

mom

eter

s.

Rea

ding

ana

logu

e th

erm

omet

ers

requ

ires

lear

ners

to re

ad th

e te

mpe

ratu

re o

n nu

mbe

red

and

un-n

umbe

red

grad

atio

n lin

es.

In th

erm

omet

ers

desi

gned

to re

ad th

e en

viro

nmen

tal t

empe

ratu

res

the

unnu

mbe

red

grad

atio

n lin

es o

ften

refe

r to

who

le

degr

ees.

In th

erm

omet

ers

desi

gned

to re

ad h

uman

bod

y te

mpe

ratu

re th

e un

num

bere

d gr

adat

ion

lines

ofte

n re

fer t

o fra

ctio

ns

of d

egre

es.

Rec

ordi

ng a

nd re

port

ing

on te

mpe

ratu

re m

easu

rem

ents

51MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Lear

ners

sho

uld

reco

rd a

nd re

port

on te

mpe

ratu

re m

easu

rem

ents

they

hav

e re

ad o

ff th

erm

omet

ers

in w

hole

num

bers

. Thi

s m

ay in

volv

e ro

undi

ng u

p or

dow

n. T

hey

can

also

reco

rd a

nd re

port

tem

pera

ture

s by

usi

ng fr

actio

n no

tion.

G

rade

6 le

arne

rs s

houl

d do

cal

cula

tions

with

and

con

vers

ions

bet

wee

n tim

e zo

nes.

The

y sh

ould

be

able

to:

• re

ad ti

me

zone

map

s an

d do

cal

cula

tions

usi

ng z

oned

map

s. H

elp

lear

ners

to u

nder

stan

d w

hy th

ere

are

time

zone

di

ffere

nces

bet

wee

n di

ffere

nt p

lace

s in

the

wor

ld

• ca

lcul

ate

time

diffe

renc

es w

hen

give

n cl

ock

face

s sh

owin

g th

e tim

es in

diff

eren

t pla

ces.

Cal

cula

tions

and

pro

blem

-sol

ving

with

tim

e an

d te

mpe

ratu

re in

clud

e •

calc

ulat

ion

of th

e nu

mbe

r of d

ays

betw

een

any

two

date

s w

ithin

the

sam

e or

con

secu

tive

year

s •

calc

ulat

ion

of ti

me

inte

rval

s w

here

tim

e is

giv

en in

min

utes

and

/or h

ours

onl

y •

calc

ulat

ions

lim

ited

to w

hole

num

bers

and

com

mon

frac

tions

for G

rade

4 a

nd 5

(Gra

de 6

sho

uld

do p

robl

ems

that

invo

lve

de

cim

al fr

actio

ns).

LTSM

Res

ourc

es

Anal

ogue

and

dig

ital c

lock

s, c

alen

dar,

stop

wat

ch, c

ellp

hone

, wor

ld ti

me

zone

cha

rts, t

herm

omet

er, n

ewsp

aper

wea

ther

repo

rts,

vide

o cl

ips,

E le

arni

ng o

nlin

e re

sour

ces,

Mat

hs g

ames

,

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

ook

refe

renc

e W

B 1

: pp

50

– 56

Ac

tiviti

es: 1

8a&b

,19a

&b

WB

1:

pp. 6

0 -6

4

Activ

ities

: (20

a&b;

21)

W

B 2

: pp

. 68

- 70

Activ

ities

: 94

, 95

WB

1:

pp. 4

6 - 5

2 ; 1

44

Activ

ities

: 16a

&b; 1

7a&b

, 54

WB

2:

pp. 6

2 –

66

Activ

ity: 8

5a&b

, 86

DB

E Te

xtbo

ok re

fere

nce

Te

rm 1

: Uni

t 6: T

ime,

p. 7

3

Term

1: U

nit 6

: Tim

e, p

. 69

Term

3: U

nit 8

: Tem

pera

ture

, p. 2

52

Term

1: U

nit 5

: Tim

e, p

. 72

Term

3: U

nit 7

: Tem

pera

ture

, p. 2

58

HO

MEW

OR

K

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

52 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

7

TER

M 1

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

……

T

ime:

6 H

OU

RS

GR

ADE

4 C

APS

pp. 5

7 - 5

8 5

CAP

S pp

. 145

-146

6

CAP

S pp

. 233

TO

PIC

D

ata

hand

ling

Con

cept

s, S

kills

and

kn

owle

dge

• C

olle

ct d

ata

usin

g ta

lly m

arks

and

ta

bles

for r

ecor

ding

•

Dra

w a

var

iety

of g

raph

s to

dis

play

an

d in

terp

ret d

ata

incl

udin

g:

- pi

ctog

raph

s (o

ne-

to-o

ne

repr

esen

tatio

n)

- ba

r gra

phs

• C

ritic

ally

read

and

inte

rpre

t dat

a re

pres

ente

d in

-

wor

ds

- pi

ctog

raph

s -

bar g

raph

s -

pie

char

ts

• C

olle

ct d

ata

usin

g ta

lly m

arks

and

ta

bles

for r

ecor

ding

•

Ord

er d

ata

from

sm

alle

st g

roup

to

larg

est g

roup

•

Dra

w a

var

iety

of g

raph

s to

dis

play

an

d in

terp

ret d

ata

incl

udin

g -

pict

ogra

phs

(man

y-to

-one

co

rresp

onde

nce)

-

bar g

raph

s •

Crit

ical

ly re

ad a

nd in

terp

ret d

ata

repr

esen

ted

in

- w

ords

-

pict

ogra

phs

- ba

r gra

phs

- pi

e ch

arts

• C

olle

ct d

ata

- U

se ta

lly m

arks

and

tabl

es fo

r re

cord

ing

- U

se s

impl

e qu

estio

nnai

res

(yes

/no

type

resp

onse

) •

Ord

er d

ata

from

sm

alle

st g

roup

to

larg

est g

roup

•

Dra

w a

var

iety

of g

raph

s to

dis

play

an

d in

terp

ret d

ata

incl

udin

g -

pict

ogra

phs

with

man

y-to

-one

repr

esen

tatio

ns

- ba

r gra

phs

and

doub

le b

ar

grap

hs

• C

ritic

ally

read

and

inte

rpre

t dat

a re

pres

ente

d in

-

wor

ds

- pi

ctog

raph

s -

bar g

raph

s -

doub

le b

ar g

raph

s -

pie

char

ts

53MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• An

alys

e da

ta b

y an

swer

ing

ques

tions

re

late

d to

dat

a ca

tego

ries

• An

alys

e da

ta b

y an

swer

ing

ques

tions

re

late

d to

- da

ta c

ateg

orie

s -

data

sou

rces

and

con

text

s •

Sum

mar

ise

data

ver

bally

and

in s

hort

writ

ten

para

grap

hs th

at in

clud

e -

draw

ing

conc

lusi

ons

abou

t the

da

ta

- m

akin

g pr

edic

tions

bas

ed o

n th

e da

ta

• Ex

amin

e un

grou

ped

num

eric

al d

ata

to d

eter

min

e th

e m

ost f

requ

ently

oc

curri

ng s

core

in th

e da

ta s

et

(mod

e)

• An

alys

e da

ta b

y an

swer

ing

ques

tions

rela

ted

to

-

data

cat

egor

ies,

incl

udin

g da

ta

inte

rval

s -

data

sou

rces

and

con

text

s -

cent

ral t

ende

ncie

s –

(mod

e an

d m

edia

n)

• Su

mm

aris

e da

ta v

erba

lly a

nd in

sh

ort w

ritte

n pa

ragr

aphs

that

incl

ude

- dr

awin

g co

nclu

sion

s ab

out t

he

data

-

mak

ing

pred

ictio

ns b

ased

on

the

data

LTSM

R

esou

rces

Ex

empl

ers

of p

icto

grap

hs, b

ar g

raph

s, p

ie c

harts

– (c

onte

xt s

ensi

tive)

, IC

T –

Exce

l wor

kshe

ets

with

gra

phs,

vid

eo

clip

s, E

lear

ning

onl

ine

reso

urce

s, M

aths

gam

es,

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

1:

pp. 5

8 -6

2 Ac

tiviti

es: 2

0; 2

1a&b

WB

1:

pp. 6

6 –

68

Activ

ities

: 22a

&b

WB

1:

pp. 7

8 –

88

Activ

ities

: 92a

&b, 9

3, 9

4, 9

5, 9

5 D

BE

Text

book

refe

renc

e Te

rm 1

: Uni

t 7: D

ata

hand

ling,

p. 8

6

Term

1: U

nit 7

: Dat

a ha

ndlin

g, p

. 81

Term

1: U

nit 7

: Dat

a ha

ndlin

g, p

. 97

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

54 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

8

TER

M 1

EDU

CAT

OR

: ……

……

……

……

……

DAT

E : f

rom

…

……

to

……

…

Tim

e: 6

HO

UR

S G

RAD

E 4

CAP

S pp

. 59

- 61

5 C

APS

pp. 1

47 -1

49

6 C

APS

pp. 2

29 –

232

TO

PIC

Pr

oper

ties

of 2

D s

hape

s C

once

pts,

Ski

lls a

nd

know

ledg

e Sh

apes

lear

ners

nee

d to

kno

w a

nd

nam

e •

Reg

ular

and

irre

gula

r pol

ygon

s:

- tri

angl

es

- sq

uare

s, re

ctan

gles

-

othe

r qua

drila

tera

ls

- pe

ntag

ons

- he

xago

ns

• C

ircle

s Th

e ch

arac

teris

tics

whi

ch le

arne

rs

use

to d

istin

guis

h, d

escr

ibe,

sor

t an

d co

mpa

re s

hape

s •

stra

ight

and

/cur

ved

side

s •

num

ber o

f sid

es

Shap

es le

arne

rs n

eed

to k

now

and

na

me

• R

egul

ar a

nd ir

regu

lar p

olyg

ons

- tri

angl

es, s

quar

es, r

ecta

ngle

s, o

ther

qu

adril

ater

als,

pen

tago

ns, h

exag

ons,

he

ptag

ons,

•

Circ

les

• Si

mila

ritie

s an

d di

ffere

nces

be

twee

n sq

uare

s an

d re

ctan

gles

C

hara

cter

istic

s le

arne

rs u

se t

o di

stin

guis

h, d

escr

ibe,

sor

t an

d co

mpa

re s

hape

s •

Stra

ight

and

/ cu

rved

sid

es

• N

umbe

r of s

ides

•

Leng

th o

f sid

es

• An

gles

: lim

ited

to

- rig

ht a

ngle

s -

angl

es s

mal

ler t

han

right

ang

les

- an

gles

gre

ater

than

righ

t ang

les

Shap

es le

arne

rs n

eed

to k

now

and

na

me

• R

egul

ar a

nd ir

regu

lar p

olyg

ons

- tria

ngle

s, s

quar

es, r

ecta

ngle

s,

para

llelo

gram

s, o

ther

qua

drila

tera

ls,

pent

agon

s, h

exag

ons,

hep

tago

ns,

octa

gons

•

Sim

ilarit

ies

and

diffe

renc

es

betw

een

rect

angl

es a

nd

para

llelo

gram

s

Feat

ures

of s

hape

s D

escr

ibe,

sor

t and

com

pare

2-D

sh

apes

in te

rms

of

• nu

mbe

r of s

ides

•

leng

th o

f sid

es

• si

ze o

f ang

les

- ac

ute

- rig

ht

- ob

tuse

-

stra

ight

55MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Furt

her a

ctiv

ities

to fo

cus

on th

e ch

arac

teris

tics

of s

hape

s D

raw

2-D

sha

pes

on g

rid p

aper

Furt

her a

ctiv

ities

to fo

cus

lear

ners

on

cha

ract

eris

tics

of s

hape

s D

raw

2-D

sha

pes

on g

rid p

aper

A

ngle

s

Lim

ited

to

• rig

ht a

ngle

s •

angl

es s

mal

ler t

han

right

ang

les

• A

ngle

s gr

eate

r tha

n rig

ht a

ngle

s

- re

flex

-

revo

lutio

n

Furt

her a

ctiv

ities

•

Dra

w 2

-D s

hape

s on

grid

pap

er

• D

raw

circ

les,

pat

tern

s in

circ

les

and

patte

rns

with

circ

les

usin

g a

pair

of c

ompa

sses

Ang

les

Rec

ogni

ze a

nd n

ame

the

follo

win

g an

gles

in 2

-D s

hape

s:

-

acut

e

- rig

ht

-

obtu

se

-

stra

ight

- re

flex

-

revo

lutio

n

Furt

her a

ctiv

ities

to fo

cus

lear

ners

on

cha

ract

eris

tics

of s

hape

s •

Dra

w

2-D

sh

apes

on

gr

id

pape

r •

Dra

w

circ

les,

pa

ttern

s in

ci

rcle

s an

d pa

ttern

s w

ith

circ

les

usin

g a

pair

of

com

pass

es

LTSM

R

esou

rces

Va

riety

of r

egul

ar a

nd ir

regu

lar p

olyg

ons:

( To

cut

out

) - t

riang

les,

squ

ares

, rec

tang

les,

par

alle

logr

ams,

oth

er q

uadr

ilate

rals

, pen

tago

ns, h

exag

ons,

hep

tago

ns, o

ctag

ons

- Tan

gram

s pa

ir of

com

pass

es, g

eobo

ards

56 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

1:

pp 6

4 -6

6 Ac

tiviti

es: 2

2a&b

WB

1:

pp 7

0 –

72

Activ

ities

: 22a

, 23

a&b

WB

1:

pp 5

4 –

58

Activ

ities

: 18a

, b, c

D

BE

Text

book

refe

renc

e Te

rm 1

: Uni

t 8: P

rope

rties

of

two

- di

imen

sion

al s

hape

s, p

. 96

Term

1: U

nit 8

: Pro

perti

es o

f tw

o -

diim

ensi

onal

sha

pes,

p. 9

1

Term

1: U

nit 6

: Pro

perti

es o

f tw

o -

diim

ensi

onal

sha

pes,

p. 8

6

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

57MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LE

SSO

N P

LAN

S: T

ERM

2

58 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

1

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 3

HO

UR

S G

RAD

E 4

CAP

S pp

. 68

5 C

APS

pp. 1

56

6 C

APS

pp. 2

40

TOPI

C

Who

le n

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

C

ount

forw

ards

and

bac

kwar

ds in

2s,

3s

, 5s,

10s

, 25s

, 50s

, 100

s be

twee

n 0

and

at le

ast 1

0 00

0

O

rder

, com

pare

and

repr

esen

t

nu

mbe

rs to

at l

east

4-d

igit

num

bers

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its in

w

hole

num

bers

to a

t lea

st 4

-dig

it nu

mbe

rs

R

ound

off

to th

e ne

ares

t 10,

100

,

1 00

0

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

6-d

igit

num

bers

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

.

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its in

w

hole

num

bers

to a

t lea

st 6

dig

it nu

mbe

rs.

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00

and

1 00

0

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

9-d

igit

num

bers

Rep

rese

nt p

rime

num

bers

to a

t lea

st

100

R

ecog

nizi

ng th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 9

-dig

it nu

mbe

rs

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00,

1 00

0

SUG

GES

TED

Se

e Te

rm 1

not

es:

59MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MET

HO

DO

LOG

Y Ex

plai

n fro

m k

now

n (lo

w n

umbe

r ran

ge) t

o an

incr

ease

d nu

mbe

r ran

ge; w

ork

with

con

cret

e LT

SM

Gra

de 4

:

Cou

ntin

g nu

mbe

r ran

ge in

crea

sed

to 1

0 00

0

Rou

ndin

g of

f to

the

near

est 1

0 an

d 10

0

Num

ber r

ange

for p

lace

val

ue, o

rder

ing,

com

parin

g an

d re

pres

entin

g nu

mbe

rs in

crea

sed

to 4

dig

its.

Gra

de 5

and

6:

C

ount

ing

num

ber r

ange

incr

ease

d –

lear

ners

cou

nt fo

rwar

ds a

nd b

ackw

ards

in 2

s, 3

s, 5

s, 1

0s, 2

5s, 5

0s, 1

00s

betw

een

0 an

d at

leas

t 10

000.

Lear

ners

als

o co

unt i

n fra

ctio

ns (a

fter t

he to

pic

of fr

actio

ns h

as b

een

cove

red

in th

e m

ain

less

on –

see

com

men

t in

that

se

ctio

n ab

out c

ount

ing

in fr

actio

ns)

R

ound

ing

off t

o th

e ne

ares

t 10,

100

, 1 0

00

N

umbe

r ran

ge fo

r pla

ce v

alue

, ord

erin

g, c

ompa

ring

and

repr

esen

ting

num

bers

incr

ease

d to

6 d

igits

Se

e fu

rther

not

es in

Ter

m 1

, but

be

awar

e th

at n

umbe

r ran

ges

have

incr

ease

d in

Ter

m 2

. The

incr

ease

d nu

mbe

r ran

ges

are

show

n in

the

colu

mn

on th

e le

ft.

LTSM

R

esou

rces

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

card

s, B

ase

10 b

lock

s, 1

00 c

harts

W

orkb

ook

refe

renc

e W

B 1

: pp

. 8 –

12

Ac

tiviti

es: 3

, 4, 5

WB

1:

pp. 3

– 8

Ac

tiviti

es: 2

, 3

WB

1:

pp. 6

– 8

; 80

- 82

Activ

ities

: 25a

&b; 2

6; 2

7

DB

E Te

xtbo

ok re

fere

nce

Te

rm 2

: Uni

t 1: W

hole

Num

bers

, p. 1

19

Te

rm 2

: Uni

t 1: W

hole

Num

bers

, p.

115

Term

2: U

nit 1

: Who

le N

umbe

rs, p

. 11

9 H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

60 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

2

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 3

HO

UR

S G

RAD

E 4

CAP

S pp

. 69

5 C

APS

pp. 1

57

6 C

APS

pp. 2

22

TOPI

C

Addi

tion

and

Subt

ract

ion

of W

hole

num

bers

C

once

pts,

Ski

lls a

nd

know

ledg

e N

umbe

r ra

nge

for c

alcu

latin

g Ad

ditio

n an

d su

btra

ctio

n of

who

le

num

bers

of a

t lea

st 4

-dig

its.

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal c

alcu

latio

ns w

ith w

hole

nu

mbe

rs in

clud

ing

• es

timat

ion

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

• us

ing

a nu

mbe

r lin

e •

usin

g ad

ditio

n an

d su

btra

ctio

n as

inve

rse

oper

atio

ns

Prop

ertie

s of

who

le n

umbe

rs

Rec

ogni

ze a

nd u

se th

e co

mm

utat

ive

and

asso

ciat

ive

prop

ertie

s of

who

le n

umbe

rs

Num

ber

rang

e fo

r cal

cula

tions

Ad

ditio

n an

d su

btra

ctio

n of

who

le

num

bers

with

at l

east

5-d

igit

num

bers

C

alcu

latio

n te

chni

ques

U

sing

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal c

alcu

latio

ns o

f who

le

num

bers

incl

udin

g:

• es

timat

ion

• ad

ding

and

sub

tract

ing

in c

olum

ns

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

usin

g a

num

ber l

ine

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• do

ublin

g an

d ha

lvin

g •

usin

g ad

ditio

n an

d su

btra

ctio

n as

inve

rse

oper

atio

ns

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e;

dist

ribut

ive

prop

ertie

s of

who

le

Num

ber

rang

e fo

r cal

cula

tions

•

Addi

tion

and

subt

ract

ion

of w

hole

nu

mbe

rs o

f at l

east

6-d

igit

num

bers

•

mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out b

rack

ets

Cal

cula

tion

tech

niqu

es

Usi

ng a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l cal

cula

tions

of w

hole

nu

mbe

rs in

clud

ing:

•

estim

atio

n

61MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Solv

ing

prob

lem

s So

lve

prob

lem

s in

con

text

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

fina

ncia

l co

ntex

ts

num

bers

0 in

term

s of

its

addi

tive

prop

erty

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng th

e fo

llow

ing:

•

finan

cial

con

text

s •

mea

sure

men

t con

text

s SU

GG

ESTE

D

MET

HO

DO

LOG

Y Al

l gra

des:

In

Ter

m 2

, lea

rner

s ad

d an

d su

btra

ct n

umbe

rs u

p to

4 d

igits

. R

ound

ing-

off i

nclu

des

roun

ding

off

to th

e ne

ares

t 1 0

00 a

s a

way

of e

stim

atin

g an

swer

s.

Lear

ners

sho

uld

solv

e pr

oble

ms

in c

onte

xts

and

do c

onte

xt fr

ee c

alcu

latio

ns

Lear

ners

con

tinue

to

• ch

eck

thei

r sol

utio

ns th

emse

lves

by

usin

g th

e in

vers

e op

erat

ion

• ju

dge

the

reas

onab

lene

ss o

f the

ir so

lutio

ns b

y ro

undi

ng o

ff nu

mbe

rs a

nd e

stim

atin

g an

swer

s.

The

calc

ulat

ion

tech

niqu

es c

ontin

ue to

mos

tly in

volv

e br

eaki

ng d

own

num

bers

. As

the

num

bers

lear

ners

wor

k w

ith g

et la

rger

, lea

rner

s m

ay b

egin

to lo

se tr

ack

of s

ome

num

bers

whe

n th

ey b

reak

up

num

bers

to

do c

alcu

latio

ns. U

sing

bra

cket

s is

hel

pful

to s

how

gro

upin

g of

num

bers

and

so

help

s le

arne

rs k

eep

track

of w

hat t

hey

are

doin

g.

Sinc

e th

e op

erat

ions

in b

rack

ets

have

to b

e do

ne fi

rst,

it re

mov

es a

ny c

onfu

sion

abo

ut th

e or

der o

f ope

ratio

ns. L

earn

ers

thus

do

not h

ave

to le

arn

rule

s su

ch a

s BO

DM

AS if

bra

cket

s ar

e us

ed ro

utin

ely

to in

dica

te w

hich

ope

ratio

ns h

ave

to b

e do

ne fi

rst.

• B

reak

ing

dow

n al

l num

bers

acc

ordi

ng to

pla

ce v

alue

par

ts to

add

E

xam

ple

Cal

cula

te 5

362

+ 2

486

5 36

2 +

2 48

6 =

5 00

0 +

300

+ 60

+ 2

+ 2

000

+ 4

00 +

80

+ 6

2

+

6 =

8

=

5 00

0 +

2 00

0 +

300

+ 40

0 +

60 +

80

+ 2

+ 6

o

r

an

d 60

+

80 =

1

40

= 7

000

+ 70

0 +

140

+ 8

and

300

+ 4

00 =

70

0 =

7 84

8

and

5 00

0 +

2 00

0 =

7 00

0 m

eans

5 3

62 +

2 4

86 =

7 8

48

62 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• A

ddin

g on

by

brea

king

dow

n th

e nu

mbe

r to

be a

dded

Ex

ampl

e C

alcu

late

5 3

62 +

2 4

86

5 36

2 +

2 00

0 →

7 3

62 +

400

→ 7

762

+ 8

0 →

7 6

842

+ 6

→ 7

848

•

Filli

ng u

p te

ns b

y br

eaki

ng d

own

the

num

ber t

o be

add

ed.

This

can

als

o be

cal

led

roun

ding

off

and

com

pens

atin

g. H

ere,

com

pens

atin

g m

eans

that

wha

teve

r is

adde

d, m

ust b

e su

btra

cted

aga

in s

o th

at th

e st

atem

ents

rem

ain

equi

vale

nt.

Exa

mpl

e C

alcu

late

2 4

86 +

48

2

486

+ 48

= (2

486

+ 1

4) –

14

+ 48

= 2

500

+ (4

8 –1

4) =

2 5

00 +

34

= 2

534

• B

reak

ing

dow

n bo

th n

umbe

rs to

sub

trac

t Ex

ampl

e C

alcu

late

4 6

87 –

2 1

43

4 68

7 –

2 14

3 =

4 00

0 +

600

+ 80

+ 7

– 2

000

- 10

0 –

40 –

3

o

r

7

–

3 =

4

= (4

000

– 2

000

) + (6

00 –

100

) + (8

0 –

40) +

(7 –

3)

an

d 80

–

40 =

40

=

2 00

0 +

500

+ 40

+ 4

a

nd 6

00 –

100

= 5

00

= 2

544

and

4 0

00 –

2 0

00 =

2 0

00

This

mea

ns th

at:

4

687

– 2

143

= 2

000

+ 50

0 +

40 +

4

= 2

544

• B

reak

ing

dow

n al

l the

num

bers

to a

dd u

sing

com

pens

atio

n (c

ount

erba

lanc

e)

Lear

ners

can

not s

ubtra

ct 4

from

3 o

r 80

from

40.

Inst

ead

of b

reak

ing

dow

n 74

3 in

to 7

00 +

40

+ 3

they

will

brea

k do

wn

743

into

600

+ 1

30 +

13.

The

n th

ey c

an s

ubtra

ct 4

from

13

and

80 fr

om 1

30.

C

alcu

late

: 8 7

43 –

5 6

84

8 74

3 –

5 68

4 =

(8 0

00 +

700

+ 4

0 +

3) –

5 0

00 –

600

– 8

0 –

4 =

(8 0

00 +

600

+ 1

30 +

13)

– 5

000

– 6

00 –

80

– 4

(bre

akin

g up

743

into

600

+ 1

30 +

13)

=

(8 0

00 –

5 0

00) +

(600

– 6

00) +

(130

– 8

0) +

(13

– 4)

= 3

000

+ 0

+ 5

0

+ 9

= 3

059

•

Subt

ract

ing

by b

reak

ing

dow

n th

e nu

mbe

r to

be s

ubtr

acte

d C

alcu

late

4 6

87 –

2 1

43

4 68

7 –

2 00

0 →

2 6

87 –

100

→ 2

587

– 4

0 →

2 5

47 –

3 →

2 5

44

63MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Kin

ds o

f pro

blem

s •

Sum

mat

ion,

Incr

ease

and

dec

reas

e, c

ompa

rison

by

diffe

renc

e; c

ompa

rison

by

ratio

•

See

the

desc

riptio

n of

pro

blem

type

s at

the

end

of th

e gr

ade

note

s G

rade

5 a

nd 6

: •

In T

erm

2, l

earn

ers

add

and

subt

ract

num

bers

with

up

to 5

dig

its.

• R

ound

ing

off a

s a

way

of e

stim

atin

g an

swer

s to

incl

ude

roun

ding

off

to th

e ne

ares

t 1 0

00 a

s w

ell a

s ro

undi

ng o

ff to

the

near

est 1

0, 1

00

Le

arne

rs s

houl

d so

lve

prob

lem

s in

con

text

s an

d do

con

text

free

cal

cula

tions

As

num

ber r

ange

s ge

t lar

ger m

any

lear

ners

tend

to lo

se th

e pa

rts o

f the

num

ber t

hat t

hey

brea

k up

, whe

n th

ey tr

y to

co

mbi

ne a

gain

. Thi

s is

esp

ecia

lly th

e ca

se w

hen

mor

e th

an tw

o 5-

digi

t num

bers

are

bei

ng a

dded

. It i

s fo

r thi

s re

ason

that

co

lum

n ad

ditio

n an

d co

lum

n su

btra

ctio

n ar

e in

trodu

ced

in G

rade

5. I

n Te

rm 2

one

can

stil

l enc

oura

ge le

arne

rs to

exp

and

the

num

bers

as

they

writ

e th

em in

col

umns

. In

Term

1, a

n op

tion

of a

col

umn

met

hod

was

pro

vide

d, b

ut it

con

sist

ed o

f pu

tting

diff

eren

t pla

ce v

alue

s in

to d

iffer

ent r

ows.

Le

arne

rs c

ontin

ue to

: •

chec

k th

eir s

olut

ions

them

selv

es e

.g. b

y us

ing

the

inve

rse

oper

atio

n •

judg

e th

e re

ason

able

ness

of t

heir

solu

tions

e.g

. by

roun

ding

off

num

bers

and

est

imat

ing

answ

ers

Exa

mpl

e:

Cal

cula

te: 5

6 42

3 +7

581

+21

479

•

Brea

king

dow

n al

l the

num

bers

to a

dd

Addi

ng in

a ro

w (h

oriz

onta

lly)

50 0

00+6

000

+400

+20+

3+7

000+

500+

80+1

+20

000+

1 00

0+40

0 +7

0+9

= 50

000

+20

000+

6 0

00+7

000

+1 0

00+4

00+5

00 +

4 00

+20+

80

+70+

3+1

+9

= 70

000

+ 1

4 00

0 +

1 30

0 +

170

+ 14

=

70 0

00 +

10

000

+ 4

000

+ 1

000

+ 30

0 +

100

+ 70

+ 1

0 +

4 =

80 0

00 +

5 0

00 +

400

+ 8

0+ 4

=

8548

4

The

horiz

onta

l met

hod

may

get

unw

ield

y w

hen

mor

e th

an tw

o 5-

digi

t num

bers

are

add

ed. T

he a

ltern

ativ

e is

to u

se th

e ex

pand

ed v

ertic

al m

etho

d.

•

Expa

nded

ver

tical

met

hod

56

423

= 5

0 00

0 +

6 00

0 +

400

+

20

+ +

7

581

=

7 00

0 +

500

+

80

+ 1

+ 21

479

= 2

0 00

0+ 1

000

+

400

+ 7

0 +

9

70 0

00+

14 0

00 +

130

0 +

170

+ 10

64 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

= 70

000

+ 1

0 00

0 +

5 00

0 +

400

+ 80

+ 4

= 85

484

• Ad

ding

on

(by

brea

king

dow

n th

e nu

mbe

r to

be a

dded

) C

alcu

late

: 56

423

+ 7

581

56

423

+ 7

000

→ 6

3 42

3 +

500

→ 6

3 92

3 +

80 →

64

003

+ 1

→ 6

4 00

4 Th

is te

nds

to w

ork

bette

r if o

nly

two

num

bers

are

add

ed. I

f a th

ird o

r fou

rth n

umbe

r is

adde

d, th

ey c

an b

e br

oken

up

and

adde

d on

e at

a ti

me,

but

the

expa

nded

col

umn

met

hod

is m

ore

effic

ient

. •

Bre

akin

g do

wn

all t

he n

umbe

rs c

ordi

ng to

pla

ce v

alue

par

ts to

sub

trac

t usi

ng c

ompe

nsat

ion

(cou

nter

bala

nce)

E

xam

ple:

C

alcu

late

: 8 7

43 –

5 6

84

8 74

3 –

5 68

4 =

8 00

0 +

700

+ 40

+ 3

– 5

000

– 6

00 –

80

– 4

= 8

000

+ 60

0 +

130

+ 13

– 5

000

– 6

00 –

80

– 4

(by

brea

king

up

743

into

600

+ 1

30 +

13)

=

8 00

0 –

5 00

0 +

600

– 60

0 +

130

– 80

+ 1

3 –

4 =

3 00

0

+

0

+

50

= 3

059

• B

reak

ing

dow

n nu

mbe

rs a

nd u

sing

the

expa

nded

col

umn

met

hod

Exam

ple:

C

alcu

late

: 98

743

– 45

684

Le

arne

rs c

anno

t sub

tract

4 fr

om 3

or 8

0 fro

m 4

0. In

stea

d of

bre

akin

g do

wn

743

into

700

+ 4

0 +

3 th

ey w

ill br

eak

dow

n 74

3 in

to 6

00 +

130

+ 1

3. T

hen

they

can

sub

tract

4 fr

om 1

3 an

d 80

from

130

.

600

130

13

9

8 7

4 3

= 9

0 00

0+

8 0

00 +

700

+

40

+

3 –

4 5

6 8

4

=

40

000+

5

000

+

6

00 +

8

0 +

4

50 00

0+

3 0

00 +

0

+

50 +

9 =

53

059

• Su

btra

ctin

g by

bre

akin

g do

wn

num

ber t

o be

sub

trac

ted

Exam

ple:

C

alcu

late

74

687

– 52

143

74

687

– 5

0 00

0→24

687

– 2

000

→ 2

2 68

7 –

100

→ 2

2 58

7 –

40 →

22

547

– 3

= 22

544

OR

65MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

25

746

– 1

0 00

0 –

4 00

0 –

500

– 30

– 2

= (1

5 74

6 –

4 00

0) –

500

– 3

0 –

2

= (1

1 74

6 –

500)

– 3

0 –

2

= (1

1 24

6 –

30) –

2

= 11

216

– 2

=

11 2

14

This

tend

s to

wor

k be

tter i

f onl

y on

e nu

mbe

r is

subt

ract

ed fr

om a

noth

er. I

f a s

econ

d or

third

num

ber i

s su

btra

cted

, the

y ca

n be

br

oken

up

and

subt

ract

ed o

ne a

t a ti

me,

but

the

expa

nded

col

umn

met

hod

is m

ore

effic

ient

. Pr

oble

ms

• Su

mm

atio

n, in

crea

se a

nd d

ecre

ase,

com

paris

on b

y di

ffere

nce;

com

paris

on b

y ra

tio

• Se

e th

e de

scrip

tion

of p

robl

em ty

pes

at th

e en

d of

the

grad

e no

tes

Gra

de 6

: C

heck

ing

solu

tions

Le

arne

rs s

houl

d kn

ow th

at th

ey c

an

• ch

eck

an a

dditi

on c

alcu

latio

n by

sub

trac

tion.

Ex

ampl

e: If

45

362

+ 32

488

= 7

7 84

8; th

en 7

7 84

8 –

32 4

88 =

45

362

• ch

eck

a su

btra

ctio

n ca

lcul

atio

n by

add

ition

Ex

ampl

e: If

54

687

– 32

134

= 2

2 54

4, th

en 2

2 54

4 +

32 1

34 =

54

687

Usi

ng th

e in

vers

e op

erat

ion

to c

heck

sol

utio

ns is

one

reas

on fo

r tea

chin

g ad

ditio

n an

d su

btra

ctio

n si

mul

tane

ousl

y.

Anot

her r

easo

n fo

r doi

ng th

e tw

o op

erat

ions

at t

he s

ame

time

is th

at w

hen

lear

ners

sol

ve p

robl

ems,

it is

som

etim

es p

ossi

ble

to

solv

e th

e sa

me

prob

lem

by

doin

g ei

ther

add

ition

or s

ubtra

ctio

n.

Ex

ampl

e: V

eli’s

sho

ppin

g co

sts

R16

3. H

e pa

ys w

ith a

R20

0 no

te. H

ow m

uch

chan

ge d

oes

he g

et”?

So

me

lear

ners

may

add

on

from

R16

3 to

get

R20

0 as

follo

ws:

R

163

+ R

7= R

170,

then

R17

0 +

R30

= R

200.

Ve

li ge

ts R

37 c

hang

e.

E

xam

ple:

C

alcu

late

: 56

423

+ 7

581

+ 2

1 47

9 •

Col

umn

met

hod

for a

ddin

g By

Gra

de 6

lear

ners

sho

uld

have

had

eno

ugh

expe

rienc

e w

ith b

reak

ing

up n

umbe

rs to

add

and

sub

tract

them

. The

hor

izon

tal

66 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

met

hod

of e

xpan

ding

num

bers

bef

ore

addi

ng th

em c

an g

et u

nwie

ldy

whe

n m

ore

than

two

5-di

git n

umbe

rs a

re a

dded

. Ter

m 1

le

arne

rs c

an re

visi

t the

exp

ande

d ve

rtica

l met

hod,

and

then

mov

e on

to th

e tra

ditio

nal c

olum

n m

etho

d •

Expa

nded

ver

tical

col

umn

met

hod

to a

dd

56

423

=

50

000

+

6

000

+

400

+

20

+

3

+7

581

=

7

000

+

5

00

+

80

+

1

+21

479

=

20

000

+

1

000

+

400

+

70

+

9

Tot

al

=

70 0

00

+

14 0

00

+

1 3

00

+

170

+

13

This

can

be

writ

ten

as 7

0 00

0 +

10 0

00 +

5 0

00 +

400

+ 8

0 +

3 =

85 4

83

• T

he v

ertic

al c

olum

n m

etho

d to

add

.

1 1

1 1

5

6 42

3

+ 21

479

+

7 58

1

85 48

3 •

Expa

nded

ver

tical

col

umn

met

hod

to s

ubtr

act

Exam

ple:

Cal

cula

te: 9

8 74

3 –

45 6

84

600

130

13

9

8 74

3 =

90 0

00

+

8 0

00

+

700

+

40

+

3

–

45 6

84

=

40

000

+

5

000

+

6

00

+

80

+

4

Tot

al =

50 0

00

+

3 0

00

+

0 +

50

+

9

Th

eref

ore

50 0

00 +

3 0

00 +

0 +

50

+ 9

= 53

059

•

The

ver

tical

col

umn

met

hod

to s

ubtr

act

6

13

13

98 7

43

– 45

684

53

059

Prob

lem

s •

Sum

mat

ion,

incr

ease

and

dec

reas

e, c

ompa

rison

by

diffe

renc

e; c

ompa

rison

by

ratio

•

See

the

desc

riptio

n of

pro

blem

type

s at

the

end

of th

e G

rade

6 n

otes

67MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Wor

king

with

cal

cula

tors

•

The

men

tal m

athe

mat

ics

prog

ram

me

cont

ains

wor

k on

num

ber c

once

pt, n

umbe

r fac

ts a

nd m

enta

l cal

cula

tion

tech

niqu

es.

Dai

ly w

ork

on m

enta

l Mat

hem

atic

s co

mbi

ned

with

dai

ly w

ritte

n ca

lcul

atio

ns w

ill pr

even

t lea

rner

s fro

m b

ecom

ing

depe

nden

t on

cal

cula

tors

and

not

kno

win

g ho

w to

cal

cula

te w

ithou

t the

m.

• C

alcu

lato

rs a

re a

use

ful w

ay fo

r lea

rner

s to

exp

lore

num

ber p

atte

rns

and

whe

n w

orki

ng w

ith v

ery

larg

e nu

mbe

rs.

LTS

M

Res

ourc

es

Num

ber l

ines

: Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, F

lash

car

ds,

Base

10

bloc

ks, 1

00 c

harts

,

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

ook

refe

nce

WB

1:

pp. 2

6 –

38

Activ

ities

: 9, 1

0a&b

; 12

pp

. 92

– 96

Ac

tiviti

es: 3

2a&b

; 33

WB

1:

pp 3

0 –

32

Activ

ities

: 10a

&b

pp 9

0 –

96

Activ

ities

: 29a

&b, 3

0a&b

WB

1:

pp.

14 -

24

Activ

ities

: 8a&

b

DB

E Te

xtbo

ok re

fere

nce

Term

2: U

nit 2

: Who

le N

umbe

rs:

Addi

tion

and

subt

ract

ion,

p. 1

28

Term

2: U

nit 2

: Who

le N

umbe

rs:

Addi

tion

and

subt

ract

ion,

p. 1

21

Term

1: U

nit 3

: Who

le N

umbe

rs:

Addi

tion

and

subt

ract

ion,

p. 2

5 H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent –

Te

st

68 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

3

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 3

HO

UR

S G

RAD

E 4

CAP

S pp

. 67

- 77

5 C

APS

pp. 1

66

6 C

APS

pp. 2

40

TOPI

C

Mul

tiplic

atio

n of

Who

le N

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 2

-di

git b

y 2-

digi

t num

bers

C

alcu

latio

n te

chni

ques

U

se a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l cal

cula

tions

with

who

le

num

bers

incl

udin

g:

• es

timat

ion

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

Num

ber

rang

e fo

r cal

cula

tions

M

ultip

licat

ion

of a

t lea

st w

hole

3-d

igit

by

2-di

git n

umbe

rs

Cal

cula

tion

tech

niqu

es

Usi

ng a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l cal

cula

tions

of w

hole

nu

mbe

rs in

clud

ing:

•

estim

atio

n •

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• us

ing

a nu

mbe

r lin

e •

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 4

-di

git b

y 3-

digi

t num

bers

•

Mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out

brac

kets

C

alcu

latio

n te

chni

ques

incl

ude

• es

timat

ion

• m

ultip

lyin

g in

col

umns

•

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• us

ing

a ca

lcul

ator

69MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s M

ultip

les

of 1

-dig

it nu

mbe

rs to

at l

east

10

0 Prop

ertie

s of

who

le n

umbe

rs

R

ecog

nize

and

use

the

com

mut

ativ

e;

asso

ciat

ive;

and

dis

tribu

tive

prop

ertie

s of

who

le n

umbe

rs

So

lvin

g pr

oble

ms

So

lve

prob

lem

s in

con

text

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

fina

ncia

l co

ntex

ts

So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng

co

mpa

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)

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igits

who

le

num

bers

to a

t lea

st 1

00

• Fa

ctor

s of

2-d

igit

who

le n

umbe

rs

to a

t lea

st 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

an

d us

e th

e co

mm

utat

ive;

as

soci

ativ

e an

d di

strib

utiv

e pr

oper

ties

with

who

le

num

bers

•

0 in

term

s of

its

addi

tive

prop

erty

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

finan

cial

con

text

s

mea

sure

men

t con

text

s

So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng c

ompa

ring

two

or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igit

and

3-di

git

num

bers

•

Fact

ors

of 2

-dig

it an

d 3-

digi

t who

le

num

bers

•

Prim

e fa

ctor

s of

num

bers

to a

t lea

st

100

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e;

dist

ribut

ive

prop

ertie

s of

w

hole

num

bers

•

0 in

term

s of

its

addi

tive

prop

erty

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

and

dec

imal

fra

ctio

ns, i

nclu

ding

-

finan

cial

con

text

s -

mea

sure

men

t con

text

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

, inc

ludi

ng

co

mpa

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 k

inds

(rat

e)

70 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LTSM

R

esou

rces

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,

Fl

ash

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, mul

tiplic

atio

n ta

ble

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

1:

pp. 1

26 –

128

Ac

tiviti

es:

46,4

7 p.

162

Ac

tiviti

es: 6

4

WB

1:

pp. 5

0 –

52

Activ

ities

: 17a

&b

pp. 1

40

Activ

ity: 4

9 pp

. 172

- 17

4

Activ

ities

: 63,

64

WB

1:

pp. 8

8 - 9

2 Ac

tiviti

es: 3

0; 3

1; 3

2;

p. 1

26

Activ

ities

: 47

DB

E Te

xtbo

ok re

fere

nce

Term

2: U

nit 5

: Who

le N

umbe

rs:

Mul

tiplic

atio

n, p

. 161

Term

2: U

nit 2

: Who

le N

umbe

rs:

Mul

tiplic

atio

n, p

. 157

Term

2: U

nit 2

: Who

le N

umbe

rs:

Mul

tiplic

atio

n, p

. 125

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

71MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 4

TE

RM

: 2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 3

HO

UR

S G

RAD

E 4

CAP

S pp

. 84

- 85

5 C

APS

pp. 1

72 -

173

6 C

APS

pp. 2

50 -

251

TOPI

C

Div

isio

n of

Who

le N

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

Num

ber

rang

e fo

r cal

cula

tions

•

Div

isio

n of

at l

east

who

le 3

-dig

it by

1-

digi

t num

bers

. Cal

cula

tion

tech

niqu

es

• U

se a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l cal

cula

tions

of w

hole

nu

mbe

rs in

clud

ing

- es

timat

ion

- bu

ildin

g u

p a

nd

brea

king

do

wn

num

bers

-

roun

ding

off

and

com

pens

atin

g -

doub

ling

and

halv

ing

- us

ing

mul

tiplic

atio

n an

d di

visi

on

as in

vers

e op

erat

ions

N

umbe

r ra

nge

for m

ultip

les

and

fact

ors

• M

ultip

les

of 1

-dig

it nu

mbe

rs to

at

leas

t 100

Num

ber

rang

e fo

r cal

cula

tions

D

ivis

ion

of a

t lea

st w

hole

3-d

igit

by

2-di

git n

umbe

rs

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal c

alcu

latio

ns w

ith w

hole

nu

mbe

rs in

clud

ing

• es

timat

ion

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

usin

g m

ultip

licat

ion

and

divi

sion

as

inve

rse

oper

atio

ns

Num

ber

rang

e fo

r co

untin

g, o

rder

ing

and

repr

esen

ting,

and

pla

ce v

alue

of

digi

ts

• Rec

ogni

ze th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 6

-dig

it nu

mbe

rs.

• R

ound

off

to th

e ne

ares

t 10,

100

, 1

000

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igit

num

bers

to a

t

Num

ber

rang

e fo

r cal

cula

tions

•

Div

isio

n of

at l

east

who

le 4

-dig

it by

3-

digi

t num

bers

•

mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out b

rack

ets

Cal

cula

tion

tech

niqu

es

• es

timat

ion

• us

ing

the

reci

proc

al re

latio

nshi

p be

twee

n m

ultip

licat

ion

and

divi

sion

•

long

div

isio

n •

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• us

ing

a ca

lcul

ator

N

umbe

r ra

nge

for m

ultip

les

and

fact

ors

• M

ultip

les

of 2

-dig

it an

d 3-

digi

t

72 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e; a

nd

dist

ribut

ive

prop

ertie

s of

who

le

num

bers

So

lvin

g pr

oble

ms

• So

lve

prob

lem

s in

co

ntex

ts

invo

lvin

g w

hole

num

bers

-

finan

cial

con

text

s -

mea

sure

men

t con

text

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

-

grou

ping

and

equ

al s

harin

g w

ith

rem

aind

ers

- co

mpa

ring

two

or m

ore

quan

titie

s of

th

e sa

me

kind

(rat

io)

- co

mpa

ring

two

quan

titie

s of

diff

eren

t ki

nds

(rate

)

leas

t 100

•

Fact

ors

of 2

-dig

it w

hole

num

bers

to

at l

east

100

M

ultip

licat

ion

fact

s •

Uni

ts b

y m

ultip

les

of 1

0 •

Uni

ts b

y m

ultip

les

of 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e; a

nd

dist

ribut

ive

prop

ertie

s of

who

le

num

bers

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

in

cont

exts

in

volv

ing

who

le

num

bers

, in

clud

ing

finan

cial

con

text

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

-

com

parin

g tw

o or

mor

e qu

antit

ies

of

the

sam

e ki

nd (r

atio

) - c

ompa

ring

two

quan

titie

s of

diff

eren

t ki

nds

(rate

) - g

roup

ing

and

equa

l sha

ring

with

re

mai

nder

s

num

bers

•

Fact

ors

of 2

-dig

it an

d 3-

digi

t who

le

num

bers

•

Prim

e fa

ctor

s of

num

bers

up

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

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

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs a

nd d

ecim

al fr

actio

ns,

incl

udin

g -

finan

cial

con

text

s -

mea

sure

men

t con

text

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

-

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(rat

io)

- co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

- gr

oupi

ng a

nd e

qual

sha

ring

with

re

mai

nder

s

73MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LTSM

Res

ourc

es

Num

ber l

ines

: Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, F

lash

car

ds,

Base

10

bloc

ks, 1

00 c

harts

, mul

tiplic

atio

n ta

ble

G

rade

4

Gra

de 5

G

rade

6

W

B 1

: p.

162

Ac

tiviti

es 6

4

WB

1:

pp. 1

40

Activ

ity: 4

9 pp

. 172

- 17

4

Activ

ities

: 63,

64

WB

1:

p. 1

26

Activ

ities

: 47

DB

E TE

XTB

OO

K

REF

EREN

CE

Term

2: U

nit 1

0: W

hole

Num

bers

: D

ivis

ion,

p. 2

00

Term

2: U

nit 9

: Who

le N

umbe

rs:

Div

isio

n, p

. 187

Term

2: U

nit 6

: Who

le N

umbe

rs:

Div

isio

n, p

. 165

HO

MEW

OR

K

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent –

Te

st

74 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

5

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 9

HO

UR

S G

RAD

E 4

CAP

S pp

. 71

- 72

5 C

APS

pp. 1

72 -

173

6 C

APS

pp. 2

80 (F

ract

ions

) C

APS

pp. 2

52 (D

ecim

al F

ract

ions

) TO

PIC

C

omm

on F

ract

ions

and

Dec

imal

Fra

ctio

ns

Con

cept

s, S

kills

and

kn

owle

dge

Des

crib

ing

and

orde

ring

frac

tions

•

Com

pare

and

ord

er c

omm

on fr

actio

ns

of d

iffer

ent d

enom

inat

ors

(hal

ves,

th

irds,

qua

rters

, fift

hs, s

ixth

s, s

even

ths,

ei

ghth

s)

• D

escr

ibe

and

com

pare

com

mon

fra

ctio

ns in

dia

gram

for

m

Cal

cula

tions

with

frac

tions

: •

Rec

ogni

ze, d

escr

ibe

and

use

the

equi

vale

nce

of d

ivis

ion

and

fract

ions

•

Addi

tion

of c

omm

on fr

actio

ns w

ith

sam

e de

nom

inat

ors

Solv

ing

prob

lem

s

• D

escr

ibin

g an

d or

derin

g fr

actio

ns

• C

ount

forw

ards

and

bac

kwar

ds in

fra

ctio

ns

• C

ompa

re a

nd o

rder

com

mon

fra

ctio

ns to

at l

east

twel

fths

Cal

cula

tions

with

frac

tions

•

Addi

tion

of c

omm

on fr

actio

ns w

ith th

e sa

me

deno

min

ator

•

Rec

ogni

ze, d

escr

ibe

and

use

the

equi

vale

nce

of d

ivis

ion

and

fract

ions

So

lvin

g pr

oble

ms

Des

crib

ing

and

orde

ring

frac

tions

•

Com

pare

and

ord

er c

omm

on

fract

ions

, inc

ludi

ng s

peci

fical

ly te

nths

an

d hu

ndre

dths

C

alcu

latio

ns u

sing

frac

tions

• A

dditi

on a

nd s

ubtra

ctio

n of

com

mon

fra

ctio

ns w

ith d

enom

inat

ors

whi

ch a

re

mul

tiple

s of

eac

h ot

her.

• Ad

ditio

n an

d su

btra

ctio

n of

mix

ed

num

bers

•

Frac

tions

of w

hole

num

bers

So

lvin

g pr

oble

ms

75MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g fra

ctio

ns, i

nclu

ding

gro

upin

g an

d eq

ual

shar

ing

Equi

vale

nt fo

rms:

R

ecog

nize

and

use

equ

ival

ent f

orm

s of

co

mm

on fr

actio

ns (d

enom

inat

ors

whi

ch

are

mul

tiple

s of

eac

h ot

her)

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, i

nclu

ding

gro

upin

g an

d sh

arin

g Eq

uiva

lent

form

s:

Rec

ogni

ze a

nd u

se e

quiv

alen

t for

ms

of

com

mon

frac

tions

with

den

omin

ator

s w

hich

are

mul

tiple

s of

eac

h ot

her.

• So

lve

prob

lem

s in

con

text

s in

volv

ing

com

mon

frac

tions

, inc

ludi

ng g

roup

ing

and

shar

ing

Perc

enta

ges

• C

alcu

late

per

cent

ages

of w

hole

nu

mbe

rs

Equi

vale

nt fo

rms:

•

Rec

ogni

ze a

nd u

se e

quiv

alen

t fo

rms

of c

omm

on fr

actio

ns w

ith

1-di

git o

r 2-d

igit

deno

min

ator

s w

ith

deno

min

ator

s w

hich

are

mul

tiple

s of

ea

ch o

ther

•

Rec

ogni

ze e

quiv

alen

ce b

etw

een

com

mon

frac

tion

and

deci

mal

frac

tion

form

s of

the

sam

e nu

mbe

r • R

ecog

nize

equ

ival

ence

bet

wee

n co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r. D

ecim

al fr

actio

ns

Rec

ogni

zing

, ord

erin

g an

d pl

ace

valu

e of

dec

imal

frac

tions

•

Cou

nt fo

rwar

ds a

nd b

ackw

ards

in

deci

mal

frac

tions

to a

t lea

st tw

o de

cim

al p

lace

s •

Com

pare

and

ord

er d

ecim

al fr

actio

ns

to a

t lea

st tw

o de

cim

al p

lace

s •

Plac

e va

lue

of d

igits

to a

t lea

st tw

o de

cim

al p

lace

s C

alcu

latio

ns w

ith d

ecim

al fr

actio

ns

• Ad

ditio

n an

d su

btra

ctio

n of

dec

imal

fra

ctio

ns o

f at l

east

two

deci

mal

pl

aces

•

Mul

tiply

dec

imal

frac

tions

by

10 a

nd

100

76 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Solv

ing

prob

lem

s So

lve

prob

lem

s in

con

text

invo

lvin

g de

cim

al fr

actio

ns

SUG

GES

TED

M

ETH

OD

OLO

GY

All l

earn

ers

can

star

t of w

ith th

e ea

sier

pro

blem

s. G

rade

4 le

arne

rs a

re tr

eatin

g th

e w

ork

as n

ew w

hile

the

Gra

de 5

an

d G

rade

6 le

arne

rs c

an m

ove

on to

mor

e co

mpl

ex p

robl

ems

as in

dica

ted

in th

e C

APS

polic

y do

cum

ent.

All g

rade

s:

Seve

nths

are

new

. Th

ere

are

diffe

rent

way

s to

und

erst

and

fract

ions

. Thi

s m

eans

that

lear

ners

sho

uld

deve

lop

the

conc

ept o

f fra

ctio

ns in

a v

arie

ty

of w

ays.

Pro

blem

-sol

ving

con

text

s ca

n he

lp l

earn

ers

to u

nder

stan

d m

any

way

s of

thi

nkin

g ab

out

fract

ions

. A

varie

ty o

f pr

oble

ms

shou

ld b

e gi

ven

to le

arne

rs. S

ee th

e ty

pes

of fr

actio

ns p

robl

ems

stat

ed a

t the

end

of t

he G

rade

not

es. T

he c

once

pt

of a

frac

tion

shou

ld fi

rst b

e de

velo

ped

befo

re le

arne

rs fo

cus

on e

quiv

alen

ce a

nd c

alcu

latin

g.

Lear

ners

can

als

o w

ork

with

app

arat

us a

nd d

iagr

ams.

Diff

eren

t di

agra

ms

or a

ppar

atus

dev

elop

diff

eren

t w

ays

of th

inki

ng

abou

t fra

ctio

ns:

R

egio

n or

are

a m

odel

s de

velo

p th

e co

ncep

t of f

ract

ions

as

part

of a

who

le. I

f use

d in

par

ticul

ar w

ays

they

can

als

o de

velo

p th

e co

ncep

t of a

frac

tion

as a

mea

sure

. Exa

mpl

es o

f are

a m

odel

s in

clud

e ci

rcle

s cu

t int

o fra

ctio

n pi

eces

or

diag

ram

s of

pie

s, re

ctan

gles

or o

ther

geo

met

ric s

hape

s di

vide

d in

to fr

actio

n pi

eces

(pap

er fo

ldin

g), f

ract

ions

usi

ng

squa

re o

r dot

ty g

rid p

aper

, geo

boar

ds

Le

ngth

or m

easu

rem

ent m

odel

s ca

n be

use

d to

dev

elop

the

conc

ept o

f fra

ctio

ns a

s pa

rt of

a w

hole

and

if u

sed

in

parti

cula

r way

s al

so fr

actio

n as

a m

easu

re. E

xam

ples

of l

engt

h m

odel

s in

clud

e fra

ctio

n st

rips,

Cui

sena

ire ro

ds,

num

ber l

ines

Set m

odel

s de

velo

p th

e co

ncep

t of a

frac

tion

of a

col

lect

ion

of o

bjec

ts a

nd c

an la

y th

e ba

sis

for t

hink

ing

abou

t a

fract

ion

of a

num

ber e

.g.1 3

of 1

2. E

xam

ples

of s

et m

odel

s in

clud

e co

unte

rs o

f any

kin

d in

diff

eren

t arra

ngem

ents

.

Lear

ners

sho

uld

not

only

wor

k w

ith o

ne k

ind

of m

odel

, be

caus

e th

is c

an li

mit

thei

r un

ders

tand

ing

of f

ract

ions

. Fo

r ex

ampl

e,

fract

ions

in d

iagr

am fo

rms

shou

ld in

clud

e re

gion

mod

els

(circ

les

and

othe

r geo

met

ric s

hape

s di

vide

d in

to fr

actio

n pa

rts),

leng

th

mod

els

(incl

udin

g nu

mbe

r lin

es) a

nd s

et m

odel

s (w

hich

sho

w c

olle

ctio

ns o

f obj

ects

). In

Ter

m 1

lear

ners

sho

uld

revi

se a

nd c

onso

lidat

e w

hat t

hey

lear

ned

abou

t fra

ctio

ns in

Gra

de 3

. Le

arne

rs s

houl

d so

lve

prob

lem

s as

wel

l as

wor

k w

ith a

ppar

atus

and

dia

gram

s in

volv

ing

area

, len

gth

and

set m

odel

s to

ens

ure

that

they

:

unde

rsta

nd th

e re

latio

nshi

p be

twee

n fra

ctio

ns a

nd d

ivis

ion

i.e. i

f you

sha

re a

mon

gst 3

lear

ners

you

will

be

mak

ing

third

s

are

able

to n

ame

fract

ions

. Ter

min

olog

y lik

e “3

ove

r 4” s

houl

d be

avo

ided

as

it te

nds

to e

ncou

rage

lear

ners

to

thin

k ab

out e

ach

fract

ion

as tw

o di

ffere

nt n

umbe

rs, r

athe

r tha

n 3 4

bein

g a

num

ber w

hich

is g

reat

er 1 2

than

but

less

77MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

than

1.

Whe

n na

min

g fra

ctio

n pa

rts it

is u

sefu

l for

lear

ners

to ra

ther

use

the

form

“3 q

uarte

rs”.

Lear

ners

sho

uld,

thro

ugh

wor

k w

ith a

ppar

atus

, dia

gram

s an

d so

lvin

g pr

oble

ms,

lear

n th

e ne

w f

ract

ions

that

the

y w

ill de

al

with

in G

rade

4.

G

rade

5 a

nd 6

: W

hat i

s di

ffere

nt to

Gra

de 4

?

Nin

ths,

tent

hs, e

leve

nths

and

twel

fths

Le

arne

rs c

ount

in fr

actio

ns

Su

btra

ctio

n of

frac

tions

with

the

sam

e de

nom

inat

ors

Ad

ditio

n an

d su

btra

ctio

n of

mix

ed n

umbe

rs

Fr

actio

ns o

f who

le n

umbe

rs th

at re

sult

in w

hole

num

bers

M

ost

of t

he n

ew w

ork

men

tione

d ab

ove

can

be d

evel

oped

in

Term

s 3

and

4. H

owev

er,

lear

ners

can

beg

in t

o co

unt

in

fract

ions

Le

arne

rs s

houl

d de

velo

p th

e co

ncep

t of f

ract

ions

in a

var

iety

of w

ays.

Pro

blem

sol

ving

con

text

s ca

n he

lp le

arne

rs to

und

erst

and

man

y w

ays

of th

inki

ng a

bout

frac

tions

. A v

arie

ty o

f pro

blem

s sh

ould

be

give

n to

lear

ners

. See

the

type

s of

frac

tions

pro

blem

s st

ated

at t

he e

nd o

f the

gra

de n

otes

. Le

arne

rs c

an a

lso

wor

k w

ith a

ppar

atus

and

dia

gram

s. D

iffer

ent d

iagr

ams

or a

ppar

atus

dev

elop

diff

eren

t way

s of

thin

king

abo

ut

fract

ions

.

Reg

ion

or a

rea

mod

els

deve

lop

the

conc

ept o

f fra

ctio

ns a

s pa

rt of

a w

hole

. If u

sed

in p

artic

ular

way

s th

ey c

an a

lso

deve

lop

the

conc

ept

of f

ract

ion

as a

mea

sure

. Ex

ampl

es o

f ar

ea m

odel

s in

clud

e ci

rcle

s cu

t in

to f

ract

ion

piec

es (

or

diag

ram

s of

pie

s),

rect

angl

es o

r ot

her

geom

etric

sha

pes

divi

ded

into

fra

ctio

n pi

eces

(pa

per

fold

ing)

, fra

ctio

ns u

sing

sq

uare

or d

otty

grid

pap

er, g

eobo

ards

.

Leng

th o

r m

easu

rem

ent

mod

els

can

be u

sed

to d

evel

op t

he c

once

pt o

f fra

ctio

ns a

s pa

rt of

a w

hole

and

if u

sed

in

parti

cula

r way

s al

so fr

actio

n as

a m

easu

re. E

xam

ples

of l

engt

h m

odel

s in

clud

e fra

ctio

n st

rips,

Cui

sena

ire ro

ds, n

umbe

r lin

es.

Se

t mod

els

deve

lop

the

conc

ept o

f fra

ctio

n of

a c

olle

ctio

n of

obj

ects

(and

can

lay

the

basi

s fo

r thi

nkin

g ab

out a

frac

tion

of a

num

ber e

.g. 1 3

of 1

2). E

xam

ples

of s

et m

odel

s in

clud

e co

unte

rs o

f any

kin

d in

diff

eren

t arra

ngem

ents

. Le

arne

rs s

houl

d no

t on

ly w

ork

with

one

kin

d of

mod

el,

beca

use

this

can

lim

it th

eir

unde

rsta

ndin

g of

fra

ctio

ns.

For

exam

ple

fract

ions

in d

iagr

am fo

rms

shou

ld in

clud

e re

gion

mod

el (c

ircle

s an

d ot

her g

eom

etric

sha

pes

divi

ded

into

frac

tion

parts

), le

ngth

m

odel

s (in

clud

ing

num

ber l

ines

) and

set

mod

els

(whi

ch s

how

col

lect

ions

of o

bjec

ts).

In T

erm

2 le

arne

rs s

houl

d re

vise

and

con

solid

ate

wha

t the

y le

arne

d ab

out f

ract

ions

in G

rade

4. T

his

is d

escr

ibed

bel

ow, b

ut

lear

ners

can

als

o co

unt i

n fra

ctio

ns.

78 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Cou

ntin

g in

frac

tions

can

hap

pen

• as

lear

ners

pla

ce d

own

fract

ion

piec

es

• on

the

num

ber l

ine

• or

in n

umbe

r cha

ins

like

the

one

show

n be

low.

+

1 2

+1 2

+1 2

+ 1 2

+1 2

+1 2

+

1 2 Eq

uiva

lenc

e, c

ompa

ring

and

orde

ring

Equi

vale

nce

shou

ld b

e ap

proa

ched

usi

ng a

ppar

atus

, dia

gram

s or

pro

blem

con

text

s. L

earn

ers

are

not e

xpec

ted

to b

e ab

le to

giv

e eq

uiva

lent

frac

tions

in s

ymbo

lic (n

umbe

r) fo

rm w

ithou

t hav

ing

diag

ram

s to

whi

ch th

ey c

an re

fer o

r a p

robl

em c

onte

xt in

whi

ch to

m

ake

sens

e of

the

equ

ival

ence

. O

nce

lear

ners

are

com

forta

ble

with

equ

ival

ence

, it

is e

asy

for

them

to

com

pare

and

ord

er

fract

ions

. C

alcu

latio

ns w

ith fr

actio

ns:

Cal

cula

tions

with

frac

tions

in th

e fir

st te

rm c

an fo

cus

on:

• m

akin

g fra

ctio

ns th

roug

h gr

oupi

ng o

r sh

arin

g w

hich

is li

nked

with

und

erst

andi

ng th

e re

latio

nshi

p be

twee

n di

visi

on a

nd

fract

ions

e.g

. If c

hild

ren

shar

e sw

eets

equ

ally,

they

will

each

get

1 5 o

f the

sw

eets

•

addi

ng fr

actio

ns w

ith th

e sa

me

deno

min

ator

s C

alcu

latio

ns a

s w

ith o

ther

asp

ects

of

fract

ions

sho

uld

be d

evel

oped

eith

er t

hrou

gh p

robl

em c

onte

xts

or w

ith t

he u

se o

f ap

para

tus

or d

iagr

ams.

Lea

rner

s sh

ould

be

give

n pr

oble

m c

onte

xts

in w

hich

they

nee

d to

add

frac

tion

parts

. Lea

rner

s sh

ould

al

so b

e gi

ven

eith

er f

ract

ion

piec

es t

o co

unt

e.g.

3 8 + 4 8

can

be

done

by

coun

ting

out

and

coun

ting

on i

n ei

ghth

s w

ith

appa

ratu

s or

by

colo

ring

in d

iagr

ams

or b

y “h

oppi

ng” i

n ei

ghth

s on

a n

umbe

r lin

e.

Gra

de 6

It

is u

sefu

l to

prac

tice

equi

vale

nce

betw

een

the

com

mon

frac

tion,

dec

imal

frac

tions

and

per

cent

age

form

s of

the

sam

e nu

mbe

r. D

ecim

al fr

actio

n is

a n

ew to

pic

for G

rade

6 le

arne

rs.

Lear

ners

sho

uld

alre

ady

have

wor

ked

with

tent

hs a

nd h

undr

edth

s in

com

mon

frac

tion

form

. The

y sh

ould

sta

rt by

rew

ritin

g an

d co

nver

ting

tent

hs a

nd h

undr

edth

s in

com

mon

fra

ctio

n fo

rm t

o de

cim

al f

ract

ions

. Whe

re d

enom

inat

ors

of o

ther

fra

ctio

ns a

re

fact

ors

of 1

0 e.

g. 2

, 5 o

r fac

tors

of 1

00 e

.g. 2

, 4, 2

5, 2

0, 5

0 le

arne

rs c

an c

onve

rt th

ese

to h

undr

edth

s us

ing

wha

t the

y kn

ow a

bout

3

79MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

equi

vale

nce.

D

ivid

ing

who

le n

umbe

rs b

y 10

, 100

, 1 0

00, e

tc. h

elps

to b

uild

lear

ners

’ und

erst

andi

ng o

f the

pla

ce v

alue

of t

he d

igits

in d

ecim

al

fract

ions

. Cal

cula

tors

can

be

usef

ul to

ols

for

lear

ners

to le

arn

abou

t pat

tern

s w

hen

mul

tiply

ing

or d

ivid

ing

deci

mal

frac

tions

by

10, 1

00, e

tc.

Cou

ntin

g in

dec

imal

s Le

arne

rs s

houl

d no

t spe

nd a

lot o

f tim

e do

ing

verb

al c

ount

ing

in d

ecim

als.

A m

ore

usef

ul e

xerc

ise

is u

sing

num

ber c

hain

s lik

e th

e on

e be

low

: The

se c

ount

ing

or “a

ddin

g on

” exe

rcis

es o

ften

help

lear

ners

to in

crea

se th

eir u

nder

stan

ding

of p

lace

val

ue.

Exer

cise

s lik

e th

e on

e ab

ove

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.

Equi

vale

nce

betw

een

com

mon

frac

tions

and

dec

imal

frac

tion

form

s Le

arne

rs a

re n

ot e

xpec

ted

to b

e ab

le to

con

vert

all c

omm

on fr

actio

n in

to it

s de

cim

al fr

actio

n fo

rm, m

erel

y to

see

the

rela

tions

hip

betw

een

tent

hs a

nd h

undr

edth

s in

thei

r dec

imal

form

s.

Cal

cula

ting

usin

g de

cim

als

Lear

ners

add

and

sub

tract

dec

imal

frac

tions

. Lea

rner

sho

uld

estim

ate

thei

r ans

wer

s be

fore

cal

cula

ting.

The

y sh

ould

be

able

to

judg

e th

e re

ason

able

ness

of a

nsw

ers

and

also

che

ck th

eir

own

answ

ers.

Und

erst

andi

ng p

lace

val

ue o

f dig

its in

dec

imal

s w

ill he

lp le

arne

rs w

hen

addi

ng a

nd s

ubtra

ctin

g. L

earn

ers

can

use

the

colu

mn

met

hod

as th

ey d

o w

ith w

hole

num

bers

. All

prob

lem

ty

pes

that

are

use

d fo

r who

le n

umbe

rs c

an b

e us

ed fo

r dec

imal

frac

tions

. D

urin

g le

sson

s on

mea

sure

men

t, le

arne

rs c

an p

ract

ise

wha

t th

ey k

now

abo

ut d

ecim

als.

LT

SM

Res

ourc

es

Num

ber l

ines

: Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, fr

actio

n w

alls

, fra

ctio

n ci

rcle

s, d

iffer

ent m

odel

s of

frac

tions

, fra

ctio

n st

rips,

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

ook

refe

renc

e

WB

1 p

p. 9

8 -

108

Ac

tiviti

es 3

4 - 3

9

WB

1 p

p. 1

04 –

114

Ac

tiviti

es 3

4 - 3

9

WB

1 p

p. 2

6 –

40, 4

2 ,4

4 A

ctiv

ities

9a

&b; 1

0a-c

; 11-

13; 1

4, 1

5 pp

. 128

– 1

38 A

ctiv

itite

s 48

, 49;

50

a&b;

51a

&b

DB

E Te

xtbo

ok re

fere

nce

Term

2: U

nit 3

: Com

mon

Fra

ctio

ns, p

. 13

6

Term

2: U

nit 3

: Com

mon

Fra

ctio

ns, p

. 12

9

Term

1: U

nit 4

: Com

mon

Fra

ctio

ns, p

. 55

Te

rm 2

: Uni

t 7: D

ecim

als,

p. 1

77

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

80 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

81MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

6

TER

M 2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

..to

……

.…

Tim

e: 2

HO

UR

S G

RAD

E 4

CAP

S p.

119

5

CAP

S p.

208

6

CAP

S p.

249

TO

PIC

Sy

mm

etry

C

once

pts,

Ski

lls a

nd

know

ledg

e Sy

mm

etry

Rec

ogni

ze, d

raw

and

des

crib

e lin

e(s)

of

sym

met

ry in

2-D

sha

pes.

Sym

met

ry

R

ecog

nize

, dra

w a

nd d

escr

ibe

line(

s)

of s

ymm

etry

in 2

-D s

hape

s.

Sym

met

ry

R

ecog

nize

, dra

w a

nd d

escr

ibe

line(

s) o

f sym

met

ry in

2-D

sha

pes.

SUG

GES

TED

M

ETH

OD

OLO

GY

Gra

de 4

– 6

Th

is s

houl

d in

clud

e sh

apes

in w

hich

ther

e is

mor

e th

an o

ne li

ne o

f sym

met

ry.

Dra

win

gs o

f 2-D

sha

pes

shou

ld in

clud

e th

ose

whe

re th

e lin

e of

sym

met

ry is

not

nec

essa

rily

verti

cal.

LTSM

Res

ourc

es

Varie

ty o

f 2D

sha

pes/

3D

obj

ects

, cha

rts w

ith le

tters

, sym

bols

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

ook

refe

renc

e

W

B 1

pp.

140

– 1

42

Activ

ities

52,

54

WB

1 p

p 15

8 –

160

Ac

tiviti

es 5

8a&b

W

B 1

pp.

104

– 1

06

Activ

ities

38,

39

DB

E Te

xtbo

ok re

fere

nce

Term

2: U

nit 8

: Sym

met

ry, p

. 189

Term

2: U

nit 8

: Sym

met

ry, p

. 181

Term

2: U

nit 5

: Sym

met

ry, p

. 161

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al

asse

ssm

ent –

Tes

t

82 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

7

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 5

HO

UR

S G

RAD

E 4

CAP

S pp

. 73

- 75

5 C

APS

pp. 1

63 -

165

6 C

APS

pp. 2

72 -

274

TOPI

C

Leng

th

Con

cept

s, S

kills

and

kn

owle

dge

Prac

tical

mea

surin

g of

2-d

sh

apes

and

3-d

obj

ects

by:

•

estim

atin

g •

mea

surin

g •

reco

rdin

g •

com

parin

g an

d or

derin

g M

easu

ring

inst

rum

ents

ru

lers

, met

re s

ticks

, tap

e m

easu

res,

trun

dle

whe

els

Uni

ts:

milli

met

res

(mm

), ce

ntim

etre

s (c

m),

met

res

(m),

kilo

met

res

(km

) C

alcu

latio

ns a

nd p

robl

em-

solv

ing

rela

ted

to le

ngth

So

lve

prob

lem

s in

con

text

s re

late

d to

le

ngth

C

onve

rsio

ns in

clud

e co

nver

ting

betw

een

Prac

tical

mea

surin

g of

2-d

sh

apes

and

3-d

obj

ects

by:

•

estim

atin

g •

mea

surin

g •

reco

rdin

g •

com

parin

g an

d or

derin

g M

easu

ring

inst

rum

ents

ru

lers

, met

re s

ticks

, tap

e m

easu

res,

trun

dle

whe

els

Uni

ts:

milli

met

res

(mm

), ce

ntim

etre

s (c

m),

met

res

(m),

kilo

met

res

(km

) C

alcu

latio

ns a

nd p

robl

em-

solv

ing

rela

ted

to le

ngth

So

lve

prob

lem

s in

con

text

rela

ted

to

leng

th

Con

vers

ions

incl

ude

conv

ertin

g be

twee

n an

y of

the

follo

win

g

Prac

tical

mea

surin

g of

2-d

sh

apes

and

3-d

obj

ects

by:

•

estim

atin

g •

mea

surin

g •

reco

rdin

g •

com

parin

g an

d or

derin

g M

easu

ring

inst

rum

ents

ru

lers

, met

re s

ticks

, tap

e m

easu

res,

trun

dle

whe

els

Uni

ts:

milli

met

res

(mm

), ce

ntim

etre

s (c

m),

met

res

(m),

kilo

met

res

(km

) C

alcu

latio

ns a

nd p

robl

em-

solv

ing

rela

ted

to le

ngth

So

lve

prob

lem

s in

con

text

s re

late

d to

le

ngth

C

onve

rsio

ns in

clud

e co

nver

ting

betw

een

any

of th

e fo

llow

ing

units

:

83MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• m

illim

etre

s (m

m),

and

cent

imet

res

(cm

) •

cent

imet

res

(cm

) and

met

res

(m)

• m

etre

s (m

) and

kilo

met

res

(km

) C

onve

rsio

ns a

re li

mite

d to

who

le

num

bers

and

frac

tions

units

:

milli

met

res

(mm

), ce

ntim

etre

s (c

m),

met

res

(m) a

nd

kilo

met

res

(km

) C

onve

rsio

ns li

mite

d to

who

le n

umbe

rs

and

fract

ions

m

illim

etre

s (m

m),

cent

imet

res

(cm

), m

etre

s (m

) and

ki

lom

etre

s (k

m)

Con

vers

ions

sho

uld

incl

ude

fract

ion

and

deci

mal

form

s (to

2 d

ecim

al p

lace

s)

SUG

GES

TED

M

ETH

OD

OLO

GY

All l

earn

ers

can

star

t off

with

the

easi

er p

robl

ems.

Gra

de 4

lear

ners

are

trea

ting

the

wor

k as

new

whi

le th

e G

rade

5 an

d G

rade

6 le

arne

rs c

an m

ove

on to

mor

e co

mpl

ex p

robl

ems

as in

dica

ted

in th

e C

APS

polic

y do

cum

ent.

All g

rade

s:

In G

rade

3 le

arne

rs w

ork

with

non

-sta

ndar

d or

info

rmal

uni

ts w

hen

mea

surin

g. T

hey

are

intro

duce

d to

met

res

and

cent

imet

res.

The

y us

e ru

lers

to m

easu

re in

cen

timet

res

only.

In G

rade

3 le

arne

rs u

se m

etre

stic

ks o

r len

gths

of s

tring

to

mea

sure

in m

etre

s. T

hey

do n

ot le

arn

that

ther

e ar

e 10

0 cm

in 1

m. T

hey

do n

ot d

o co

nver

sion

s be

twee

n un

its. I

n G

rade

4

lear

ners

wor

k w

ith n

ew m

easu

ring

inst

rum

ents

. Milli

met

res

and

kilo

met

res

are

intro

duce

d an

d le

arne

rs d

o co

nver

sion

s

betw

een

units

. Gra

de 4

lear

ners

nee

d to

und

erst

and

and

lear

n th

e re

latio

nshi

p be

twee

n m

etre

s an

d ce

ntim

etre

s,

cent

imet

res

and

milli

met

res,

met

res

and

kilo

met

res.

Rea

ding

inst

rum

ents

for m

easu

ring

leng

ths

Lear

ners

sho

uld

mea

sure

leng

ths

usin

g

• ru

lers

(mm

, cm

)

• m

etre

stic

ks (m

)

• ta

pe m

easu

res

(m, c

m, m

m)

• tru

ndle

whe

els

(m)

84 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Lear

ners

find

rule

rs e

asy

to u

se fo

r mea

surin

g be

caus

e:

• ce

ntim

etre

s ar

e al

way

s nu

mbe

red

• th

ere

are

alw

ays

10m

m d

ivis

ions

in a

cen

timet

re

In G

rade

4 le

arne

rs n

orm

ally

reco

rd th

eir m

easu

rem

ents

with

rule

rs a

s m

illim

etre

s or

cen

timet

res

or m

illim

etre

s an

d

cent

imet

res

e.g.

the

penc

il is

11

cent

imet

res

and

3 m

illim

etre

s lo

ng.

Lear

ners

can

som

etim

es r

ecor

d th

eir

mea

sure

men

ts in

cen

timet

res

and

fract

ions

of

cent

imet

ers

e.g.

the

eras

er is

2 1 2 c

m

long

. Thi

s is

eas

y to

do

beca

use

on a

rule

r, th

e 5t

h m

illim

etre

gra

datio

n lin

e is

nor

mal

ly lo

nger

. Onc

e le

arne

rs h

ave

lear

ned,

from

read

ing

com

mer

cial

mas

s an

d ca

paci

ty p

acka

ging

, tha

t is

the

sam

e as

2,5

, the

y w

ill al

so b

e ab

le to

use

the

deci

mal

5 in

thei

r rec

ordi

ng i.

e. 2

,5 c

m lo

ng.

Che

ck th

at le

arne

rs k

now

to s

tart

mea

surin

g fro

m z

ero,

or t

o su

btra

ct th

e in

itial

mea

sure

men

t fro

m th

e fin

al

mea

sure

men

t. Th

is is

illu

stra

ted

belo

w.

Exa

mpl

e:

Th

e er

aser

is 2

cm +

7m

m o

r 20

mm

+ 7

mm

or 2

7mm

long

85MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

The

eras

er is

(3cm

–1c

m )

+ 7m

m =

2cm

+ 7

mm

or 2

0mm

+ 7

mm

or 2

7mm

long

Onc

e le

arne

rs h

ave

som

e ex

perie

nce

of m

easu

ring

in e

ach

unit,

they

sho

uld

estim

ate

befo

re e

very

mea

sure

men

t. It

is

usef

ul to

hav

e ev

eryd

ay re

fere

nts

as c

ompa

rison

s e.

g. th

e w

idth

of a

doo

r and

hei

ght o

f a w

indo

w a

re o

ften

1 m

, the

wid

th

of a

mat

ch is

ofte

n 1

mm

.

Tape

mea

sure

s th

at a

re lo

nger

than

1 m

and

2 m

sho

uld

also

be

used

e.g

. bui

lder

tape

s or

sur

veyo

r tap

es c

an b

e m

ore

than

10 m

etre

s. T

he lo

nger

mea

surin

g ta

pes

are

mor

e di

fficu

lt to

use

. Lea

rner

s ca

nnot

onl

y re

ad o

ff th

e nu

mbe

r cor

resp

ondi

ng w

ith

the

final

mea

sure

men

t. Th

ey a

lso

need

to k

now

for h

ow m

any

met

res

they

hav

e un

rolle

d th

e ta

pe, e

.g, t

he d

ista

nce

may

be

4

m a

nd 7

8 cm

, but

the

tape

may

onl

y sh

ow th

e nu

mbe

r 78.

Whe

n us

ing

the

long

er m

easu

ring

tape

s, e

stim

atio

n be

com

es e

ven

mor

e im

porta

nt.

Com

pare

and

ord

er le

ngth

s up

to 4

dig

its in

mm

, cm

, m, k

m

In G

rade

s R

to 2

lear

ners

pla

ce o

bjec

ts n

ext t

o ea

ch o

ther

and

dis

cuss

whi

ch is

long

er o

r sho

rter.

In th

e In

term

edia

te P

hase

lear

ners

nee

d to

com

pare

leng

ths

and

heig

hts

whe

n gi

ven

draw

ings

of o

bjec

ts w

ith s

peci

fied

leng

ths,

or w

ritte

n de

scrip

tions

of o

bjec

ts w

ith s

peci

fied

leng

ths.

At f

irst l

earn

ers

can

com

pare

leng

th g

iven

in th

e sa

me

units

, but

onc

e th

ey k

now

how

to

conv

ert b

etw

een

units

, the

y ca

n co

mpa

re le

ngth

s an

d he

ight

s of

obj

ects

whi

ch a

re s

peci

fied

in d

iffer

ent u

nits

.

Cal

cula

tions

(inc

ludi

ng c

onve

rsio

ns) a

nd p

robl

em –

solv

ing

Mea

sure

men

t pro

vide

s a

cont

ext i

n w

hich

to p

ract

ise

skills

acq

uire

d in

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

. The

ski

lls,

oper

atio

ns a

nd n

umbe

r ran

ges

that

lear

ners

hav

e w

orke

d w

ith s

o fa

r in

the

year

, are

giv

en b

elow

:

86 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Estim

ate

and

calc

ulat

e us

ing

mm

, cm

, m, k

m

• ro

undi

ng n

umbe

rs u

p or

dow

n to

the

appr

opria

te u

nit o

f len

gth

• ro

undi

ng o

ff to

10,

100

, 1 0

00

• ad

ditio

n an

d su

btra

ctio

n of

up

to 4

-dig

it nu

mbe

rs

• m

ultip

licat

ion

of 2

-dig

it by

1-d

igit

num

bers

• di

visi

on o

f 2-d

igit

by 1

-dig

it nu

mbe

rs

• ad

d fra

ctio

ns in

mea

sure

men

t con

text

s (u

sing

onl

y ha

lves

, thi

rds,

qua

rters

, fift

hs, s

ixth

s, s

even

ths

and

eigh

ths)

By th

e en

d of

the

year

the

num

ber r

ange

s an

d op

erat

ions

can

be

incr

ease

d to

incl

ude

ever

ythi

ng th

at is

cov

ered

und

er

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

.

Solv

e pr

oble

ms

rela

ting

to d

ista

nce

and

leng

th

Incl

ude

rate

and

ratio

pro

blem

s

Con

vers

ions

bet

wee

n un

its: m

m ↔

cm

; cm

↔ m

; m ↔

km

Con

verti

ng b

etw

een

the

units

of m

easu

rem

ent a

bove

pro

vide

s a

cont

ext f

or p

ract

isin

g m

ultip

lyin

g an

d di

vidi

ng b

y 10

; 100

and

1

000.

Con

vers

ions

sho

uld

be li

mite

d to

who

le n

umbe

rs a

nd fr

actio

ns g

iven

onl

y as

hal

ves,

third

s, q

uarte

rs, f

ifths

, six

ths,

sev

enth

s,

eigh

ths.

In G

rade

4 le

arne

rs d

o no

t cal

cula

te u

sing

dec

imal

s. W

hen

doin

g di

visi

on th

ey s

omet

imes

hav

e a

rem

aind

er e

.g. 3

7 ÷

4 =

9

rem

aind

er 1

. Sim

ilarly

whe

n co

nver

ting

betw

een

units

, the

y m

ay g

ive

thei

r ans

wer

s in

a c

ombi

natio

n of

uni

ts e

.g.

• 35

mm

= 3

cm a

nd 5

mm

or 3

1 2 cm

• 52

6cm

= 5

m a

nd 2

6cm

• 2

500m

= 2

m a

nd 5

00cm

• 4

1 2 km

= 4

500

m

87MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Gra

de 5

and

6:

In G

rade

5 le

arne

rs w

ork

with

the

sam

e un

its o

f len

gth

that

they

wor

ked

with

in G

rade

4. T

hey

also

wor

k w

ith th

e sa

me

mea

surin

g in

stru

men

ts. C

heck

whe

ther

lear

ners

hav

e a

sens

e of

whi

ch u

nits

and

inst

rum

ents

are

app

ropr

iate

for

mea

surin

g va

rious

leng

ths,

hei

ghts

and

dis

tanc

es

Lear

ners

sho

uld

have

a s

ense

of w

hich

uni

ts a

re a

ppro

pria

te fo

r mea

surin

g di

ffere

nt le

ngth

s. F

or e

xam

ple,

they

nee

d to

kno

w

whi

ch u

nits

to u

se to

sta

te

• th

e le

ngth

and

wid

th o

f a d

esk

• th

e di

stan

ce to

the

next

tow

n

• th

e le

ngth

of a

nai

l

Lear

ners

sho

uld

have

a s

ense

of w

hich

inst

rum

ents

are

app

ropr

iate

for m

easu

ring

diffe

rent

leng

ths.

For

exa

mpl

e, th

ey n

eed

to k

now

whi

ch in

stru

men

ts to

use

to m

easu

re:

• th

e le

ngth

and

wid

th o

f a d

esk

• th

e le

ngth

of t

he c

lass

room

• th

e le

ngth

of a

rugb

y fie

ld

Rea

ding

inst

rum

ents

for m

easu

ring

leng

ths

Lear

ners

sho

uld

mea

sure

leng

ths

usin

g

• ru

lers

(mm

, cm

)

• m

etre

stic

ks (m

)

• ta

pe m

easu

res

(m, c

m, m

m)

• tru

ndle

whe

els

(m)

88 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Lear

ners

find

rule

rs e

asy

to u

se fo

r mea

surin

g. T

his

is b

ecau

se

• ce

ntim

etre

s ar

e al

way

s nu

mbe

red

• th

ere

are

alw

ays

10m

m d

ivis

ions

in a

cen

timet

re.

Stat

ing

and

reco

rdin

g le

ngth

mea

sure

men

ts

In G

rade

5 le

arne

rs c

ontin

ue to

reco

rd th

eir m

easu

rem

ents

usi

ng ru

lers

, as

milli

met

res

or c

entim

etre

s or

milli

met

res

and

cent

imet

res

e.g.

the

penc

il is

11

cent

imet

res

and

3 m

illim

etre

s lo

ng. T

hey

can

som

etim

es re

cord

thei

r mea

sure

men

ts in

cent

imet

res

and

fract

ions

of c

entim

etre

s e.

g. th

e er

aser

is 2

1 2 cm

long

. Thi

s is

eas

y to

do

beca

use

on a

rule

r, th

e 5t

h m

illim

etre

grad

atio

n lin

e is

nor

mal

ly lo

nger

. Onc

e le

arne

rs h

ave

lear

ned,

from

read

ing

com

mer

cial

mas

s an

d ca

paci

ty p

acka

ging

, tha

t 2 1 2

is th

e sa

me

as 2

,5 ,

they

will

also

be

able

to u

se th

e de

cim

al ',

5' in

thei

r rec

ordi

ng i.

e. 2

,5cm

long

.

Tape

mea

sure

s th

at a

re lo

nger

than

1m

and

2m

sho

uld

also

be

used

e.g

. bui

lder

tape

s or

sur

veyo

r tap

es c

an b

e m

ore

than

10

met

res.

The

long

er m

easu

ring

tape

s ar

e m

ore

diffi

cult

to u

se. L

earn

ers

cann

ot o

nly

read

off

the

num

ber c

orre

spon

ding

with

the

final

mea

sure

men

t. Th

ey a

lso

need

to k

now

for h

ow m

any

met

res

they

hav

e un

rolle

d th

e ta

pe, e

.g, t

he d

ista

nce

may

be

4m a

nd 7

8cm

, but

the

tape

may

onl

y sh

ow th

e nu

mbe

r 78.

Whe

n us

ing

the

long

er lo

nger

mea

surin

g ta

pes,

est

imat

ion

beco

mes

eve

n m

ore

impo

rtant

.

Com

pare

and

ord

er le

ngth

s up

to 6

dig

its in

mm

, cm

, m, k

m

In th

e In

term

edia

te P

hase

lear

ners

nee

d to

wor

k w

ith d

raw

ings

of o

bjec

ts w

ith s

peci

fied

leng

ths,

or w

ritte

n de

scrip

tions

of

obje

cts

with

spe

cifie

d le

ngth

s. A

t firs

t lea

rner

s ca

n co

mpa

re le

ngth

giv

en in

the

sam

e un

its, b

ut o

nce

they

kno

w h

ow to

con

vert

betw

een

units

, the

y ca

n co

mpa

re le

ngth

s an

d he

ight

s of

obj

ects

whi

ch a

re s

peci

fied

in d

iffer

ent u

nits

89MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Cal

cula

tions

(inc

ludi

ng c

onve

rsio

ns) a

nd p

robl

em-s

olvi

ng

Mea

sure

men

t pro

vide

s a

cont

ext i

n w

hich

to p

ract

ise

skills

acq

uire

d in

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

. The

ski

lls,

oper

atio

ns a

nd n

umbe

r ran

ges

requ

ired

are

give

n be

low.

Estim

ate

and

calc

ulat

e us

ing

• R

ound

num

bers

up

or d

own

to th

e ap

prop

riate

uni

t of l

engt

h

• R

ound

ing

off t

o 5,

10,

100

and

1 00

0

• Ad

ditio

n an

d su

btra

ctio

n up

to 5

-dig

it nu

mbe

rs

• M

ultip

licat

ion:

3-d

igit

num

ber b

y 2-

digi

t num

ber

• D

ivis

ion:

3-d

igit

num

ber b

y 2-

digi

t num

ber

• Ad

d co

mm

on fr

actio

ns in

the

cont

ext o

f mea

sure

men

t (us

ing

only

hal

ves,

third

s, q

uarte

rs, f

ifths

, six

ths,

sev

enth

s an

d

eigh

ths)

By th

e en

d of

the

year

the

num

ber r

ange

s an

d op

erat

ions

can

be

incr

ease

d to

incl

ude

ever

ythi

ng th

at is

cov

ered

und

er

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

.

Solv

e pr

oble

ms

rela

ting

to d

ista

nce

and

leng

th in

clud

ing

rate

and

ratio

pro

blem

s.

Con

vers

ions

bet

wee

n un

its: m

m

↔

cm

; cm

↔

m;

m ↔

km

Con

verti

ng b

etw

een

the

units

of m

easu

rem

ent a

bove

pro

vide

s a

cont

ext f

or p

ract

isin

g m

ultip

licat

ion

and

divi

sion

by

10, 1

00, 1

000

Con

vers

ions

sho

uld

be li

mite

d to

who

le n

umbe

rs a

nd fr

actio

ns g

iven

onl

y as

hal

ves

/ thi

rds

/ qua

rters

/ fif

ths

/ six

ths

/ sev

enth

s /

eigh

ths.

90 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

In G

rade

5 le

arne

rs d

o no

t cal

cula

te u

sing

dec

imal

s. W

hen

doin

g di

visi

on th

ere

will

som

etim

es b

e a

rem

aind

er in

the

answ

er,

e.g.

37

÷ 4

= 9

rem

aind

er 1

. Sim

ilarly

whe

n co

nver

ting

betw

een

units

, ans

wer

s m

ay b

e st

ated

in a

com

bina

tion

of u

nits

e.g

.

• 35

cm =

3cm

and

5m

m o

r 3 1 2

cm

• 52

6cm

= 5

m a

nd 2

6cm

• 2

500m

= 2

m a

nd 5

00cm

• 4

1 2 km

= 4

500

m

Gra

de 6

: D

ecim

als

are

intr

oduc

ed.

This

allo

ws

lear

ners

to e

xpre

ss c

onve

rsio

ns a

nd p

arts

of m

easu

res

in d

ecim

al fr

actio

n fo

rm to

one

or t

wo

deci

mal

pla

ces.

Use

the

cont

exts

of l

engt

h m

easu

rem

ent t

o pr

actis

e th

e re

adin

g, w

ritin

g an

d un

ders

tand

ing

of d

ecim

al fr

actio

ns, a

nd fo

r

roun

ding

off,

con

verti

ng, a

ddin

g an

d su

btra

ctin

g w

ith d

ecim

al fr

actio

ns.

Rea

ding

inst

rum

ents

for m

easu

ring

leng

ths

Lear

ners

sho

uld

mea

sure

leng

ths

usin

g

• ru

lers

(mm

, cm

)

• m

etre

stic

ks (m

)

• ta

pe m

easu

res

(m, c

m, m

m)

• tru

ndle

whe

els

(in m

)

Lear

ners

find

rule

rs e

asy

to u

se fo

r mea

surin

g be

caus

e

• ce

ntim

etre

s ar

e al

way

s nu

mbe

red

• th

ere

are

alw

ays

10m

m d

ivis

ions

in a

cen

timet

er

•

91MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Stat

ing

and

reco

rdin

g le

ngth

mea

sure

men

ts

In G

rade

6 le

arne

rs s

houl

d be

giv

en o

ppor

tuni

ties

to re

cord

thei

r mea

sure

men

ts u

sing

rule

rs, i

n de

cim

al fr

actio

n fro

m e

.g.

e.g.

the

eras

er is

2,5

cm lo

ng.

Tape

mea

sure

s th

at a

re lo

nger

than

1m

and

2m

sho

uld

also

be

used

e.g

. bui

lder

tape

s or

sur

veyo

r tap

es c

an b

e m

ore

than

10 m

etre

s. T

he lo

nger

mea

surin

g ta

pes

are

mor

e di

fficu

lt to

use

. Lea

rner

s ca

n’t o

nly

read

off

the

num

ber a

t the

end

of t

he

dist

ance

. The

y al

so n

eed

to k

now

how

man

y m

etre

s th

ey h

ave

unro

lled

the

tape

. For

exa

mpl

e, th

e di

stan

ce m

ay b

e 4m

and

78cm

, but

at t

he e

nd o

f the

obj

ect /

dis

tanc

e th

e ta

pe m

ay o

nly

show

the

num

ber 7

8. W

ith th

ese

long

er ta

pe m

easu

res

estim

atio

n be

com

es e

ven

mor

e im

porta

nt. R

ecor

ding

this

in o

ne u

nit o

f mea

sure

men

t can

als

o be

com

e m

ore

com

plex

: in

this

exa

mpl

e 4,

78m

or 4

78cm

. But

if th

e m

easu

rem

ent i

s 4m

and

7cm

, lea

rner

s ne

ed to

rem

embe

r to

conv

ert c

orre

ctly

into

4,07

m o

r 407

cm

Com

pare

and

ord

er le

ngth

s up

to 9

dig

its in

mm

, cm

, m, k

m

In th

e In

term

edia

te P

hase

lear

ners

nee

d to

wor

k w

ith d

raw

ings

of o

bjec

ts w

ith s

peci

fied

leng

ths,

or w

ritte

n de

scrip

tions

of

obje

cts

with

spe

cifie

d le

ngth

s. In

Gra

de 6

the

focu

s is

on

com

parin

g le

ngth

s gi

ven

in d

ecim

al fo

rm

Cal

cula

tions

(inc

ludi

ng c

onve

rsio

ns) a

nd p

robl

em-s

olvi

ng

Mea

sure

men

t pro

vide

s a

cont

ext i

n w

hich

to p

ract

ise

skills

acq

uire

d in

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

. The

ski

lls,

oper

atio

ns a

nd n

umbe

r ran

ges

requ

ired

are

give

n be

low.

Estim

ate

and

calc

ulat

e us

ing

mm

, cm

, m, k

m

• ro

und

num

bers

up

or d

own

to th

e ap

prop

riate

uni

t of l

engt

h

• ro

undi

ng o

ff to

5, 1

0, 1

00, 1

000

(rea

ding

mea

sure

men

ts fr

om ru

lers

and

tape

mea

sure

s ca

n he

lp le

arne

rs to

und

erst

and

the

mea

ning

beh

ind

roun

ding

up

or d

own)

92 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• ad

ditio

n an

d su

btra

ctio

n ca

lcul

atio

ns c

an in

clud

e ca

lcul

atio

ns w

ith c

omm

on fr

actio

ns a

nd d

ecim

al fr

actio

ns to

2 d

ecim

al

plac

es

• m

ultip

licat

ion

of 4

-dig

it by

3-d

igit

num

bers

• di

visi

on o

f 4-d

igit

by 3

-dig

it nu

mbe

rs

• fin

d pe

rcen

tage

s of

who

le n

umbe

rs

• m

ultip

le o

pera

tions

with

or w

ithou

t bra

cket

s

Solv

e pr

oble

ms

rela

ting

to d

ista

nce

and

leng

th

• In

clud

e ra

te a

nd ra

tio p

robl

ems.

• Pr

oble

ms

with

dec

imal

s sh

ould

be

limite

d to

add

ing

and

subt

ract

ing

the

num

bers

.

Con

vers

ions

bet

wee

n un

its

• m

m ↔

cm

• cm

↔ m

• m

↔ k

m

• m

m ↔

m

• m

m ↔

km

• cm

↔ k

m ,

usin

g w

hole

num

bers

, com

mon

frac

tions

and

dec

imal

frac

tions

. Thi

s pr

ovid

es a

con

text

for l

earn

ers

to p

ract

ise

mul

tiply

ing

and

divi

ding

by

10, 1

00 a

nd 1

000

.

If co

nver

sion

s re

quire

mor

e th

an 2

dec

imal

pla

ces

e.g.

3 2

45m

con

verte

d to

kilo

met

res,

lear

ners

can

con

tinue

to w

rite

this

as

3km

and

245

m a

s th

ey h

ave

in p

revi

ous

grad

es. O

n th

e w

hole

thou

gh e

xam

ples

sho

uld

be c

hose

n to

avo

id th

is p

robl

em.

LTSM

Res

ourc

es

Mea

surin

g ta

pe, r

uler

s, s

trin

g, tr

undl

e w

heel

G

rade

4

Gra

de 5

G

rade

6

93MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Wor

book

refe

renc

e

WB

1 p

p. 1

10 –

114

Ac

tiviti

es 4

0; 4

1; 4

2

WB

1 p

p. 1

16 –

126

Ac

tiviti

es

40; 4

1a&b

; 42a

&b; 4

3

WB

1 p

p. 9

6 –

106

Activ

ities

10

0a&b

, 101

- 10

4

DB

E Te

xtbo

ok re

fere

nce

Term

2: U

nit 4

: Len

gth,

p. 1

47

Term

2: U

nit 4

: Len

gth,

p. 1

43

Term

3: U

nit 1

1: L

engt

h, p

. 285

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– te

st

94 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

8

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 5

HO

UR

S G

RAD

E 4

CAP

S pp

. 108

- 11

0 5

CAP

S pp

. 178

- 18

0 6

CAP

S pp

. 259

- 26

1 TO

PIC

M

ass

Con

cept

s, S

kills

and

kn

owle

dge

Prac

tical

mea

surin

g of

3-D

ob

ject

s by

•

estim

atin

g •

mea

surin

g, re

cord

ing

• co

mpa

ring

and

orde

ring

• 3-

D o

bjec

ts u

sing

mas

s M

easu

ring

inst

rum

ents

ba

thro

om s

cale

s, k

itche

n sc

ales

an

d ba

lanc

es

Uni

ts

gra

ms

(g) a

nd k

ilogr

ams

(kg)

Cal

cula

tions

and

pro

blem

-so

lvin

g w

ith m

ass

incl

ude

Solv

e pr

oble

ms

in c

onte

xts

with

mas

s co

nver

ting

betw

een

gram

s an

d ki

logr

ams

limite

d to

exa

mpl

es w

ith w

hole

num

bers

an

d fra

ctio

ns

Prac

tical

mea

surin

g of

3-D

ob

ject

s by

•

estim

atin

g •

mea

surin

g, re

cord

ing

• co

mpa

ring

and

orde

ring

Mea

surin

g in

stru

men

ts

bath

room

sca

les,

kitc

hen

scal

es

and

bala

nces

U

nits

gr

ams

(g) a

nd k

ilogr

ams

(kg)

; C

alcu

latio

ns a

nd p

robl

em-

solv

ing

rela

ted

to m

ass

incl

ude

Solv

e pr

oble

ms

in c

onte

xts

rela

ted

to

mas

s co

nver

ting

betw

een

gram

s an

d ki

logr

ams

limite

d to

exa

mpl

es w

ith

who

le n

umbe

rs a

nd fr

actio

n

Prac

tical

mea

surin

g of

3-D

ob

ject

s by

•

estim

atin

g •

mea

surin

g, re

cord

ing

• co

mpa

ring

and

orde

ring

Mea

surin

g in

stru

men

ts

bath

room

sca

les

(ana

logu

e an

d di

gita

l), k

itche

n sc

ales

(ana

logu

e an

d di

gita

l) an

d ba

lanc

es

Uni

ts

gram

s (g

) and

kilo

gram

s (k

g)

Cal

cula

tions

and

pro

blem

-so

lvin

g re

late

d to

mas

s in

clud

e So

lve

prob

lem

s in

con

text

usi

ng m

ass

conv

ertin

g be

twee

n gr

ams

and

kilo

gram

s co

nver

sion

s sh

ould

incl

ude

fract

ion

and

deci

mal

form

s (to

2 d

ecim

al p

lace

s).

95MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

All l

earn

ers

can

star

t off

with

the

easi

er p

robl

ems.

Gra

de 4

lear

ners

are

trea

ting

the

wor

k as

new

whi

le th

e G

rade

5

and

Gra

de 6

lear

ners

can

mov

e on

to m

ore

com

plex

pro

blem

s as

indi

cate

d in

the

CAP

S po

licy

docu

men

t. Al

l gra

des:

In

Gra

de 3

lear

ners

wor

k w

ith n

on-s

tand

ard

or in

form

al u

nits

whe

n m

easu

ring

mas

s. T

hey

also

wor

k w

ith k

ilogr

ams

and

gram

s. T

hey

read

bat

hroo

m s

cale

s bu

t onl

y re

ad th

e m

ass

at th

e nu

mbe

red

calib

ratio

n lin

es. T

hey

do n

ot le

arn

that

ther

e ar

e 1

000g

in 1

kg. T

hey

do n

ot c

onve

rt be

twee

n un

its. T

he G

rade

4 le

arne

rs m

ust l

earn

the

rela

tions

hip

betw

een

the

two

units

. G

rade

4 le

arne

rs n

eed

to:

co

nsol

idat

e th

eir s

ense

of h

ow m

uch

1kg

is

fu

rther

dev

elop

a s

ense

of h

ow m

uch

1g is

unde

rsta

nd a

nd k

now

the

rela

tions

hip

betw

een

gram

s an

d ki

logr

am

co

nver

t bet

wee

n gr

ams

and

kilo

gram

s

read

mea

sure

men

ts o

n sc

ales

indi

cate

d on

bot

h nu

mbe

red

and

unnu

mbe

red

calib

ratio

n lin

es.

Rea

ding

inst

rum

ents

and

mea

surin

g m

ass

Lear

ners

nee

d to

: •

estim

ate

mas

s in

gra

ms

and

kilo

gram

s •

read

the

mas

ses

stip

ulat

ed o

n pa

ckag

ing

• re

ad th

e m

ass

on p

ictu

res

of k

itche

n sc

ales

(in

g &

kg) a

nd b

athr

oom

sca

les

(in k

g) a

nd b

alan

ce s

cale

s (in

g &

kg)

•

read

the

mas

s on

real

kitc

hen

scal

es in

(g &

kg)

and

bat

hroo

m s

cale

s (in

kg)

and

bal

ance

sca

les

(in g

& k

g).

Rea

ding

the

mas

s on

kitc

hen

and

bath

room

sca

les

invo

lves

: •

know

ing

whe

re to

sta

nd to

read

the

scal

e co

rrect

ly

• kn

owin

g ho

w to

rea

d th

e nu

mbe

red

grad

atio

n lin

es a

nd to

cal

cula

te w

hat t

he u

n-nu

mbe

red

grad

atio

n lin

es m

ean.

Lea

rner

s ne

ed to

read

:

◊ di

ffere

nt k

inds

of m

ass

met

ers

◊ m

ass

met

ers

on w

hich

the

num

bere

d in

terv

als/

gra

datio

n lin

es /

calib

ratio

n re

pres

ent d

iffer

ent i

nter

vals

/mas

ses

◊ ap

para

tus

whi

ch h

ave

diffe

rent

num

bers

of u

n-nu

mbe

red

inte

rval

s w

ithin

eac

h nu

mbe

red

inte

rval

.

96 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Lea

rner

s ne

ed to

pra

ctic

e w

ith e

xam

ples

in w

hich

the

num

bere

d in

terv

als

are

divi

ded

into

:

- 2

un-n

umbe

red

inte

rval

s

- 4

un-n

umbe

red

inte

rval

s

- 5

un-n

umbe

red

inte

rval

s

- 10

un-

num

bere

d in

terv

als

Exa

mpl

e:

Her

e th

e nu

mbe

red

lines

sho

w 1

00g

inte

rval

s: 1

00g,

200

g, 3

00g,

400

g, 5

00g,

600

g, 7

00g,

800

g, 9

00g,

1 0

00g

It

is s

omet

imes

use

ful t

o co

nver

t the

circ

ular

dia

l int

o a

num

ber l

ine

for l

earn

ers

600

g

70

0 g

Ther

e ar

e 10

spa

ces

betw

een

each

100

g.

Each

100

g in

terv

al h

as b

een

divi

ded

into

10

smal

ler s

pace

s.

This

mea

ns th

at e

ach

un-n

umbe

red

inte

rval

sho

ws

100g

÷ 1

0 =

10g

97MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Com

pare

mas

ses

with

up

to 4

dig

its in

gra

ms

and

kilo

gram

s

Lear

ners

sho

uld

sequ

ence

con

tain

ers

mar

ked

in g

ram

s an

d/ki

logr

ams.

Her

e le

arne

rs n

eed

to b

e ab

le to

tran

slat

e th

e de

cim

al

num

bers

on

som

e pa

ckag

ing

into

frac

tions

e.g

. 2,5

kg o

f flo

ur is

the

sam

e as

2 1 2

kg o

f flo

ur. O

ne s

houl

d al

so c

hoos

e ex

ampl

es

that

allo

w le

arne

rs to

real

ize

that

the

size

of a

con

tain

er o

r the

vol

ume

it co

ntai

ns is

not

dire

ctly

pro

porti

onal

to th

e m

ass:

som

e su

bsta

nces

hav

e a

grea

ter d

ensi

ty th

an o

ther

s.

Cal

cula

tions

(inc

ludi

ng c

onve

rsio

ns) a

nd p

robl

em-s

olvi

ng

Mea

sure

men

t pro

vide

s a

cont

ext i

n w

hich

to p

ract

ice

skills

acq

uire

d in

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

. The

ski

lls,

op

erat

ions

and

num

ber r

ange

s re

quire

d ar

e gi

ven

belo

w.

Cal

cula

te a

nd e

stim

ate

(usi

ng g

ram

s an

d ki

logr

ams)

•

roun

d nu

mbe

rs u

p or

dow

n to

the

appr

opria

te u

nit o

f mas

s •

roun

ding

to 1

0, 1

00, 1

000

•

addi

tion

and

subt

ract

ion

of u

p to

4-d

igit

num

bers

•

mul

tiplic

atio

n 2-

digi

t by

2-di

git n

umbe

rs

• di

visi

on: 3

-dig

it by

1-d

igit

num

bers

•

add

fract

ions

in c

onte

xt (u

sing

onl

y ha

lves

, thi

rds,

qua

rters

, fift

hs, s

ixth

s, s

even

ths

and

eigh

ths)

So

lve

prob

lem

s re

latin

g to

mas

s:

• in

clud

e ra

te e

spec

ially

rand

s pe

r kilo

gram

s an

d ra

tio p

robl

ems

e.g.

incr

easi

ng o

r dec

reas

ing

the

mas

s of

ingr

edie

nts

in a

re

cipe

by

a se

t rat

io

• w

rite

num

ber s

ente

nces

to d

escr

ibe

prob

lem

s C

onve

rt b

etw

een

units

: g ↔

kg

Con

verti

ng b

etw

een

the

units

of m

easu

rem

ent a

bove

pro

vide

s a

cont

ext f

or p

ract

isin

g m

ultip

lyin

g an

d di

vidi

ng b

y

1

00

0.

Con

vers

ions

sho

uld

be li

mite

d to

who

le n

umbe

rs a

nd fr

actio

ns g

iven

onl

y as

hal

ves

/ thi

rds

/ qua

rters

/ fif

ths

/ six

ths

/ se

vent

hs /

eigh

ths.

Con

vers

ions

can

als

o in

clud

e co

nver

ting

the

deci

mal

hal

f to

the

com

mon

frac

tion

form

of h

alf.

Whe

n le

arne

rs d

o di

visi

on in

Gra

de 4

the

answ

ers

may

hav

e re

mai

nder

s e.

g.11

5 ÷

25 =

4 re

mai

nder

15.

Sim

ilarly

whe

n co

nver

ting

gram

s to

kilo

gram

s, le

arne

rs m

ay g

et p

art o

f the

ans

wer

in k

ilogr

ams

and

stat

e th

e re

mai

ning

par

t in

gram

s e.

g.

4 25

0g =

4kg

and

250

g

98 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Rec

ordi

ng m

asse

s Be

caus

e le

arne

rs w

ill on

ly w

ork

with

dec

imal

frac

tions

in G

rade

6, t

hey

shou

ld re

cord

mas

ses

in:

• ki

logr

ams

only

e.g

. 5kg

•

gram

s on

ly e

.g. 2

50g

Sinc

e le

arne

rs w

ill be

read

ing

half

kilo

gram

s in

dec

imal

form

on

som

e pa

ckag

ing,

they

can

als

o w

rite

half

kilo

gram

s in

the

deci

mal

form

. How

ever

this

is n

ot a

requ

irem

ent i

n th

is g

rade

. G

rade

5:

Let t

he G

rade

4, 5

and

6 le

arne

rs d

o th

e sa

me

wor

k w

ith th

e di

ffere

nces

as

indi

cate

d be

low

. R

eadi

ng in

stru

men

ts a

nd m

easu

ring

mas

s Le

arne

rs n

eed

to:

• es

timat

e m

ass

in g

ram

s an

d ki

logr

ams,

incl

udin

g be

ing

able

to m

atch

obj

ects

to th

e ap

prop

riate

uni

t of m

easu

rem

ent

befo

re m

easu

ring

them

. •

choo

se, w

ith re

ason

s, th

e m

ost a

ppro

pria

te s

cale

to u

se fo

r par

ticul

ar o

bjec

ts fr

om a

rang

e of

sca

les

prov

ided

E

xam

ple

Her

e th

e nu

mbe

red

lines

sho

w 1

00g

inte

rval

s: 1

00g;

200

g; 3

00g;

400

g; 5

00g;

600

g; 7

00g;

It

is s

omet

imes

use

ful t

o co

nver

t a c

ircul

ar d

ial i

nto

a nu

mbe

r lin

e fo

r lea

rner

s

100

2

00

300

4

00

500

6

00

700

Th

ere

are

10 s

pace

s be

twee

n ea

ch 1

00g.

Ea

ch 1

00g

inte

rval

has

bee

n di

vide

d in

to 1

0 sm

alle

r spa

ces.

Th

is m

eans

that

eac

h un

-num

bere

d in

terv

al s

how

s 10

0g ÷

10

= 10

g C

ompa

re m

asse

s w

ith u

p to

6-d

igits

in g

ram

s an

d ki

logr

ams.

99MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Gra

de 6

: It

mak

es s

ense

to le

t lea

rner

s w

ork

with

dig

ital s

cale

s, p

artic

ular

ly o

nes

that

giv

e re

adin

gs u

p to

one

or t

wo

deci

mal

pla

ces.

Pr

oble

ms,

cal

cula

tions

and

con

vers

ions

aro

und

mas

s pr

ovid

e a

cont

ext f

or p

ract

isin

g ca

lcul

atin

g w

ith d

ecim

al fr

actio

ns.

Supe

rmar

kets

with

ele

ctro

nic

scal

es o

ften

prin

t the

mas

s la

bels

incl

udin

g de

cim

al p

lace

s e.

g. 2

,25k

g po

tato

es. T

hese

co

ntex

ts c

an b

e us

ed to

pra

ctis

e th

e re

adin

g, w

ritin

g an

d un

ders

tand

ing

of d

ecim

al fr

actio

ns, a

nd fo

r rou

ndin

g of

f, co

nver

ting,

ad

ding

and

sub

tract

ing

deci

mal

frac

tions

. In

Gra

de 6

lear

ners

wor

k w

ith th

e sa

me

units

of m

ass

they

wor

ked

with

in G

rade

s 4

and

5. T

hey

also

wor

k w

ith th

e sa

me

mea

surin

g in

stru

men

ts.

Lear

ners

nee

d to

: •

cons

olid

ate

thei

r sen

se o

f how

muc

h is

1kg

•

cons

olid

ate

thei

r sen

se o

f how

muc

h is

1g

• to

und

erst

and

and

know

the

rela

tions

hip

betw

een

kilo

gram

s an

d gr

ams.

Le

arne

rs s

houl

d ha

ve a

sen

se o

f whi

ch u

nits

are

app

ropr

iate

for m

easu

ring

whi

ch d

iffer

ent m

asse

s. F

or e

xam

ple,

they

nee

d to

kno

w w

hich

uni

ts to

use

to s

tate

the

mas

s of

: •

a co

w

• a

baby

•

flour

for b

akin

g a

cake

•

thei

r ow

n m

ass

Rea

ding

sca

les

and

bala

nces

Le

arne

rs n

eed

to:

• es

timat

e m

ass

in g

ram

s an

d ki

logr

ams

• re

ad k

itche

n sc

ales

(gra

ms

and

kilo

gram

s) b

athr

oom

sca

les

(kilo

gram

s) a

nd b

alan

ces

scal

es (g

ram

s an

d ki

logr

ams)

Th

is in

clud

es re

adin

g th

e m

ass

on:

• re

al d

igita

l sca

les

• pi

ctur

es o

f dec

imal

sca

les

• re

al a

nalo

gue

scal

es

• pi

ctur

es o

f ana

logu

e sc

ales

100 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

The

skills

invo

lved

in re

adin

g an

alog

ue s

cale

s in

clud

e:

• kn

owin

g w

here

to s

tand

to re

ad th

e sc

ales

cor

rect

ly

• K

now

ing

how

to re

ad th

e nu

mbe

red

grad

atio

n lin

es a

nd to

cal

cula

te w

hat t

he u

n-nu

mbe

red

grad

atio

n lin

es in

dica

te.

Lear

ners

nee

d to

read

: •

diffe

rent

kin

ds o

f mea

surin

g ap

para

tus

• ap

para

tus

in w

hich

the

num

bere

d in

terv

als,

gra

datio

n lin

es o

r cal

ibra

tion

repr

esen

t diff

eren

t int

erva

ls

• A

ppar

atus

in w

hich

ther

e ar

e a

diffe

rent

num

ber o

f un-

num

bere

d in

terv

als

with

in e

ach

num

bere

d in

terv

al. L

earn

ers

need

pr

actic

e w

ith e

xam

ples

in w

hich

the

num

bere

d in

terv

als

are

divi

ded

into

:

- 2

un-n

umbe

red

inte

rval

s -

4 un

-num

bere

d in

terv

als

- 5

un-n

umbe

red

inte

rval

s -

10 u

n-nu

mbe

red

inte

rval

s

Exa

mpl

e:

H

ere

the

num

bere

d lin

es s

how

100

g in

terv

als:

100

g; 2

00g;

300

g; 4

00g;

500

g; 6

00g;

700

g.

It is

som

etim

es u

sefu

l to

conv

ert t

he c

ircul

ar d

ial i

nto

a nu

mbe

r lin

e Th

ere

are

10 s

pace

s be

twee

n ea

ch 1

00g.

Eac

h 10

0g in

terv

al h

as b

een

divi

ded

into

10

smal

ler s

pace

s. T

his

mea

ns th

at e

ach

un-n

umbe

red

inte

rval

sho

ws

100g

÷ 1

0 =

10g

101MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Com

pare

, ord

er, s

eque

nce

mas

ses

of u

p to

9 d

igits

in g

ram

s an

d ki

logr

ams

If le

arne

rs h

ave

not i

n pr

evio

us g

rade

s se

quen

ced

cont

aine

rs m

arke

d in

gra

ms

and

kilo

gram

s, it

is w

orth

doi

ng. C

hoos

e ex

ampl

es th

at a

llow

lear

ners

to re

aliz

e th

at th

e si

ze o

f a c

onta

iner

or t

he v

olum

e it

cont

ains

is n

ot d

irect

ly p

ropo

rtion

al to

the

mas

s be

caus

e so

me

subs

tanc

es h

ave

a gr

eate

r den

sity

than

oth

ers.

Lea

rner

s sh

ould

do

exer

cise

s fro

m th

eir t

extb

ook

that

as

k th

em to

ord

er a

nd c

ompa

re th

e m

ass

of o

bjec

ts in

clud

ing

groc

ery

item

s la

belle

d in

gra

ms

and

kilo

gram

s.

Lear

ners

sho

uld

also

com

pare

, ord

er, s

eque

nce

mas

ses

stat

ed in

diff

eren

t uni

ts.

Cal

cula

tions

(inc

ludi

ng c

onve

rsio

ns) a

nd p

robl

em-s

olvi

ng

Mea

sure

men

t pro

vide

s a

cont

ext i

n w

hich

to p

ract

ise

skills

acq

uire

d in

Num

bers

, Ope

ratio

ns a

nd R

elat

ions

hips

. The

ski

lls,

oper

atio

ns a

nd n

umbe

r ran

ges

usin

g gr

ams

and

kilo

gram

s re

quire

d ar

e gi

ven

belo

w.

• R

ound

ing

num

bers

up

or d

own

to th

e m

ost a

ppro

pria

te u

nit o

f mas

s

• R

ound

ing

off t

o 5,

10,

100

and

1 0

00 M

easu

rem

ent e

spec

ially

whe

n fo

cusi

ng o

n re

adin

g an

alog

ue m

easu

ring

inst

rum

ents

can

help

lear

ners

to u

nder

stan

d th

e m

eani

ng b

ehin

d ro

undi

ng u

p or

dow

n

• Ad

ditio

n an

d su

btra

ctio

n C

alcu

latio

ns a

nd p

robl

ems

shou

ld in

clud

e fra

ctio

nal p

arts

of k

ilogr

ams

expr

esse

d ei

ther

as

com

mon

frac

tions

or d

ecim

al fr

actio

ns- u

p to

2 d

ecim

al p

lace

s

• M

ultip

licat

ion

of u

p to

4-d

igit

by 3

-dig

it w

hole

num

bers

• D

ivis

ion

of u

p to

4-d

igit

by 3

-dig

it w

hole

num

bers

• Fi

nd p

erce

ntag

es o

f who

le n

umbe

rs

• m

ultip

le o

pera

tions

with

or w

ithou

t bra

cket

s

Solv

e pr

oble

ms

rela

ting

to m

ass

incl

udin

g:

• ra

te e

.g. p

rice

per k

ilogr

am a

nd ra

tio p

robl

ems

• pr

oble

ms

with

dec

imal

s sh

ould

be

limite

d to

add

ition

and

sub

tract

ion

C

onve

rt b

etw

een

units

: g↔

kg

Con

vers

ions

sho

uld

be g

iven

in th

e fo

llow

ing

form

s: w

hole

num

bers

, com

mon

frac

tions

, dec

imal

frac

tions

up

to 2

dec

imal

pl

aces

Thi

s pr

ovid

es a

con

text

for l

earn

ers

to p

ract

ise

mul

tiply

ing

and

divi

ding

by

1 00

0.

102 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

If co

nver

sion

s re

quire

mor

e th

an 2

dec

imal

pla

ces

e.g.

3 2

45 g

ram

s co

nver

ted

to k

ilogr

ams

lear

ners

can

con

tinue

to w

rite

this

as

3kg

and

245

g as

in p

revi

ous

grad

es. O

n th

e w

hole

thou

gh e

xam

ples

sho

uld

be c

hose

n to

avo

id th

is p

robl

em.

LT

SM

Res

ourc

es

Kitc

hen

scal

e, b

athr

oom

sca

le, b

alan

ce

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 2

pp.

112

– 1

22

Activ

ities

11

1, 1

12, 1

13a&

b, 1

14, 1

15)

WB

1 p

p. 2

0 –

26

Activ

ities

74a

&b, 7

5, 7

6

WB

2 p

p. 2

– 9

Act

iviti

es 6

5, 6

6a&b

, 67

DB

E Te

xtbo

ok re

fere

nce

Term

4: U

nit 3

: Mas

s, p

. 297

Term

3: U

nit 2

: Mas

s, p

. 211

Term

3: U

nit 1

: Mas

s, p

. 215

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

103MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

9

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 5

HO

UR

S G

RAD

E 4

CAP

S pp

. 88

- 90

5 C

APS

pp. 1

50 -

153

6 C

APS

pp. 2

53 -

256

TOPI

C

Cap

acity

and

Vol

ume

Con

cept

s, S

kills

and

kn

owle

dge

Prac

tical

mea

surin

g of

3-D

ob

ject

s by

• es

timat

ing

•

mea

surin

g

• re

cord

ing

•

com

parin

g an

d or

derin

g M

easu

ring

inst

rum

ents

m

easu

ring

spoo

n, m

easu

ring

cups

, mea

surin

g ju

gs

Uni

ts

milli

litre

(ml)

, litr

es (l

) C

alcu

latio

ns a

nd p

robl

em-s

olvi

ng

rela

ted

to c

apac

ity/v

olum

e in

clud

e:

Prac

tical

mea

surin

g of

3-D

obj

ects

by

• es

timat

ing

•

mea

surin

g

• re

cord

ing

•

com

parin

g an

d or

derin

g

• Mea

surin

g in

stru

men

ts

mea

surin

g sp

oons

, mea

surin

g cu

ps,

mea

surin

g ju

gs

Uni

ts

milli

litre

(ml)

litre

(l)

Cal

cula

tions

and

pro

blem

-sol

ving

re

late

d to

cap

acity

/ vo

lum

e in

clud

e

Prac

tical

mea

surin

g of

3-D

ob

ject

s by

• es

timat

ing

•

mea

surin

g

• re

cord

ing

•

com

parin

g an

d or

derin

g M

easu

ring

inst

rum

ents

m

easu

ring

jugs

U

nits

m

illilit

re (m

l); li

tres

(l) a

nd k

ilolit

res

(kl)

Cal

cula

tions

and

pro

blem

-sol

ving

re

late

d to

cap

acity

/vol

ume

incl

ude:

104 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• So

lve

prob

lem

s in

con

text

s us

ing

capa

city

• C

onve

rt be

twee

n lit

res

and

milli

litre

s, li

mite

d to

exa

mpl

es o

f w

hole

num

bers

and

frac

tions

• so

lvin

g pr

oble

ms

in c

onte

xts

usin

g ca

paci

ty/v

olum

e

• co

nver

ting

betw

een

litre

s an

d m

illilit

res

limite

d to

exa

mpl

es w

ith

who

le n

umbe

rs a

nd fr

actio

ns

• so

lvin

g pr

oble

ms

in c

onte

xt w

ith

capa

city

• co

nver

ting

betw

een

kilo

litre

s,

litre

s an

d m

illilit

res

•

conv

ersi

ons

shou

ld in

clud

e fra

ctio

n an

d de

cim

al fo

rms

to 2

de

cim

al p

lace

s LT

SM

Res

ourc

es

Mea

surin

g ju

gs, b

ottle

s, v

arie

ty o

f con

tain

ers

from

lear

ner’s

con

text

, cub

es

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

2 p

p. 2

– 1

2 Ac

tiviti

es 6

5; 6

6a&b

; 67

a&b;

68

pp. 1

52 –

156

Act

iviti

es 1

30, 1

31, 1

32

WB

2 p

p. 7

4 –

76 A

ctiv

ities

24a

&b

pp. 1

60 -1

64

Activ

ities

130

, - 1

32.

WB

1 p

p. 1

60 –

162

Ac

tiviti

es 6

2; 6

3 W

B 2

pp.

158

Ac

tivity

124

DB

E Te

xtbo

ok re

fere

nce

Term

3: U

nit 2

: Cap

acity

and

vol

ume,

p.

225

Term

1: U

nit 9

: Cap

acity

and

vol

ume,

p.

104

Term

2: U

nit 8

: Cap

acity

and

vol

ume,

p.

200

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

105MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

10

TER

M:

2

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E: fr

om

……

… to

……

…

Tim

e: 5

HO

UR

S G

RAD

E 4

CAP

S pp

. 78

- 79

5 C

APS

pp. 1

67 -

168

6 C

APS

pp. 2

44 -

246

TOPI

C

3D O

bjec

ts

Con

cept

s, S

kills

and

kn

owle

dge

Obj

ects

lear

ners

nee

d to

kno

w

and

nam

e •

rect

angu

lar p

rism

s •

sphe

res

• cy

linde

rs

• co

nes

• sq

uare

-bas

ed p

yram

ids

Cha

ract

eris

tics

whi

ch le

arne

rs u

se

to d

istin

guis

h, d

escr

ibe,

sor

t and

co

mpa

re o

bjec

ts

• sh

apes

of f

aces

•

flat a

nd c

urve

d su

rface

s Fu

rthe

r act

iviti

es to

focu

s le

arne

rs

on c

hara

cter

istic

s of

obj

ects

C

reat

e 3-

D m

odel

s us

ing

cut-o

ut

Pol

ygon

s

Obj

ects

lear

ners

nee

d to

kno

w

and

nam

e •

rect

angu

lar p

rism

s an

d ot

her p

rism

s •

cube

s •

cylin

ders

•

cone

s •

pyra

mid

s •

sim

ilarit

ies

and

diffe

renc

es

betw

een

cube

s an

d re

ctan

gula

r pr

ism

s C

hara

cter

istic

s le

arne

rs u

se to

di

stin

guis

h, d

escr

ibe,

sor

t and

co

mpa

re s

hape

s •

shap

e of

face

s •

num

ber o

f fac

es

• fla

t and

cur

ved

surfa

ces

Furt

her a

ctiv

ities

to fo

cus

lear

ners

on

cha

ract

eris

tics

of o

bjec

ts

• M

ake

3-D

mod

els

usin

g cu

t out

po

lygo

ns

• C

uttin

g op

en b

oxes

to tr

ace

and

de

scrib

e th

eir n

ets

Obj

ects

lear

ners

nee

d to

kno

w

and

nam

e •

rect

angu

lar p

rism

s •

cube

s •

tetra

hedr

ons

and

othe

r pyr

amid

s •

sim

ilarit

ies

and

diffe

renc

es

betw

een

tetra

hedr

ons

and

othe

r py

ram

ids

Feat

ures

lear

ners

use

to

dist

ingu

ish,

des

crib

e, s

ort a

nd

com

pare

obj

ects

D

escr

ibe,

sor

t and

com

pare

2-D

sh

apes

and

3-D

obj

ects

in te

rms

of:

• nu

mbe

r and

sha

pe o

f fac

es

• nu

mbe

r of v

ertic

es

• nu

mbe

r of e

dges

Fu

rthe

r act

iviti

es to

focu

s le

arne

rs

on c

hara

cter

istic

s of

obj

ects

M

ake

3-D

mod

els

usin

g:

• dr

inki

ng s

traw

s/to

othp

icks

, etc

. to

form

a s

kele

ton

• ne

ts

106 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Com

plex

ity c

hang

es fr

om g

rade

to g

rade

as

is s

tipul

ated

in th

e C

APS

docu

men

t. Al

l gra

des:

W

hat i

s di

ffere

nt to

Gra

de 3

? Le

arne

rs fo

cus

on th

e sa

me

3-D

geo

met

rical

obj

ects

, but

in G

rade

3 th

ey s

poke

of

bo

xes,

and

in G

rade

4 th

ey c

all t

hese

rect

angu

lar p

rism

s

• ba

ll sh

apes

and

in G

rade

4 th

ey c

all t

hese

sph

eres

Obj

ects

and

thei

r dis

tingu

ishi

ng c

hara

cter

istic

s Th

ere

are

two

way

s in

whi

ch le

arne

rs d

istin

guis

h 3-

D o

bjec

ts in

Gra

de 4

. 1.

C

heck

whe

ther

they

hav

e fla

t or c

urve

d su

rface

s. T

hree

dim

ensi

onal

obj

ects

can

be

grou

ped

as fo

llow

s:

• O

bjec

ts w

ith a

cur

ved

surfa

ce o

nly:

Exam

ple:

a s

pher

e

•

Obj

ects

with

flat

and

cur

ved

surfa

ces

C

ones

Cyl

inde

rs

107MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• O

bjec

ts w

ith o

nly

flat s

urfa

ces.

In G

rade

4 le

arne

rs o

nly

iden

tify

and

nam

e th

em.

E

xam

ples

rect

angu

lar p

rism

s.

pyr

amid

s: s

quar

e- b

ase

pyra

mid

2.

W

hen

look

ing

at th

e gr

oup

of o

bjec

ts w

ith fl

at s

urfa

ces,

lear

ners

sho

uld

know

that

the

flat s

urfa

ces

of a

3-D

obj

ect a

re

calle

d fa

ces.

The

y de

scrib

e th

ese

obje

cts

acco

rdin

g to

the

kind

s of

2-D

sha

pes

that

mak

e up

the

flat s

urfa

ces

e.g.

the

face

s of

a re

ctan

gula

r pris

m c

an a

ll be

rect

angl

es o

r som

e ca

n be

squ

ares

. Squ

are-

base

d py

ram

ids

have

one

squ

are

face

and

the

othe

r fac

es a

re tr

iang

les

Gra

de 5

and

6:

Wha

t is

diffe

rent

?

• cu

bes

are

intro

duce

d

• le

arne

rs w

ork

with

pris

ms

as a

gro

up fo

r the

firs

t tim

e

• in

the

sam

e w

ay th

at le

arne

rs d

istin

guis

h be

twee

n re

ctan

gles

and

squ

ares

, usi

ng th

e le

ngth

s of

thei

r sid

es, s

o th

ey

dist

ingu

ish

betw

een

cube

s an

d re

ctan

gula

r pris

ms

usin

g th

e sh

apes

of t

heir

face

s

• le

arne

rs c

ount

the

num

ber o

f fac

es o

n 3-

D o

bjec

ts a

nd u

se th

is a

s pa

rt of

thei

r des

crip

tions

of o

bjec

ts

Obj

ects

and

thei

r dis

tingu

ishi

ng c

hara

cter

istic

s

Ther

e ar

e th

ree

way

s in

whi

ch le

arne

rs d

istin

guis

h 3-

D o

bjec

ts in

Gra

de 5

1.

By c

heck

ing

whe

ther

they

hav

e fla

t or c

urve

d su

rface

s. T

hree

dim

ensi

onal

obj

ects

can

be

grou

ped

as fo

llow

s:

• O

bjec

ts w

ith a

cur

ved

surfa

ce o

nly:

108 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Exam

ple:

sph

ere

•

Obj

ects

with

flat

and

curv

ed s

urfa

ces

E

xam

ples

:

•

Obj

ects

with

onl

y fla

t sur

face

s. In

Gra

de 5

lear

ners

onl

y id

entif

y an

d na

me

Rec

tang

ular

pris

ms

Cub

es

109MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

2.

W

hen

look

ing

at th

e gr

oup

of o

bjec

ts w

ith fl

at s

urfa

ces,

lear

ners

sho

uld

know

that

the

flat s

urfa

ces

of a

3-D

obj

ect a

re

calle

d fa

ces.

The

y de

scrib

e th

ese

obje

cts

acco

rdin

g to

the

kind

s of

2-D

sha

pes

that

mak

e up

the

flat s

urfa

ces

e.g.

the

face

s of

a re

ctan

gula

r pris

m c

an a

ll be

rect

angl

es o

r som

e ca

n be

squ

ares

. Squ

are-

base

d py

ram

ids

have

one

squ

are

face

and

the

othe

r fac

es a

re tr

iang

les.

3.

Lear

ners

can

als

o lo

ok fo

r rig

ht a

ngle

s on

the

face

s of

obj

ects

. If t

he o

bjec

t tha

t the

y ar

e ex

amin

ing

has

face

s w

ith o

nly

right

ang

les,

then

it w

ill be

eith

er a

cub

e or

a re

ctan

gula

r pris

m.

Furt

her a

ctiv

ities

to fo

cus

lear

ners

on

char

acte

ristic

s of

obj

ects

:

• Le

arne

rs c

reat

e 3-

D o

bjec

ts b

y pu

tting

toge

ther

cut

-out

pol

ygon

s, w

hich

hel

ps to

focu

s at

tent

ion

on th

e sh

apes

of t

he

face

s of

the

3-D

obj

ects

.

• Le

arne

rs c

ut o

pen

boxe

s to

mak

e ne

ts. T

hey

desc

ribe

the

nets

of t

he b

oxes

. Int

erpr

etin

g dr

awin

gs o

f 3-D

obj

ects

and

writ

ten

exer

cise

s

Lear

ners

nee

d to

wor

k w

ith re

al o

bjec

ts. H

owev

er th

ey a

lso

need

to d

o w

ritte

n ex

erci

ses

on 3

-D o

bjec

ts. I

nter

pret

ing

pict

ures

of 3

-D o

bjec

ts is

mor

e di

fficu

lt th

an w

orki

ng w

ith th

e re

al o

bjec

ts. L

earn

ers

shou

ld p

ract

ice

inte

rpre

ting

draw

ings

of 3

-D

obje

cts.

The

y sh

ould

iden

tify

and

nam

e 3-

D o

bjec

ts in

dra

win

gs a

nd id

entif

y ev

eryd

ay o

bjec

ts th

at lo

ok li

ke g

eom

etric

obj

ects

e.g.

a m

ilk c

arto

n lo

oks

like

a re

ctan

gula

r pris

ms.

Des

crib

e th

e su

rface

s of

obj

ects

whe

n sh

own

draw

ing

of 3

-D o

bjec

ts, m

atch

110 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

the

2-D

sha

pes

that

hav

e th

e sa

me

shap

e as

the

face

s of

3-D

obj

ects

, mat

ch n

ets

of re

ctan

gula

r pris

ms

to th

e ap

prop

riate

draw

ing

of re

ctan

gula

r pris

ms

and

com

pare

3-D

obj

ects

from

dra

win

gs.

Gra

de 6

: W

hat i

s di

ffere

nt?

• Te

trahe

dron

s ar

e ne

w o

bjec

ts

• O

ther

pyr

amid

s ar

e ne

w o

bjec

ts

• Le

arne

rs d

istin

guis

h be

twee

n te

trahe

dron

s an

d ot

her p

yram

ids

by lo

okin

g at

the

shap

es o

f the

ir ba

ses,

• Le

arne

rs u

se n

ets

to b

uild

obj

ects

• Le

arne

rs m

atch

net

s w

ith d

raw

ings

of o

bjec

ts

• Le

arne

rs c

ount

the

num

ber o

f edg

es o

f 3-D

obj

ects

• Le

arne

rs b

uild

ske

leto

n ob

ject

s us

ing

drin

king

stra

ws

• Le

arne

rs c

ount

the

num

ber o

f ver

tices

of o

bjec

ts.

Obj

ects

and

thei

r dis

tingu

ishi

ng c

hara

cter

istic

s Th

ere

are

thre

e w

ays

in w

hich

lear

ners

dis

tingu

ish

3-D

obj

ects

in G

rade

6.

1.

Che

ckin

g w

heth

er th

ey h

ave

flat o

r cur

ved

surfa

ces.

Obj

ects

with

onl

y fla

t sur

face

s. In

Gra

de 6

lear

ners

onl

y id

entif

y an

d na

me

the

obje

cts.

Pris

ms

Rec

tang

ular

pris

ms

111MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Cub

es

O

ther

pris

ms

Pyra

mid

s 2.

W

hen

look

ing

at th

e gr

oup

of o

bjec

ts w

ith fl

at

surfa

ces,

lear

ners

sho

uld

know

that

the

flat s

urfa

ces

of 3

-D o

bjec

ts a

re c

alle

d fa

ces.

The

y de

scrib

e th

ese

obje

cts

acco

rdin

g to

• th

e ki

nds

and

num

bers

of 2

-D s

hape

s th

at m

ake

up th

e fla

t sur

face

s e.

g. a

rect

angu

lar p

rism

can

hav

e 6

face

s th

at a

re

rect

angl

es o

r 4 th

at a

re re

ctan

gles

and

2 th

at a

re s

quar

es.

• th

e nu

mbe

r of e

dges

• th

e nu

mbe

r of v

ertic

es

112 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

3.

Lear

ners

can

als

o lo

ok fo

r rig

ht a

ngle

s on

the

face

s of

obj

ects

. If t

he o

bjec

t tha

t the

y ar

e ex

amin

ing

has

face

s w

ith o

nly

right

ang

les,

then

it w

ill be

eith

er a

cub

e or

a re

ctan

gula

r pris

m.

Furt

her a

ctiv

ities

: mak

ing

mod

els

of 3

-d o

bjec

ts

• Le

arne

rs c

reat

e 3-

D o

bjec

ts fr

om n

ets

• Le

arne

rs c

reat

e sk

elet

ons

of 3

-D o

bjec

ts w

ith s

traw

s / t

ooth

pick

s, e

tc.

Inte

rpre

ting

draw

ings

of 3

-D o

bjec

ts a

nd w

ritte

n ex

erci

ses

Lear

ners

nee

d to

wor

k w

ith re

al o

bjec

ts. H

owev

er th

ey a

lso

need

to d

o w

ritte

n ex

erci

ses

on 3

-D o

bjec

ts. I

nter

pret

ing

pict

ures

of 3

-D o

bjec

ts is

mor

e di

fficu

lt th

an w

orki

ng w

ith th

e re

al o

bjec

ts. L

earn

ers

shou

ld p

ract

ise

inte

rpre

ting

draw

ings

of 3

-D o

bjec

ts. T

hey

shou

ld id

entif

y an

d na

me

3-D

obj

ects

in d

raw

ings

iden

tify

ever

yday

obj

ects

that

look

like

geo

met

ric

obje

cts

e.g.

a m

ilk c

arto

n lo

oks

like

a re

ctan

gula

r pris

m, m

atch

net

s of

obj

ects

to d

raw

ing

of o

bjec

ts, d

escr

ibe

3-D

obj

ects

by s

tatin

g th

e nu

mbe

r of f

lat a

nd c

urve

d su

rface

s, c

ount

the

num

ber o

f ver

tices

, edg

es, a

nd n

umbe

r and

sha

pe o

f fac

es

whe

n sh

own

draw

ings

of 3

-D o

bjec

ts.

In T

erm

2 le

arne

rs fo

cus

on th

e ki

nd o

f sur

face

, the

sha

pe a

nd th

e nu

mbe

r of f

aces

of a

3-D

obj

ect.

They

als

o bu

ild o

bjec

ts

usin

g ne

ts.

In T

erm

4 th

ey c

an c

onso

lidat

e w

hat t

hey

have

lear

ned

in T

erm

1 a

nd b

uild

ske

leto

n sh

apes

with

stra

ws

or to

othp

icks

. The

y w

ill

then

focu

s on

the

edge

s an

d ve

rtice

s of

the

obje

cts.

Thi

s m

eans

that

by

the

end

of th

e ye

ar th

ey w

ill be

abl

e to

des

crib

e

3-D

geo

met

ric o

bjec

ts a

ccor

ding

to s

urfa

ces,

face

s, e

dges

and

ver

tices

.

LTSM

Res

ourc

es

Varie

ty o

f 3D

obj

ects

from

lear

ner’s

con

text

, geo

solid

s

Gra

de 4

G

rade

5

Gra

de 6

113MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Wor

kboo

k re

fere

nce

WB

1 p

p 13

0 -1

34

Act

iviti

es 4

8, 4

9, 5

0 W

B 1

pp.

142

– 1

50 A

ctiv

ities

50

; 51;

52;

53;

54

WB

1 p

p 94

– 9

6 Ac

tiviti

es 3

3; 3

4

DB

E Te

xtbo

ok re

fere

nce

Term

2: U

nit 6

: Pro

pert

ies

of th

ree

dim

ensi

onal

obj

ects

, p. 1

73

Term

2: U

nit 6

: Pro

pert

ies

of th

ree

dim

ensi

onal

obj

ects

, p. 1

67

Term

2: U

nit 3

: Pro

pert

ies

of th

ree

dim

ensi

onal

obj

ects

, p. 1

42

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

as

sess

men

t – te

st

114 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LESS

ON

PLA

NS:

TER

M 3

115MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

1

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

….

Tim

e: 3

HO

UR

S G

RAD

E 4

CAP

S p.

92 (m

etho

dolo

gy e

xpla

ined

on

pp. 3

7 - 3

8)

5 C

APS

p.81

(met

hodo

logy

exp

lain

ed o

n pp

. 125

-126

)

6 C

APS

p. 2

62 (m

etho

dolo

gy e

xpla

ined

on

pp.

215

– 2

16)

TOPI

C

Who

le n

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

C

ount

forw

ards

and

bac

kwar

ds in

2s,

3s

, 5s,

10s

, 25s

, 50s

, 100

s be

twee

n 0

and

at le

ast 1

0 00

0

O

rder

, com

pare

and

repr

esen

t

nu

mbe

rs to

at l

east

4 d

igit

num

bers

.

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its in

w

hole

num

bers

to a

t lea

st 4

-dig

it nu

mbe

rs

R

ound

off

to th

e ne

ares

t 10,

100

, 10

00.

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

6-d

igit

num

bers

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

.

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its in

w

hole

num

bers

to a

t lea

st 6

dig

it nu

mbe

rs.

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00 ,

1 00

0 an

d 10

000

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

9-d

igit

num

bers

Rep

rese

nt p

rime

num

bers

to a

t lea

st

100

R

ecog

nizi

ng th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 9

-dig

it nu

mbe

rs

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00,

1 00

0, 1

0 00

0, 1

00 0

00, a

nd

1 00

0 00

0

116 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Ref

er b

ack

to T

erm

1

Expl

ain

from

kno

wn

(low

num

ber r

ange

) to

unkn

own

(hig

her r

ange

of n

umbe

rs);

wor

k w

ith c

oncr

ete

LTSM

C

ount

ing,

ord

erin

g, c

ompa

ring,

repr

esen

ting

of d

igits

N

ote

:Acc

ordi

ng to

CA

PS n

umbe

r ran

ge p

er g

rade

per

term

as

in c

onte

nt c

larif

icat

ion

Rea

d, s

ay a

nd w

rite

up to

at l

east

4-d

igit

num

bers

(Gr4

), 6-

digi

t (G

r5) a

nd 9

-dig

it (G

r6)

(Lea

rner

s ca

n w

rite

num

bers

with

fing

ers

in th

e ai

r whi

le th

ey a

re re

adin

g / s

ayin

g th

e w

ords

.) e.

g.,

One

hun

dred

and

tw

enty

-nin

e.

C

onve

rt fro

m w

ords

to n

umbe

rs a

nd n

umbe

rs to

wor

ds

Ar

rang

e th

e nu

mbe

rs b

elow

from

the

smal

lest

to th

e bi

gges

t:

Exam

ple:

10

111

; 1 1

01; 1

110

; 1 0

11

M

ake

the

bigg

est/s

mal

lest

num

ber y

ou c

an w

ith th

ese

digi

ts: 3

, 2, 5

, 4, 9

. O

dd a

nd e

ven

num

bers

: Ex

plor

e pr

actic

ally

by

wor

king

with

num

bers

to b

e ab

le to

des

crib

e th

e pa

ttern

e.g

. all

odd

num

bers

end

in 1

, 3, 5

, 7 o

r 9

ther

efor

e 2

201

, 2 2

03, 2

205

, 2 2

07, 2

209

are

all

odd

num

bers

. Pr

ime

num

bers

for G

rade

6

Iden

tify

prim

e nu

mbe

rs a

nd c

ompo

site

num

bers

up

to a

t lea

st 1

00

Mul

tiple

s:

I cou

nt in

mul

tiple

s of

3, u

p to

100

.

Will

I cou

nt th

e nu

mbe

r 12?

Why

?

Will

I cou

nt th

e nu

mbe

r 42?

Why

? Pl

ace

valu

e:

D

istin

guis

h be

twee

n th

e nu

mer

ic v

alue

and

the

plac

e va

lue.

Use

pla

ce v

alue

car

ds to

bui

ld u

p an

d br

eak

dow

n nu

mbe

rs

Rou

ndin

g of

f:

Rou

nd o

ff as

indi

cate

d pe

r gra

de (u

se b

ase

ten

bloc

ks, n

umbe

r lin

es, p

lace

val

ue ta

ble)

Use

con

cept

of r

ound

ing

off,

in p

robl

em s

olvi

ng c

onte

xt

117MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LTSM

R

esou

rces

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

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, aba

cus,

bea

ds

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

ferr

ence

W

B 2

pp

28 -

30 A

ctiv

ities

76,

77

WB

2 p

p 32

Act

ivity

79

WB

2 p

p 10

8 –

110

Ac

tivity

105

a&b

DB

E Te

xtbo

ok re

fere

nce

Term

3: U

nit 3

: Who

le N

umbe

rs p

. 238

Te

rm 3

: Uni

t 3: W

hole

Num

bers

p.

218

Term

3: U

nit 2

: Who

le N

umbe

rs p

. 22

1 H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

118 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

2

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

……

T

ime:

8 H

OU

RS

GR

ADE

4 C

APS

p.93

(met

hodo

logy

exp

lain

ed o

n pp

. 43

- 45)

5 C

APS

p 18

2- 1

83 (m

etho

dolo

gy

expl

aine

d on

pp.

132

– 1

35)

6 C

APS

p.26

2-26

3 (m

etho

dolo

gy

expl

aine

d on

pp.

222

– 2

25)

TOPI

C

Addi

tion

and

subt

ract

ion

of w

hole

num

bers

C

once

pts,

Ski

lls a

nd

know

ledg

e N

umbe

r ran

ge fo

r cal

cula

tions

•

Addi

tion

and

subt

ract

ion

of w

hole

nu

mbe

rs o

f at l

east

4 d

igits

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to p

erfo

rm

and

chec

k w

ritte

n an

d m

enta

l ca

lcul

atio

ns o

f who

le n

umbe

rs

incl

udin

g •

estim

atio

n •

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• do

ublin

g an

d ha

lvin

g •

usin

g a

num

ber l

ine

• us

ing

addi

tion

and

subt

ract

ion

as in

vers

e op

erat

ions

Num

ber r

ange

for c

alcu

latio

ns

Ad

ditio

n an

d su

btra

ctio

n of

who

le

num

bers

of a

t lea

st 5

dig

its

Cal

cula

tion

tech

niqu

es

Usi

ng a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l ca

lcul

atio

ns w

ith w

hole

num

bers

in

clud

ing

• es

timat

ion

• ad

ding

and

sub

tract

ing

in c

olum

ns

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

usin

g a

num

ber l

ine

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• do

ublin

g an

d ha

lvin

g •

usin

g ad

ditio

n an

d su

btra

ctio

n as

inve

rse

oper

atio

ns

Num

ber

rang

e fo

r cal

cula

tions

•

Addi

tion

and

subt

ract

ion

of w

hole

nu

mbe

rs w

ith a

t lea

st 6

-dig

it nu

mbe

r •

Mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out

brac

kets

C

alcu

latio

n te

chni

ques

U

sing

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal c

alcu

latio

ns w

ith w

hole

nu

mbe

rs in

clud

ing

• es

timat

ion

• ad

ding

, sub

tract

ing

in c

olum

ns

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

usin

g ad

ditio

n an

d su

btra

ctio

n as

inve

rse

oper

atio

ns

• us

ing

a ca

lcul

ator

119MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e an

d as

soci

ativ

e pr

oper

ties

with

who

le n

umbe

rs

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g w

hole

num

bers

, inc

ludi

ng fi

nanc

ial

cont

exts

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e, a

ssoc

iativ

e an

d di

strib

utiv

e pr

oper

ties

with

w

hole

num

bers

•

0 in

term

s of

its

addi

tive

prop

erty

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

, inc

ludi

ng

o fin

anci

al c

onte

xts

o m

easu

rem

ent c

onte

xts

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

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

So

lvin

g pr

oble

ms

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs a

nd d

ecim

al

fract

ions

, inc

ludi

ng

o fin

anci

al c

onte

xts

o m

easu

rem

ent c

onte

xts

• So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng c

ompa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd

(ratio

) LT

SM

R

esou

rces

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

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, cou

nter

s, m

aths

vid

eo c

lips,

E le

arni

ng re

sour

ces

onlin

e, M

aths

gam

es

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

2 p

p. 3

2 -4

0 ; p

p 70

– 7

6 Ac

tiviti

es

(78a

&b; 7

9-81

; 95a

&b; 9

6)

WB

2 p

p. 4

0 –

48 A

ctiv

ities

82

a&b;

83-

85

WB

2 p

p. 2

0 –

44 A

ctiv

ities

71

a&b;

72a

&b; 7

3-78

D

BE

Text

book

refe

renc

e Te

rm 2

: Uni

t 9: W

hole

Num

bers

: Ad

ditio

n an

d su

btra

ctio

n, p

. 193

Te

rm 3

: Uni

t 4: W

hole

Num

bers

: Ad

ditio

n an

d su

btra

ctio

n, p

. 243

Term

3: U

nit 4

: Who

le N

umbe

rs:

Addi

tion

and

subt

ract

ion,

p. 2

23

Term

3: U

nit 3

: Who

le N

umbe

rs:

Addi

tion

and

subt

ract

ion,

p. 2

24

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

120 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

3

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to

…

T

ime:

3 H

OU

RS

GR

ADE

4 M

ultip

licat

ion

- CAP

S pp

.101

5 M

ultip

licat

ion

- CAP

S p.

192

(Ref

er to

p.1

66

for m

etho

dolo

gy.)

6 M

ultip

licat

ion

- CAP

S p.

241-

243

TOPI

C

Mul

tiplic

atio

n of

who

le n

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 2

-di

git b

y 2-

digi

t num

bers

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 3

-dig

it by

2-

digi

t num

bers

Num

ber

rang

e fo

r cou

ntin

g,

orde

ring,

com

parin

g an

d re

pres

entin

g, a

nd p

lace

val

ue

of d

igits

•

Ord

er, c

ompa

re a

nd re

pres

ent

num

bers

to a

t lea

st 9

-dig

it nu

mbe

rs

• R

epre

sent

prim

e nu

mbe

rs to

at

leas

t 100

•

Rec

ogni

ze t

he p

lace

val

ue o

f di

gits

in

who

le n

umbe

rs t

o at

le

ast 9

-dig

it nu

mbe

rs

• R

ound

off

to th

e ne

ares

t 5, 1

0,

100

or 1

000

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 4

-di

git b

y 3-

digi

t num

bers

•

Mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out

brac

kets

121MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal c

alcu

latio

ns w

ith w

hole

nu

mbe

rs in

clud

ing

• es

timat

ion

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s M

ultip

les

of 1

-dig

it nu

mbe

rs to

at l

east

10

0 Pr

oper

ties

of w

hole

num

bers

Rec

ogni

ze a

nd u

se th

e co

mm

utat

ive;

ass

ocia

tive;

and

di

strib

utiv

e pr

oper

ties

of w

hole

nu

mbe

rs

So

lvin

g pr

oble

ms

• So

lve

prob

lem

s in

con

text

s in

volv

ing

who

le n

umbe

rs,

incl

udin

g fin

anci

al c

onte

xts

Cal

cula

tion

tech

niqu

es

Usi

ng a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l cal

cula

tions

of w

hole

num

bers

in

clud

ing:

•

estim

atio

n •

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• us

ing

a nu

mbe

r lin

e •

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igits

who

le n

umbe

rs to

at

leas

t 100

•

Fact

ors

of 2

-dig

it w

hole

num

bers

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e;

asso

ciat

ive

and

dist

ribut

ive

prop

ertie

s w

ith w

hole

num

bers

•

0 in

term

s of

its

addi

tive

prop

erty

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

Cal

cula

tion

tech

niqu

es in

clud

e •

estim

atio

n •

mul

tiply

ing

in c

olum

ns

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

usin

g a

calc

ulat

or

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igit

and

3-di

git

num

bers

•

Fact

ors

of 2

-dig

it an

d 3-

digi

t who

le

num

bers

•

Prim

e fa

ctor

s of

num

bers

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e;

dist

ribut

ive

prop

ertie

s of

w

hole

num

bers

•

0 in

term

s of

its

addi

tive

prop

erty

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

and

dec

imal

fra

ctio

ns, i

nclu

ding

122 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

•

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

, inc

ludi

ng

o co

mpa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd

(ratio

) o

com

parin

g tw

o qu

antit

ies

of

diffe

rent

kin

ds (r

ate)

o fin

anci

al c

onte

xts

o m

easu

rem

ent

cont

exts

•

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

, in

clud

ing

com

parin

g tw

o or

mor

e qu

antit

ies

of

the

sam

e ki

nd (r

atio

)

- fin

anci

al c

onte

xts

- m

easu

rem

ent c

onte

xts

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

o

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

o co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

SUG

GES

TED

M

ETH

OD

OLO

GY

Ref

er to

Ter

m1

and

Term

2 le

sson

Pla

ns

Expl

ain

from

kno

wn

(low

num

ber r

ange

); w

ork

with

con

cret

e LT

SM.

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

cont

inue

to

• ch

eck

thei

r sol

utio

ns b

y us

ing

mul

tiplic

atio

n w

hen

doin

g di

visi

on a

nd v

ice

vers

a •

judg

e th

e re

ason

able

ness

of t

heir

solu

tions

, by

estim

atin

g be

fore

cal

cula

ting

Mul

tiplic

atio

n (K

now

tim

es ta

bles

– p

ract

ice

regu

larly

up

to 1

0 x

10 fo

r Gra

des

4 an

d 5

and

12 x

12

for G

rade

6).

Follo

w th

ese

step

s in

teac

hing

mul

tiplic

atio

n:

• So

lve

prob

lem

s in

con

text

. U

nder

stan

d th

at: 8

6 +

86 +

86

is e

quiv

alen

t to

86 x

3 o

r 3 x

86

(repe

ated

add

ition

) •

mul

tiplic

atio

n by

1 le

aves

a n

umbe

r unc

hang

ed e

.g. 5

x 1

= 5

; 99

x 1

= 99

•

mul

tiplic

atio

n of

zer

o re

sults

in z

ero

e.g.

7 x

0 =

0; 3

1 x

0 =

0 U

nder

stan

d th

at m

ultip

licat

ion

is th

e in

vers

e of

div

isio

n (m

ultip

licat

ion

reve

rses

div

isio

n an

d vi

ce v

ersa

) and

use

this

to c

heck

re

sults

. e.g

. 15

x 35

= 5

25

525

÷ 2

5 =

15 o

r 525

÷ 1

5 =

25

Not

e: In

Gra

de 5

Ter

m 3

lear

ners

are

intro

duce

d to

col

umn

met

hods

of m

ultip

licat

ion

123MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

DIV

ISIO

N

Men

tal d

ivis

ion

: Tur

ning

the

tabl

es –

pra

ctic

e re

gula

rly (e

.g. 5

x 8

= 4

0 4

0 ÷

8 =

5 / 4

0 ÷

5 =

8) (A

lso

calle

d in

vers

e op

erat

ion

or

reci

proc

al o

pera

tion.

) M

ake

lear

ners

aw

are

that

div

isio

n is

non

-com

mut

ativ

e i.e

. 81

÷ 9

is n

ot th

e sa

me

as 9

÷ 8

1.

With

the

conc

ept g

rasp

ed, t

each

ing

divi

sion

will

beco

me

mor

e ab

out g

uide

d pr

actic

e to

hel

p th

e le

arne

rs to

bec

ome

fam

iliar w

ith

the

divi

sion

ope

ratio

n (a

lthou

gh it

’s re

ally

goi

ng to

be

a di

ffere

nt ty

pe o

f mul

tiplic

atio

n pr

actic

e.) S

tart

by p

ract

isin

g di

visi

on b

y 1,

2

and

3 an

d th

en g

radu

ally

mov

e up

to 9

. N

ote:

Gra

de 4

and

5 fo

cus

is o

n us

ing

the

clue

boa

rd –

met

hod

ON

LY! G

rade

6 is

the

intro

duct

ion

of lo

ng d

ivis

ion

in th

e th

ird

term

LT

SM

Res

ourc

es

Num

ber l

ines

: 0 –

120

, Stru

ctur

ed -,

sem

i- st

ruct

ured

- , u

nstru

ctur

ed n

umbe

r lin

es; C

ount

ers,

Pic

ture

s, a

rrays

/ dia

gram

s, F

lash

ca

rds,

Bas

e 10

blo

cks,

100

cha

rts, m

ultip

licat

ion

tabl

e m

aths

vid

eo c

lips,

E le

arni

ng re

sour

ces

onlin

e, M

aths

gam

es, c

ount

ers

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 2

pp.

78

– 90

Act

iviti

es

98 a

&b, 9

9a&b

; 100

-101

W

B 2

pp.

90

– 94

Act

iviti

es

103a

&b; 1

04

WB

2 p

p. 1

22 –

130

Act

iviti

es

110a

&b; 1

11; 1

12a&

b D

BE

refe

renc

e Te

rm 3

: Uni

t 10

Who

le N

umbe

rs:

Mul

ti[pl

icat

ion,

p. 2

75

Term

3: U

nit 1

1: W

hole

Num

bers

: M

ulti[

plic

atio

n, p

. 269

Te

rm 4

: Uni

t 2: W

hole

Num

bers

: M

ulti[

plic

atio

n, p

. 300

5 H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

as

sess

men

t – T

est

124 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

4

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

…..…

T

ime:

4 H

OU

RS

GR

ADE

4 D

ivis

ion

- CAP

S p.

84-8

5 (M

etho

dolo

gy

refe

r to

pp. 8

4 - 8

5

5 D

ivis

ion

- CAP

S p.

200

(Met

hodo

logy

refe

r to

pp.1

72)

6 D

ivis

ion

- CAP

S p.

250

– 2

51

(Met

hodo

logy

refe

r to

pp.1

72)

TOPI

C

Div

isio

n of

who

le n

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

Num

ber

rang

e fo

r cal

cula

tions

D

ivis

ion

of a

t lea

st w

hole

3-d

igit

by

1-di

git n

umbe

rs.

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to p

erfo

rm

and

chec

k w

ritte

n an

d m

enta

l ca

lcul

atio

ns o

f who

le n

umbe

rs

incl

udin

g -

estim

atio

n -

build

ing

up

and

br

eaki

ng

dow

n nu

mbe

rs

- ro

undi

ng o

ff an

d co

mpe

nsat

ing

- do

ublin

g an

d ha

lvin

g -

usin

g m

ultip

licat

ion

and

divi

sion

as

inve

rse

oper

atio

ns

Num

ber

rang

e fo

r cal

cula

tions

D

ivis

ion

of a

t lea

st w

hole

3-d

igit

by

2-di

git n

umbe

rs

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to p

erfo

rm

and

chec

k w

ritte

n an

d m

enta

l ca

lcul

atio

ns w

ith w

hole

num

bers

in

clud

ing

• es

timat

ion

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

usin

g m

ultip

licat

ion

and

divi

sion

as

inve

rse

oper

atio

ns

Num

ber

rang

e fo

r cal

cula

tions

D

ivis

ion

of a

t lea

st w

hole

4-d

igit

by

3-di

git n

umbe

rs: m

ultip

le o

pera

tions

on

who

le n

umbe

rs w

ith o

r with

out b

rack

ets

Cal

cula

tion

tech

niqu

es

• es

timat

ion

• us

ing

the

reci

proc

al re

latio

nshi

p be

twee

n m

ultip

licat

ion

and

divi

sion

•

long

div

isio

n •

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• us

ing

a ca

lcul

ator

125MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

1-d

igit

num

bers

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e;

asso

ciat

ive;

and

dis

tribu

tive

prop

ertie

s of

who

le n

umbe

rs

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g w

hole

num

bers

-

finan

cial

con

text

s -

mea

sure

men

t con

text

s

• So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng

- gr

oupi

ng a

nd e

qual

sha

ring

w

ith re

mai

nder

s -

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(rat

io)

- co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

Num

ber

rang

e fo

r cou

ntin

g,

orde

ring

and

repr

esen

ting,

and

pl

ace

valu

e of

dig

its

• Rec

ogni

ze th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 6

-dig

it nu

mbe

rs.

• R

ound

off

to th

e ne

ares

t 10,

100

, 1

000

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igit

num

bers

to a

t le

ast 1

00

• Fa

ctor

s of

2-d

igit

who

le n

umbe

rs

to a

t lea

st 1

00

Mul

tiplic

atio

n fa

cts

• U

nits

by

mul

tiple

s of

10

• U

nits

by

mul

tiple

s of

100

Pr

oper

ties

of w

hole

num

bers

•

Rec

ogni

ze a

nd u

se th

e co

mm

utat

ive;

as

soci

ativ

e; a

nd d

istri

butiv

e pr

oper

ties

of w

hole

num

bers

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g w

hole

num

bers

, inc

ludi

ng fi

nanc

ial

cont

exts

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igit

and

3-di

git

num

bers

•

Fact

ors

of 2

-dig

it an

d 3-

digi

t who

le

num

bers

•

Prim

e fa

ctor

s of

num

bers

up

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

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

•

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs a

nd d

ecim

al fr

actio

ns,

incl

udin

g -

finan

cial

con

text

s -

mea

sure

men

t con

text

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

-

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(rat

io)

- co

mpa

ring

two

quan

titie

s of

-

diffe

rent

kin

ds (r

ate)

-

grou

ping

and

equ

al s

harin

g w

ith

rem

aind

ers

126 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng

- co

mpa

ring

two

or m

ore

quan

titie

s of

th

e sa

me

kind

(rat

io)

- co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

- gr

oupi

ng a

nd e

qual

sha

ring

with

re

mai

nder

s SU

GG

ESTE

D

MET

HO

DO

LOG

Y R

efer

to T

erm

1 a

nd T

erm

2 l

esso

n pl

ans

for c

larif

icat

ion

Expl

ain

from

kno

wn

(low

num

ber r

ange

); w

ork

with

con

cret

e LT

SM.

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

cont

inue

to

• ch

eck

thei

r sol

utio

ns th

emse

lves

, by

usin

g m

ultip

licat

ion

whe

n do

ing

divi

sion

and

vic

e ve

rsa

• ju

dge

the

reas

onab

lene

ss o

f the

ir so

lutio

ns, b

y es

timat

ing

befo

re c

alcu

latin

g

Mul

tiplic

atio

n (K

now

tim

es ta

bles

– p

ract

ice

regu

larly

up

to 1

0 x

10 fo

r Gra

de 4

and

5 a

nd 1

2 x

12 fo

r Gra

de 6

).

Follo

w th

ese

step

s in

teac

hing

mul

tiplic

atio

n:

- So

lve

prob

lem

s in

con

text

.

Und

erst

and

that

: 86

+ 86

+ 8

6 is

equ

ival

ent t

o 86

x 3

or 3

x 8

6 (re

peat

ed a

dditi

on)

- m

ultip

licat

ion

by 1

leav

es a

num

ber u

ncha

nged

e.g

. 5 x

1 =

5; 9

9 x

1 =

99

- m

ultip

licat

ion

of z

ero

resu

lts in

zer

o e.

g. 7

x 0

= 0

; 31

x 0

= 0

127MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Und

erst

and

that

mul

tiplic

atio

n is

the

inve

rse

of d

ivis

ion

(mul

tiplic

atio

n re

vers

es d

ivis

ion

and

vice

ver

sa) a

nd u

se th

is to

che

ck

resu

lts. e

.g. 1

5 x

35 =

525

5

25 ÷

25

= 15

or 5

25 ÷

15

= 25

Not

e: In

Gra

de 5

Ter

m 3

lear

ners

are

intro

duce

d to

col

umn

met

hods

of m

ultip

licat

ion

DIV

ISIO

N

Men

tal d

ivis

ion

: Tur

ning

the

tabl

es –

pra

ctic

e re

gula

rly (e

.g. 5

x 8

= 4

0 4

0 ÷

8 =

5 / 4

0 ÷

5 =

8)

(

Also

cal

led

inve

rse

oper

atio

n or

reci

proc

al o

pera

tion.

)

Mak

e le

arne

rs a

war

e th

at d

ivis

ion

is n

on-c

omm

utat

ive

i.e. 8

1 ÷

9 is

not

the

sam

e as

9 ÷

81.

With

the

conc

ept g

rasp

ed, t

each

ing

divi

sion

will

beco

me

mor

e ab

out g

uide

d pr

actic

e to

hel

p yo

ur le

arne

rs to

bec

ome

fam

iliar

with

the

divi

sion

ope

ratio

n (a

lthou

gh it

’s re

ally

goi

ng to

be

a di

ffere

nt ty

pe o

f mul

tiplic

atio

n pr

actic

e.) S

tart

by p

ract

isin

g di

visi

on

by 1

, 2 a

nd 3

and

then

gra

dual

ly m

ove

up to

9.

Not

e: G

rade

s 4

and

5 fo

cus

is o

n us

ing

the

clue

boa

rd –

met

hod

ON

LY! I

n G

rade

6 th

e lo

ng d

ivis

ion

is in

trodu

ced

in th

e th

ird

term

LTSM

Res

ourc

es

Te

xt b

ooks

; Int

erne

t Web

site

s; N

umbe

r boa

rd; F

lard

car

ds; B

ase

10 b

lock

s / D

iene

s bl

ocks

; Con

cret

e m

ater

ial,

e.g.

cou

nter

s;

Num

ber l

ines

+ e

mpt

y on

es; H

undr

ed c

hart,

mul

tiplic

atio

n ta

ble,

mat

hs v

ideo

clip

s, E

lear

ning

reso

urce

s on

line,

Mat

hs g

ames

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

oo re

fere

nce

WB

2 p

. 142

Act

iviti

es 1

25

WB

2 p

p. 1

38 –

144

Ac

tiviti

es 1

20 -

123

WB

2 p

p. 1

64 –

174

Act

iviti

es

126;

127

a7b;

128

; 129

a&b

DB

E Te

xtbo

ok re

fere

nce

Term

4: U

nit 6

: Who

le N

umbe

rs:

Div

isio

n, p

. 313

Te

rm 3

: Uni

t 5: W

hole

Num

bers

: D

ivis

ion,

p. 3

10

Term

4: U

nit 7

: Who

le N

umbe

rs:

Div

isio

n, p

. 336

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

128 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

129MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

5

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

……

T

ime:

6 H

OU

RS

GR

ADE

4

Num

eric

pat

tern

s: p

p. 9

7 - 9

9

Geo

met

ric p

atte

rns:

pp.

117

m

etho

d ex

plai

ned

on p

p. 8

0 -8

2

5

Num

eric

pat

tern

s: p

p. 1

89 –

191

,

Geo

met

ric p

atte

rns:

p. 2

06,

met

hod

expl

aine

d on

p. 1

69

6

Num

eric

pat

tern

s: p

p. 2

70 -

272

G

eom

etric

pat

tern

s: p

p. 2

47 -

249

TOPI

C

Geo

met

ric a

nd N

umer

ic p

atte

rns

N

ote:

Geo

met

ric p

atte

rns

are

bein

g do

ne a

s ex

tend

ed c

onso

lidat

ion

in th

is c

ycle

. C

once

pts,

Ski

lls a

nd

know

ledg

e N

umer

ic p

atte

rns:

•

Inve

stig

ate

and

exte

nd n

umer

ic

patte

rns

look

ing

for r

elat

ions

hips

or

rule

s of

pat

tern

s:

o se

quen

ces

invo

lvin

g a

cons

tant

diff

eren

ce o

r rat

io

o of

lear

ner’s

ow

n cr

eatio

n o

repr

esen

ted

in ta

bles

•

Des

crib

e th

e ge

nera

l rul

es fo

r the

ob

serv

ed re

latio

nshi

ps

Inpu

t and

out

put v

alue

s •

Det

erm

ine

inpu

t val

ues,

out

put

valu

es a

nd ru

les

for t

he p

atte

rns

and

rela

tions

hips

usi

ng:

o flo

w d

iagr

ams

o ta

bles

Equi

vale

nt fo

rms

Det

erm

ine

equi

vale

nce

of d

iffer

ent

desc

riptio

ns o

f the

sam

e re

latio

nshi

p or

ru

le p

rese

nted

Num

eric

pat

tern

s:

• In

vest

igat

e an

d ex

tend

num

eric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps o

r ru

les

of p

atte

rns:

o

sequ

ence

s no

t lim

ited

to a

co

nsta

nt d

iffer

ence

or r

atio

o

of le

arne

r’s o

wn

crea

tion

•

Des

crib

e th

e ge

nera

l rul

es fo

r the

ob

serv

ed re

latio

nshi

ps

Inpu

t and

out

put v

alue

s •

Det

erm

ine

inpu

t val

ues,

out

put

valu

es a

nd ru

les

for t

he p

atte

rns

and

rela

tions

hips

usi

ng:

o flo

w d

iagr

ams

o ta

bles

Equi

vale

nt fo

rms

Det

erm

ine

equi

vale

nce

of d

iffer

ent

desc

riptio

ns o

f the

sam

e re

latio

nshi

p or

ru

le p

rese

nted

Num

eric

pat

tern

s :

• In

vest

igat

e an

d ex

tend

num

eric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps o

r ru

les

of p

atte

rns:

o

sequ

ence

s no

t lim

ited

to 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

the

gene

ral r

ules

for t

he

obse

rved

rela

tions

hips

In

put a

nd o

utpu

t val

ues

• D

eter

min

e in

put v

alue

s, o

utpu

t va

lues

and

rule

s fo

r the

pat

tern

s an

d re

latio

nshi

ps u

sing

: o

flow

dia

gram

s o

tabl

es

Equi

vale

nt fo

rms

Det

erm

ine

equi

vale

nce

of d

iffer

ent

desc

riptio

ns o

f the

sam

e re

latio

nshi

p or

ru

le p

rese

nted

130 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• ve

rbal

ly

• in

a fl

ow d

iagr

am

• in

a ta

ble

• by

a n

umbe

r sen

tenc

e G

eom

etric

pat

tern

s •

Inve

stig

ate

and

exte

nd g

eom

etric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps o

r ru

les

of p

atte

rns

o re

pres

ente

d in

phy

sica

l or

diag

ram

form

o

sequ

ence

s no

t lim

ited

to a

co

nsta

nt d

iffer

ence

or r

atio

o

of le

arne

r’s o

wn

crea

tion

• D

escr

ibe

obse

rved

rela

tions

hips

or

rule

s in

lear

ners

’ ow

n w

ords

Inpu

t and

out

put v

alue

s •

Det

erm

ine

inpu

t val

ues,

out

put

valu

es a

nd ru

les

for t

he p

atte

rns

and

rela

tions

hips

usi

ng

flow

dia

gram

s Eq

uiva

lent

form

s •

Det

erm

ine

equi

vale

nce

of d

iffer

ent

desc

riptio

ns o

f the

sam

e re

latio

nshi

p or

rule

pre

sent

ed

o ve

rbal

ly

o in

a fl

ow d

iagr

am

o by

a n

umbe

r sen

tenc

e

• ve

rbal

ly

• in

a fl

ow d

iagr

am

• in

a ta

ble

• by

a n

umbe

r sen

tenc

e G

eom

etric

pat

tern

s •

Inve

stig

ate

and

exte

nd g

eom

etric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps o

r ru

les

of p

atte

rns

o re

pres

ente

d in

phy

sica

l or

diag

ram

form

o

sequ

ence

s no

t lim

ited

to a

co

nsta

nt d

iffer

ence

or r

atio

o

of le

arne

r’s o

wn

crea

tion

• D

escr

ibe

obse

rved

rela

tions

hips

or

rule

s in

lear

ners

’ ow

n w

ords

In

put a

nd o

utpu

t val

ues

• D

eter

min

e in

put v

alue

s, o

utpu

t va

lues

and

rule

s fo

r the

pat

tern

s an

d re

latio

nshi

ps u

sing

flo

w d

iagr

ams

Equi

vale

nt fo

rms

• D

eter

min

e eq

uiva

lenc

e of

diff

eren

t de

scrip

tions

of t

he s

ame

rela

tions

hip

or ru

le p

rese

nted

o

verb

ally

o

in a

flow

dia

gram

o

by a

num

ber s

ente

nce

• ve

rbal

ly

• in

a fl

ow d

iagr

am

• in

a ta

ble

• by

a n

umbe

r sen

tenc

e

G

eom

etric

pat

tern

s •

Inve

stig

ate

and

exte

nd g

eom

etric

pa

ttern

s lo

okin

g fo

r rel

atio

nshi

ps o

r ru

les

of p

atte

rns

o re

pres

ente

d in

phy

sica

l or

diag

ram

form

o

sequ

ence

s no

t lim

ited

to 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

•

Des

crib

e th

e ge

nera

l rul

es fo

r the

ob

serv

ed re

latio

nshi

ps

Inpu

t and

out

put v

alue

s •

Det

erm

ine

inpu

t val

ues,

out

put

valu

es a

nd ru

les

for t

he p

atte

rns

and

rela

tions

hips

usi

ng

o flo

w d

iagr

ams

o ta

bles

Eq

uiva

lent

form

s •

Det

erm

ine

equi

vale

nce

of d

iffer

ent

desc

riptio

ns o

f the

sam

e re

latio

nshi

p or

rule

pre

sent

ed

o ve

rbal

ly

o in

a fl

ow d

iagr

am

o in

a ta

ble

o by

a n

umbe

r sen

tenc

e

131MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Ref

er b

ack

to th

e le

sson

pla

n fo

r Ter

m 1

on

Num

eric

Pat

tern

s fo

r rev

isio

n N

umer

ic P

atte

rns:

Th

e fo

cus

shou

ld b

e on

inpu

t-out

put f

low

dia

gram

s th

at h

elp

lear

ners

to u

nder

stan

d an

d le

arn

abou

t:

• th

e in

vers

e op

erat

ion

betw

een

mul

tiplic

atio

n an

d di

visi

on

• th

e m

ultip

licat

ion

of u

nits

by

mul

tiple

s of

ten

in G

rade

4; a

nd 1

0, 1

00 a

nd 1

000

in G

rade

5 a

nd G

rade

6

• th

e as

soci

ativ

e pr

oper

ty w

ith w

hole

num

bers

and

how

we

can

use

this

pro

perty

whe

n w

e m

ultip

ly b

y m

ultip

les

of 1

0.

Usi

ng fl

ow d

iagr

ams

help

lear

ners

to u

nder

stan

d an

d us

e th

e fa

ct th

at m

ultip

licat

ion

and

divi

sion

are

inve

rse

oper

atio

ns

Lear

ners

are

not

exp

ecte

d to

use

the

expr

essi

on “i

nver

se o

pera

tions

”. Th

ey a

re e

xpec

ted

to k

now

that

:

• th

ey c

an u

se m

ultip

licat

ion

to c

heck

div

isio

n ca

lcul

atio

ns

• th

ey c

an u

se d

ivis

ion

to c

heck

mul

tiplic

atio

n ca

lcul

atio

ns

Then

they

can

wor

k w

ith e

xam

ples

whi

ch h

ave

a tw

o st

age

rule

e.g

. mul

tiply

and

then

add

, whe

re o

ne s

tage

is le

ft ou

t.

Sequ

ence

s of

num

bers

: In

the

Inte

rmed

iate

Pha

se le

arne

rs e

xten

d se

quen

ces

of n

umbe

rs. I

n G

rade

4 th

ey w

ork

with

two

kind

s of

seq

uenc

es.

1.

Sequ

ence

s in

volv

ing

a co

nsta

nt d

iffer

ence

Ex

ampl

es

a)

2; 4

; 6; 8

…

b)

18; 1

6; 1

4; 1

2…

In th

e ex

ampl

es a

bove

lear

ners

are

add

ing

2 or

sub

tract

ing

2 to

mak

e th

e pa

ttern

.

Lear

ners

may

des

crib

e it

as a

pat

tern

of c

ount

ing

on o

r cou

ntin

g ba

ck in

twos

.

Lear

ners

sho

uld

also

be

give

n ex

ampl

es w

hich

do

not s

tart

on a

mul

tiple

of t

he n

umbe

r the

y ar

e ad

ding

or s

ubtra

ctin

g.

132 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Exam

ples

a)

1;

4; 7

; 10…

….

b)

87; 6

6; 4

5; …

…..

2.

Sequ

ence

s in

volv

ing

a co

nsta

nt ra

tio

Exam

ple

1 60

0; 8

00; 4

00; .

..

In th

e ab

ove

exam

ple

lear

ners

are

div

idin

g by

2. A

ll th

e nu

mbe

rs in

the

sequ

ence

are

mul

tiple

s of

2.

Lear

ners

sho

uld

also

be

give

n ex

ampl

es in

whi

ch th

e nu

mbe

rs in

the

sequ

ence

are

not

mul

tiple

s of

the

num

ber b

y w

hich

they

are

mul

tiply

ing

or d

ivid

ing.

Exam

ples

a)

3;

6; 1

2; 2

4; ..

.

b)

10; 3

0; 9

0; 2

70; .

..

NB:

Tow

ards

the

end

of G

rade

6, t

he fo

cus

shou

ld b

e on

“fin

ding

the

rule

”

Geo

met

ric P

atte

rns

NB:

1.

Patte

rns

inve

stig

ated

sho

uld

incl

ude

shap

es a

s pe

r Gra

de, r

equi

red

in C

onte

nt A

rea

3. In

Gra

de 6

ther

e is

an

incr

ease

in

com

plex

ity o

f exa

mpl

es.

2.

Educ

ator

s m

ay p

refe

r to

teac

h G

eom

etric

pat

tern

s fir

st a

nd li

nk it

with

num

eric

pat

tern

s.

Lear

ners

sho

w th

e sa

me

patte

rns

in d

iffer

ent w

ays:

in a

dia

gram

, as

a ve

rbal

des

crip

tion,

as

a flo

w d

iagr

am a

nd in

a n

umbe

r

sent

ence

. Som

etim

es le

arne

rs a

re a

ble

to s

ee d

iffer

ent a

spec

ts o

f a p

atte

rn w

hen

they

cha

nge

the

form

in w

hich

the

patte

rn

is p

rese

nted

. Lea

rner

s w

ork

with

pat

tern

s th

at a

re m

ade

from

2-D

sha

pes

and

3-D

obj

ects

or f

rom

dra

win

gs/d

iagr

ams

of

133MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

thes

e sh

apes

and

obj

ects

. In

Patte

rns,

Fun

ctio

ns a

nd A

lgeb

ra w

e ch

oose

geo

met

ric p

atte

rns

that

can

be

re-d

escr

ibed

usi

ng a

num

ber p

atte

rn (t

his

does

not

mea

n th

at it

can

’t be

des

crib

ed in

wor

ds, i

n fa

ct th

e de

scrip

tion

in w

ords

is u

sual

ly th

e st

artin

g

poin

t). In

Sha

pe a

nd S

pace

lear

ners

als

o w

ork

with

vis

ual p

atte

rns

that

are

geo

met

ric. H

owev

er, i

n Sh

ape

and

Spac

e th

ey

are

only

requ

ired

to d

escr

ibe

the

patte

rns

usin

g th

e la

ngua

ge o

f geo

met

ry a

nd to

cop

y th

e pa

ttern

s. W

hile

man

y of

thes

e

patte

rns

can

be d

escr

ibed

usi

ng a

lgeb

raic

exp

ress

ions

, thi

s is

bey

ond

the

scop

e of

Inte

rmed

iate

Pha

se le

arne

rs.

Gra

de 4

,5 a

nd 6

Wha

t is

diffe

rent

to G

rade

3?

In G

rade

3 le

arne

rs c

opy,

ext

end

and

desc

ribe

patte

rns

mad

e w

ith n

umbe

rs, o

bjec

ts o

r dra

win

gs T

he d

escr

iptio

ns a

re o

nly

verb

al. T

hey

also

cre

ate

thei

r ow

n pa

ttern

s.

The

kind

s of

pat

tern

s be

com

e m

ore

com

plex

in G

rade

4.

In G

rade

4 le

arne

rs a

re in

trodu

ced

to a

new

way

to re

pres

ent p

atte

rns:

the

inpu

t- ou

tput

flow

dia

gram

(som

e le

arne

rs m

ay

have

use

d th

is in

Fou

ndat

ion

Phas

e, b

ut it

is n

ot a

spe

cific

atio

n).

Lear

ners

sho

w th

e sa

me

patte

rns

in d

iffer

ent w

ays:

in a

dia

gram

, as

a ve

rbal

des

crip

tion,

as

a flo

w d

iagr

am a

nd in

a n

umbe

r se

nten

ce. S

omet

imes

lear

ners

are

abl

e to

see

diff

eren

t asp

ects

of a

pat

tern

whe

n th

ey c

hang

e th

e fo

rm in

whi

ch th

e pa

ttern

is

pres

ente

d.

Lear

ners

wor

k w

ith p

atte

rns

that

are

mad

e fro

m 2

-D s

hape

s an

d 3-

D o

bjec

ts o

r fro

m d

raw

ings

/dia

gram

s of

thes

e sh

apes

an

d ob

ject

s. In

Pat

tern

s, F

unct

ions

and

Alg

ebra

we

choo

se g

eom

etric

pat

tern

s th

at c

an b

e re

-des

crib

ed u

sing

a n

umbe

r pa

ttern

(thi

s do

es n

ot m

ean

that

it c

an’t

be d

escr

ibed

in w

ords

, in

fact

the

desc

riptio

n in

wor

ds is

usu

ally

the

star

ting

poin

t).

In S

hape

and

Spa

ce le

arne

rs a

lso

wor

k w

ith v

isua

l pat

tern

s th

at a

re g

eom

etric

. How

ever

, in

Shap

e an

d Sp

ace

they

are

on

ly re

quire

d to

des

crib

e th

e pa

ttern

s us

ing

the

lang

uage

of g

eom

etry

and

to c

opy

the

patte

rns.

Whi

le m

any

of th

ese

patte

rns

can

be d

escr

ibed

usi

ng a

lgeb

raic

exp

ress

ions

, thi

s is

bey

ond

the

scop

e of

Inte

rmed

iate

Pha

se le

arne

rs.

Wha

t kin

ds o

f geo

met

ric p

atte

rns

shou

ld le

arne

rs w

ork

with

? •

Sim

ple

repe

atin

g pa

ttern

s –

but t

his

is re

ally

mor

e of

a fo

cus

in th

e Fo

unda

tion

Phas

e

134 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Exam

ple:

Com

plet

e th

e pa

ttern

•

Patt

erns

in w

hich

the

shap

es g

row

or d

ecre

ase

in d

iffer

ent w

ays.

We

can

desc

ribe

thes

e pa

ttern

s by

the

way

th

ey lo

ok.

patte

rns

in w

hich

the

shap

e ke

eps

its fo

rm, b

ut g

ets

larg

er (o

r sm

alle

r) in

eac

h st

age.

patte

rns

in w

hich

a s

hape

or p

art o

f a s

hape

is a

dded

at e

ach

stag

e

In

eac

h of

the

exam

ples

abo

ve th

e pa

ttern

s ar

e m

ade

by a

ddin

g on

the

sam

e nu

mbe

r of m

atch

es in

eac

h su

cces

sive

sha

pe.

In th

e to

p pa

ttern

3 m

atch

es a

re a

dded

eac

h tim

e. In

the

seco

nd p

atte

rn tw

o m

atch

es a

re a

dded

eac

h tim

e. B

oth

patte

rns

show

num

ber p

atte

rns

with

a c

onst

ant d

iffer

ence

. M

ost g

eom

etric

pat

tern

s le

arne

rs s

ee in

Gra

de 4

, will

be p

atte

rns

with

a c

onst

ant d

iffer

ence

. The

y ar

e m

ore

likel

y to

get

pa

ttern

s w

ith a

con

stan

t rat

io w

hen

wor

king

onl

y w

ith n

umbe

r seq

uenc

es.

• P

atte

rns

with

nei

ther

a c

onst

ant d

iffer

ence

nor

a c

onst

ant r

atio

.

Exa

mpl

e

135MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Wha

t sho

uld

lear

ners

do?

•

Cop

y an

d ex

tend

the

patte

rn. T

his

help

s th

em to

und

erst

and

how

the

patte

rn is

form

ed.

• D

escr

ibe

the

patte

rn in

wor

ds.

-

Diff

eren

t lea

rner

s w

ill de

scrib

e di

ffere

nt a

spec

ts o

f the

pat

tern

.

- Y

ou w

ant l

earn

ers

to d

escr

ibe

the

rela

tions

hip

betw

een

shap

es in

the

sequ

ence

or

rule

s in

thei

r ow

n w

ords

. To

do th

is,

lear

ners

nee

d to

dis

cuss

how

they

mad

e th

e pa

ttern

or t

o an

swer

the

ques

tion

“How

do

I get

from

one

sta

ge in

the

patte

rn

to th

e ne

xt?”

Le

arne

rs n

eed

to h

ave

oppo

rtuni

ties

to s

ee th

at s

omet

imes

cha

ngin

g th

e fo

rm o

f rep

rese

ntat

ion

(geo

met

ric to

ver

bal o

r to

a flo

w d

iagr

am o

r to

a ta

ble)

can

hel

p th

em to

und

erst

and

the

patte

rn in

diff

eren

t way

s. L

earn

ers

shou

ld “t

rans

late

” the

se

geom

etric

seq

uenc

es in

to o

ther

form

s of

exp

ress

ion

or re

pres

enta

tion,

nam

ely

• ve

rbal

ly d

escr

ibe

the

patte

rn

• nu

mbe

r seq

uenc

es w

hich

can

als

o be

reco

rded

in a

tabl

e fo

rm.

Exa

mpl

e:

Exte

ndin

g th

e pa

ttern

:

Des

crib

ing

the

patte

rn in

ow

n w

ords

“It

is

a p

atte

rn o

f tria

ngle

s”

“ Eac

h tri

angl

e is

big

ger t

han

the

one

befo

re”

Des

crib

ing

how

they

mad

e th

e pa

ttern

or a

nsw

erin

g th

e qu

estio

n “h

ow to

I ge

t fro

m o

ne s

tage

to th

e ne

xt?”

“I

adde

d on

e m

ore

mat

chst

ick

to e

ach

side

of e

ach

trian

gle”

“E

ach

trian

gle

has

one

mor

e m

atch

stic

k in

eac

h si

de th

an th

e tri

angl

e on

its

left”

136 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Rec

ordi

ng th

e nu

mbe

r pat

tern

in a

tabl

e.

Whe

n le

arne

rs fi

ll in

the

tabl

e lik

e th

e on

e sh

own

belo

w, th

ey w

ill se

e th

at th

e nu

mbe

r of m

atch

stic

ks u

sed

for e

ach

trian

gle

is 3

tim

es th

e po

sitio

n of

the

trian

gle

in th

e se

quen

ce. T

hey

will

see

that

the

rule

is tr

iang

le n

umbe

r tim

es 3

. Lea

rner

s ca

n th

en b

e as

ked

to p

redi

ct h

ow m

any

mat

ches

they

will

use

for t

riang

les

they

hav

e no

t bui

lt, e

.g. 1

0th,

100

th e

tc.

G

rade

6:

Wha

t is

diffe

rent

? •

Ther

e is

mor

e em

phas

is o

n pr

esen

ting

patte

rns

in ta

bles

. •

Ther

e is

mor

e em

phas

is o

n st

atin

g th

e ge

nera

l rul

e of

the

patte

rn.

Wha

t kin

ds o

f geo

met

ric p

atte

rns

shou

ld le

arne

rs w

ork

with

? Le

arne

rs c

an w

ork

with

pat

tern

s w

hich

are

mad

e fro

m re

al s

hape

s, o

r obj

ects

con

cret

e ap

para

tus.

W

hat k

inds

of p

atte

rns

shou

ld le

arne

rs w

ork

with

? Pa

ttern

s in

whi

ch th

e sh

apes

gro

w o

r dec

reas

e in

diff

eren

t way

s.

LT

SM

R

esou

rces

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

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, mat

hs v

ideo

clip

s, E

lear

ning

reso

urce

s on

line,

Mat

hs g

ames

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 2

pp.

168

– 1

70

Activ

ities

138

-139

WB

2 p

p. 1

80 –

184

Act

iviti

es

140;

141

a&b;

101;

102

) W

B 2

pp.

98

– 10

2 Ac

tiviti

es 3

5,36

,37

DB

E Te

xtbo

ok re

fere

nce

Term

4: U

nit 1

0: G

eom

etric

Pat

tern

s, p

. 33

9 Te

rm 4

: Uni

t 9: G

eom

etric

Pat

tern

s,

p. 3

38

Term

2: U

nit 4

: Geo

met

ric P

atte

rns,

p.

152

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

137MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

138 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

6

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

……

T

ime:

8 H

OU

RS

GR

ADE

4

CAP

S p.

93,

115

and

116

5

CAP

S p.

184

- 18

5 6

CAP

S p.

263

- 26

5 TO

PIC

Pr

oper

ties

of 2

-D s

hape

s.

Tran

sfor

mat

ions

; Pos

ition

and

vie

ws;

Loc

atio

n an

d di

rect

ions

Con

cept

s, S

kills

and

kn

owle

dge

2 - D

sha

pes,

R

ecog

nize

, vis

ualiz

e an

d na

me

2-D

sh

apes

in

re

gula

r and

irre

gula

r pol

ygon

s –

trian

gles

, squ

ares

, rec

tang

les,

oth

er

quad

rilat

eral

s, p

enta

gons

, hex

agon

s an

d ci

rcle

s

Cha

ract

eris

tics

of s

hape

s

stra

ight

sid

es

cu

rved

sid

es

nu

mbe

r of s

ides

2 -

D s

hape

s,

Rec

ogni

ze, v

isua

lize

and

nam

e 2-

D

shap

es in

regu

lar a

nd ir

regu

lar p

olyg

ons

– tri

angl

es, s

quar

es, r

ecta

ngle

s, o

ther

qu

adril

ater

als,

pen

tago

ns, h

exag

ons,

he

ptag

ons

and

circ

les

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n sq

uare

s an

d re

ctan

gles

C

hara

cter

istic

s of

sha

pes

st

raig

ht s

ides

curv

ed s

ides

num

ber o

f sid

es

an

gles

in s

hape

D

raw

2-D

sha

pes

on g

rid p

aper

In

volv

ing

angl

es

2 - D

sha

pes,

R

ecog

nize

, vis

ualiz

e an

d na

me

2-D

sh

apes

in

re

gula

r and

irre

gula

r pol

ygon

s -

trian

gles

, sq

uare

s, re

ctan

gles

, par

alle

logr

ams,

ot

her

quad

rilat

eral

s, p

enta

gons

, hex

agon

s,

hept

agon

s, o

ctag

ons

si

mila

ritie

s an

d di

ffere

nces

bet

wee

n re

ctan

gles

and

par

alle

logr

ams

C

hara

cter

istic

s of

sha

pes

nu

mbe

r of s

ides

leng

ths

in s

ides

size

s of

ang

les

Dra

w 2

-D s

hape

s on

grid

pap

er

Invo

lvin

g an

gles

139MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Tran

sfor

mat

ions

Build

com

posi

te s

hape

s

Tess

ella

tions

Des

crib

e pa

ttern

s Po

sitio

n an

d vi

ews

Loca

tion

and

dire

ctio

ns

Rec

ogni

ze a

nd d

escr

ibe

angl

es in

2-D

sh

apes

Tr

ansf

orm

atio

ns

U

se tr

ansf

orm

atio

ns to

mak

e co

mpo

site

sha

pes

U

se tr

ansf

orm

atio

ns to

mak

e te

ssel

latio

ns

D

escr

ibe

patte

rns

Posi

tion

and

view

s Lo

catio

n an

d di

rect

ions

Rec

ogni

ze a

nd d

escr

ibe

angl

es in

2-D

sh

apes

Tr

ansf

orm

atio

ns

U

se tr

ansf

orm

atio

ns to

mak

e co

mpo

site

sha

pes

E

nlar

gem

ent a

nd re

duct

ions

Des

crib

e pa

ttern

s Po

sitio

n an

d vi

ews

Loca

tion

and

dire

ctio

ns

SU

GG

ESTE

D

MET

HO

DO

LOG

Y Pr

oper

ties

of 2

D s

hape

s Se

e Te

rm 1

for s

ugge

sted

teac

hing

met

hodo

logy

. Le

arne

rs re

vise

and

con

solid

ate

wha

t the

y le

arne

d in

Ter

m 1

. Gra

de 6

lear

ners

dra

w c

ircle

s an

d pa

ttern

s w

ith c

ircle

s us

ing

a pa

ir of

pai

r of c

ompa

sses

. Vi

ewin

g ob

ject

s Le

arne

rs w

ork

with

sid

e vi

ews,

pla

n vi

ews

and

top

view

s of

sim

ple

sing

le o

bjec

ts s

uch

as a

cup

, hat

, sho

e, b

ox o

r app

le. T

hey

also

wor

k w

ith s

ide

view

s an

d pl

an v

iew

s of

a c

lass

room

, sim

ple

build

ings

, and

sch

ool f

ield

s.

Posi

tion

/ Loc

atio

n an

d di

rect

ion

This

link

s w

ith th

e w

ork

done

in G

eogr

aphy

in M

ap S

kills

. Cel

ls in

a g

rid a

re o

ften

labe

lled

with

a le

tter a

nd a

num

ber e

.g. D

4;

A3; E

7. T

his

is c

alle

d al

pha-

num

eric

refe

renc

ing.

The

ski

lls o

f ide

ntify

ing

ever

yday

obj

ects

and

col

lect

ions

of o

bjec

ts, a

nd th

e sk

ills d

escr

ibed

bel

ow c

an b

e de

velo

ped

in th

e G

eogr

aphy

less

on a

nd p

ract

ised

in th

e M

athe

mat

ics

less

on.

Le

arne

rs w

ork

with

alp

ha-n

umer

ic g

rid re

fere

nces

on

grid

s an

d m

aps.

Loc

ate

obje

cts

usin

g th

e gr

id re

fere

nces

. W

hen

lear

ners

wor

k w

ith g

rid re

fere

nces

they

nee

d to

lear

n

to fi

nd th

e ce

ll i.e

. to

answ

er q

uest

ions

like

“Wha

t is

in c

ell B

3?”

in

whi

ch c

ell a

n ob

ject

is i.

e. to

ans

wer

que

stio

ns li

ke “W

here

is th

e co

w?”

Tr

ansf

orm

atio

ns

Gra

de 4

lear

ners

sho

uld

focu

s on

:

build

ing

com

posi

te s

hape

s in

clud

ing

som

e sh

apes

with

line

sym

met

ry.

140 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

te

ssel

latio

ns a

nd d

escr

ibin

g pa

ttern

s in

real

life

. Te

ssel

latio

ns

Le

arne

rs u

se 2

-D s

hape

s to

cre

ate

tess

ella

tion

patte

rns.

Tilin

g pa

ttern

s ca

n be

mad

e by

pac

king

out

the

tiles

.

Le

arne

rs n

eed

to id

entif

y an

d de

scrib

e te

ssel

latio

n pa

ttern

s.

Gra

de 4

lear

ners

are

not

requ

ired

to c

reat

e th

e pa

ttern

s by

rota

ting,

tran

slat

ing

or re

flect

ing

a si

ngle

sha

pe.

Des

crib

e pa

ttern

s Le

arne

rs d

escr

ibe

patte

rns

by ta

lkin

g ab

out t

he s

hape

s th

ey s

ee in

the

patte

rn. e

.g.

th

e pa

ttern

I se

e on

the

cran

e is

mad

e of

stra

ight

line

s

the

patte

rn w

e se

e on

the

hone

ycom

b lo

oks

like

a te

ssel

latio

n pa

ttern

of h

exag

ons

th

e pa

ttern

I se

e on

the

bead

bra

cele

t loo

ks li

ke a

tess

ella

tion

patte

rn o

f tria

ngle

s Le

arne

rs d

escr

ibe

patte

rns

by d

iscu

ssin

g th

e sy

mm

etry

of s

hape

s e.

g. th

e bu

tterfl

y’s

win

gs m

ake

a sy

mm

etric

al p

atte

rn.

Lear

ners

ofte

n fin

d pa

ttern

s ea

sier

to d

escr

ibe,

onc

e th

ey h

ave

copi

ed o

r mad

e th

e pa

ttern

s. It

is u

sefu

l to

link

the

proc

ess

of

mak

ing

or c

opyi

ng p

atte

rns

with

the

desc

riptio

ns o

f pat

tern

s fro

m n

atur

e, m

oder

n ev

eryd

ay li

fe a

nd o

ur c

ultu

ral h

erita

ge. O

ften

the

geom

etric

al p

roce

ss y

ou u

se to

mak

e a

copy

of t

he p

atte

rn is

not

the

sam

e as

the

orig

inal

pro

cess

use

d to

mak

e th

e pa

ttern

. In

Gra

de 5

lear

ners

wor

k w

ith v

iew

s of

sin

gle

ever

yday

obj

ects

or c

olle

ctio

ns o

f eve

ryda

y ob

ject

s. T

hey

mat

ch v

iew

s of

the

obje

ct o

r obj

ects

with

the

posi

tion

of th

e vi

ewer

. Lea

rner

s ar

e pr

esen

ted

with

mul

tiple

vie

ws

of a

n ev

eryd

ay o

bjec

t or c

olle

ctio

n of

ev

eryd

ay o

bjec

ts o

r sce

nes

from

eve

ryda

y lif

e, a

s w

ell a

s po

sitio

ns o

f vie

wer

s in

rela

tion

to th

e ob

ject

or o

bjec

ts. T

hey

mat

ch

each

vie

w w

ith a

vie

wer

or v

iew

poin

t. Ev

eryd

ay o

bjec

ts o

ften

have

mor

e irr

egul

ar s

urfa

ces

than

geo

met

ric o

bjec

ts e

.g. c

ompa

re

a te

apot

to a

sph

ere

or a

per

son

to a

cub

e. T

his

mak

es it

eas

ier t

o id

entif

y vi

ews

and

view

poin

ts o

f eve

ryda

y ob

ject

s.

In G

rade

4 le

arne

rs c

reat

e co

mpo

site

sha

pes

by p

laci

ng 2

-D s

hape

s ne

xt to

eac

h ot

her.

In G

rade

5 le

arne

rs tr

ace

and

mov

e a

2-D

sha

pe u

sing

refle

ctio

ns, r

otat

ions

and

tran

slat

ions

to d

raw

com

posi

te s

hape

s.

In

Gra

de 4

lear

ners

cre

ate

tess

ella

tions

by

pack

ing

out s

hape

s. In

Gra

de 5

lear

ners

trac

e an

d m

ove

a 2-

D s

hape

usi

ng

refle

ctio

ns, r

otat

ions

and

tran

slat

ions

to d

raw

tess

ella

tions

. Le

arne

rs u

se a

2-D

sha

pe a

s a

tem

plat

e w

hich

they

trac

e an

d m

ove

by re

flect

ing,

tran

slat

ing

and

rota

ting

to c

reat

e co

mpo

site

sh

apes

. Som

e of

the

new

sha

pes

draw

n sh

ould

hav

e lin

es o

f sym

met

ry. L

earn

ers

desc

ribe

how

they

mov

ed th

e sh

ape

to c

reat

e th

e pa

ttern

usi

ng th

e w

ords

“rot

atio

n, tr

ansl

atio

n an

d re

flect

ion”

In

Gra

de 6

vie

win

g an

d po

sitio

n is

ext

ende

d to

geo

met

ric o

bjec

ts o

r col

lect

ions

of g

eom

etric

obj

ects

or c

ompo

site

geo

met

ric

obje

cts.

Lea

rner

s ar

e pr

esen

ted

with

mul

tiple

vie

ws

of a

n ev

eryd

ay o

r geo

met

ric o

bjec

t or c

olle

ctio

ns o

f obj

ects

or c

ompo

site

141MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

geom

etric

obj

ects

, as

wel

l as

posi

tions

of v

iew

ers

in re

latio

n to

the

obje

ct o

r obj

ects

. The

y m

atch

eac

h vi

ew w

ith a

vie

wer

or

view

poin

t. In

Gra

de 6

, lea

rner

s ar

e on

ly re

quire

d to

use

the

trans

form

atio

n co

ncep

ts in

des

crib

ing

patte

rns.

Lea

rner

s de

scrib

e pa

ttern

s by

di

scus

sing

the

shap

es th

ey s

ee in

the

patte

rn a

nd h

ow th

ey w

ould

tran

sfor

m th

at s

hape

if th

ey w

ante

d to

ext

end

the

patte

rn

Th

e pa

ttern

I se

e on

the

hone

ycom

b lo

oks

like

a te

ssel

latio

n pa

ttern

of h

exag

ons.

I ca

n m

ake

this

pat

tern

by

trans

latin

g th

e he

xago

n.

Th

e pa

ttern

I se

e on

the

bead

bra

cele

t loo

ks li

ke a

tess

ella

tion

patte

rn o

f tria

ngle

s. I

can

mak

e th

is p

atte

rn b

y re

flect

ing

the

trian

gle.

I can

mak

e a

patte

rn li

ke th

e on

e I s

ee o

n th

e do

ily b

y tra

nsla

ting

the

para

llelo

gram

. Le

arne

rs id

entif

y sy

mm

etry

in p

atte

rns.

Alth

ough

lear

ners

are

not

requ

ired

to d

raw

the

patte

rns

in G

rade

6, t

hey

ofte

n fin

d pa

ttern

s ea

sier

to d

escr

ibe,

onc

e th

ey h

ave

copi

ed o

r mad

e th

e pa

ttern

s. It

is u

sefu

l to

link

the

proc

ess

of m

akin

g or

cop

ying

pa

ttern

s w

ith th

e de

scrip

tions

of p

atte

rns

from

nat

ure,

mod

ern

ever

yday

life

and

our

cul

tura

l her

itage

. Ofte

n th

e ge

omet

rical

pr

oces

s yo

u us

e to

mak

e a

copy

of t

he p

atte

rn is

not

the

sam

e as

the

orig

inal

pro

cess

use

d to

mak

e th

e pa

ttern

. Bee

s do

not

te

ssel

late

with

hex

agon

s to

mak

e a

hone

ycom

b, b

ut if

lear

ners

tess

ella

te w

ith a

hex

agon

, the

y ca

n m

ake

a pa

ttern

that

look

s si

mila

r to

the

patte

rn th

ey s

ee in

the

hone

ycom

b.

Gra

de 6

lear

ners

als

o do

enl

arge

men

ts a

nd re

duct

ions

.

LTSM

R

esou

rces

Va

riety

of r

egul

ar a

nd ir

regu

lar p

olyg

ons:

( To

cut

out

) - t

riang

les,

squ

ares

, rec

tang

les,

par

alle

logr

ams,

oth

er q

uadr

ilate

rals

, pen

tago

ns, h

exag

ons,

hep

tago

ns, o

ctag

ons

- Tan

gram

s pa

ir of

com

pass

es, g

eobo

ards

Gra

de 4

G

rade

5

Gra

de 6

WB

2 p

p. 4

2 –

50 A

ctiv

ities

82

-85b

pp

. 158

– 1

62

Activ

ities

133

-135

)

WB

2 p

p. 5

0 –

66

Activ

ities

86-

93;

pp. 1

66 –

178

Ac

tiviti

es 1

33-1

39

WB

2 p

p. 4

6 –

48 A

ctiv

ities

79

a&b;

83;

84a

&b

DB

E Te

xtbo

ok re

fere

nce

Term

3: U

nit 6

: Pro

pert

ies

of tw

o di

men

sion

al s

hape

s, p

. 355

Te

rm 3

: Uni

t 12:

Tra

nsfo

rmat

ion,

p. 2

86

Term

4: U

nit 9

: Tra

nsfo

rmat

ion,

p. 3

33

Term

3: U

nit 5

: Vie

win

g ob

ject

s, p

. 250

Te

rm 4

: Uni

t 8: P

ositi

on a

nd m

ovem

ent,

p.

329

Term

3: U

nit 6

: Pro

pert

ies

of tw

o di

men

sion

al s

hape

s, p

. 236

Te

rm 3

: Uni

t 7: T

rans

form

atio

n, p

. 24

2 Te

rm 4

: Uni

t 8: T

rans

form

atio

n, p

. 33

3 Te

rm 3

: Uni

t 5: V

iew

ing

obje

cts,

p.

233

Term

3: U

nit 5

: Pro

pert

ies

of tw

o di

men

sion

al s

hape

s, p

. 237

Te

rm 3

: Uni

t 6: T

rans

form

atio

n, p

. 24

7 Te

rm 4

: Uni

t 9: T

rans

form

atio

n, p

. 35

0 Te

rm 3

: Uni

t 4: V

iew

ing

obje

cts,

p.

231

142 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Term

4: U

nit 7

: Pos

ition

and

m

ovem

ent,

p. 3

31

Term

4: U

nit 1

0: P

ositi

on a

nd

mov

emen

t, p

. 358

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –T

est

143MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

7

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

….…

T

ime:

4 H

OU

RS

GR

ADE

4 M

ake

use

of le

sson

pla

ns o

f Ter

m 2

5

M

ake

use

of le

sson

pla

ns o

f Ter

m 2

6

M

ake

use

of le

sson

pla

ns o

f Ter

m 2

TO

PIC

Le

ngth

, Mas

s, C

apac

ity a

nd V

olum

e C

once

pts,

Ski

lls a

nd

know

ledg

e Le

ngth

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

obj

ects

Mea

surin

g in

stru

men

ts

U

nits

Cal

cula

tions

and

pro

blem

sol

ving

in

volv

ing

leng

th

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D o

bjec

ts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng

invo

lvin

g le

ngth

Leng

th

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D o

bjec

ts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng

invo

lvin

g le

ngth

C

once

pts,

Ski

lls a

nd

know

ledg

e M

ass

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D o

bjec

ts

M

easu

ring

inst

rum

ents

Uni

ts ((

milli

litre

s, li

tres)

Cal

cula

tions

and

pro

blem

sol

ving

in

volv

ing

mas

s

Mas

s

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

obj

ects

Mea

surin

g in

stru

men

ts

U

nits

(milli

litre

s, li

tres)

Cal

cula

tions

and

pro

blem

sol

ving

in

volv

ing

mas

s

Mas

s

Prac

tical

mea

surin

g of

2-D

sha

pes

and

3-D

obj

ects

Mea

surin

g in

stru

men

ts

U

nits

(milli

litre

s, li

tres,

kilo

litre

s)

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng

invo

lvin

g m

ass

(incl

ude

fract

ion

and

deci

mal

form

s to

2 d

ecim

al p

lace

s)

Con

cept

s, S

kills

and

kn

owle

dge

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D o

bjec

ts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng

invo

lvin

g ca

paci

ty/v

olum

e

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D o

bjec

ts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng

invo

lvin

g ca

paci

ty/v

olum

e

Cap

acity

/Vol

ume

Pr

actic

al m

easu

ring

of 2

-D s

hape

s an

d 3-

D o

bjec

ts

M

easu

ring

inst

rum

ents

Uni

ts

C

alcu

latio

ns a

nd p

robl

em s

olvi

ng

invo

lvin

g ca

paci

ty/v

olum

e

144 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LTSM

Res

ourc

es

Mea

surin

g ta

pe, r

uler

s, s

trin

g, tr

undl

e w

heel

, Kitc

hen

scal

e, b

athr

oom

sca

le, b

alan

ce, M

easu

ring

jugs

, bot

tles,

var

iety

of

con

tain

ers

from

lear

ner’s

con

text

, cub

es

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 1

pp.

110

– 1

14

Activ

ities

40

– 42

M

ass

WB

2 p

p. 1

12 -

122

Ac

tiviti

es

111-

115

Volu

me/

Cap

acity

WB

2 p

p. 1

52 –

156

Ac

tiviti

es 1

30-1

32

WB

1 p

p 11

6 - 1

26

Activ

ities

40

– 43

M

ass

WB

2 p

p. 2

0 - 2

6 W

B 2

Act

iviti

es

74 -

76; 7

8 Vo

lum

e/ca

paci

ty W

B 2

pp.

160

– 1

64

Activ

ities

130

- 13

2

Leng

th W

B 1

pp.

102

- 10

6 Ac

tiviti

es

102

- 104

M

ass

WB

1 p

p. 1

0 –

12 A

ctiv

ities

68

a&b

Volu

me/

cap

acity

WB

1 p

p. 1

64 –

166

Ac

tiviti

es 6

4a&b

DB

E Te

xtbo

ok re

fere

nce

Term

2: U

nit 4

: Len

gth,

p. 1

47

Term

4: U

nit 3

: Mas

s, p

. 297

Te

rm 3

: Uni

t 2: C

apac

ity a

nd v

olum

e,

p. 2

25

Term

2: U

nit 4

: Len

gth,

p. 1

43

Term

3: U

nit 2

: Mas

s, p

. 211

Te

rm 1

: Uni

t 9: C

apac

ity a

nd v

olum

e,

p. 1

04

Term

3: U

nit 1

1: L

engt

h, p

. 285

Te

rm 3

: Uni

t 1: M

ass,

p. 2

15

Term

2: U

nit 8

: Cap

acity

and

vol

ume,

p.

200

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent –

Te

st

145MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

8

TER

M 3

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

….…

T

ime:

12

HO

UR

S G

RAD

E 4

Com

mon

Fra

ctio

ns

CAP

S p.

91

5 C

omm

on fr

actio

ns

CAP

S p.

160-

162,

176

-177

& 1

99

6 C

omm

on fr

actio

ns

CAP

S p.

226

-227

, 280

D

ecim

al fr

actio

ns -

CAP

S p.

252

, Pe

rcen

tage

s -

CAP

S p.

267

TO

PIC

C

omm

on fr

actio

ns, d

ecim

al fr

actio

ns a

nd p

erce

ntag

es

146 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Con

cept

s, S

kills

and

kn

owle

dge

Des

crib

ing

and

orde

ring

frac

tions

Com

pare

and

ord

er c

omm

on

fract

ions

with

diff

eren

t den

omin

ator

s (h

alve

s; th

irds,

qua

rters

; fift

hs;

sixt

hs; s

even

ths;

eig

hths

)

Des

crib

e an

d co

mpa

re c

omm

on

fract

ions

in d

iagr

am fo

rm

Cal

cula

tions

with

frac

tions

:

Addi

tion

of c

omm

on fr

actio

ns w

ith

the

sam

e de

nom

inat

ors

Rec

ogni

ze, d

escr

ibe

and

use

the

equi

vale

nce

of d

ivis

ion

and

fract

ions

So

lvin

g pr

oble

ms

•

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g fra

ctio

ns, i

nclu

ding

gr

oupi

ng a

nd e

qual

sha

ring

Equi

vale

nt fo

rms:

R

ecog

nize

and

use

equ

ival

ent f

orm

s of

co

mm

on fr

actio

ns (f

ract

ions

in w

hich

one

de

nom

inat

or is

a m

ultip

le o

f ano

ther

)

Des

crib

ing

and

orde

ring

frac

tions

:

Cou

nt fo

rwar

ds a

nd b

ackw

ards

in

fract

ions

Com

pare

and

ord

er c

omm

on

fract

ions

to a

t lea

st tw

elfth

s C

alcu

latio

ns w

ith fr

actio

ns:

•

Addi

tion

and

subt

ract

ion

of c

omm

on

fract

ions

with

the

sam

e de

nom

inat

ors

•

Addi

tion

and

subt

ract

ion

of m

ixed

nu

mbe

rs

•

Frac

tions

of w

hole

num

bers

whi

ch

resu

lt in

who

le n

umbe

rs

•

Rec

ogni

ze, d

escr

ibe

and

use

the

equi

vale

nce

of d

ivis

ion

and

fract

ions

So

lvin

g pr

oble

ms

•

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, i

nclu

ding

gr

oupi

ng a

nd s

harin

g Eq

uiva

lent

form

s:

Rec

ogni

ze a

nd u

se e

quiv

alen

t for

ms

of

com

mon

frac

tions

(fra

ctio

ns in

whi

ch o

ne

deno

min

ator

is a

mul

tiple

of a

noth

er)

Des

crib

ing

and

orde

ring

frac

tions

: •

Com

pare

and

ord

er c

omm

on

fract

ions

, inc

ludi

ng te

nths

and

hu

ndre

dths

C

alcu

latio

ns w

ith fr

actio

ns:

•

Addi

tion

and

subt

ract

ion

of c

omm

on

fract

ions

in w

hich

one

den

omin

ator

is

a m

ultip

le o

f ano

ther

•

Addi

tion

and

subt

ract

ion

of m

ixed

nu

mbe

rs

•

Frac

tions

of w

hole

num

bers

So

lvin

g pr

oble

ms

•

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g co

mm

on fr

actio

ns, i

nclu

ding

gr

oupi

ng a

nd s

harin

g Pe

rcen

tage

s •

Find

per

cent

ages

of w

hole

num

bers

Eq

uiva

lent

form

s:

•

Rec

ogni

ze a

nd u

se e

quiv

alen

t for

ms

of c

omm

on fr

actio

ns w

ith 1

-dig

it or

2-

digi

t den

omin

ator

s (fr

actio

ns in

whi

ch

one

deno

min

ator

is a

mul

tiple

of

anot

her)

•

Rec

ogni

ze e

quiv

alen

ce b

etw

een

147MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

com

mon

frac

tion

and

deci

mal

fra

ctio

n fo

rms

of th

e sa

me

num

ber

•

Rec

ogni

ze e

quiv

alen

ce b

etw

een

com

mon

frac

tion,

dec

imal

frac

tion

and

perc

enta

ge fo

rms

of th

e sa

me

num

ber

Rec

ogni

zing

, ord

erin

g an

d pl

ace

valu

e of

dec

imal

frac

tions

•

Cou

nt fo

rwar

ds a

nd b

ackw

ards

in

deci

mal

frac

tions

to a

t lea

st tw

o de

cim

al p

lace

s •

Com

pare

and

ord

er d

ecim

al fr

actio

ns

to a

t lea

st tw

o de

cim

al p

lace

s •

Plac

e va

lue

of d

igits

to a

t lea

st tw

o de

cim

al p

lace

s C

alcu

latio

ns w

ith d

ecim

al fr

actio

ns

•

Addi

tion

and

subt

ract

ion

of d

ecim

al

fract

ions

with

at l

east

two

deci

mal

pl

aces

•

Mul

tiply

dec

imal

frac

tions

by

10 a

nd

100

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

in c

onte

xt in

volv

ing

deci

mal

frac

tions

C

alcu

latio

ns

•

Find

per

cent

ages

of w

hole

num

bers

. SU

GG

ESTE

D

MET

HO

DO

LOG

Y R

efer

to th

e Le

sson

pla

ns in

Ter

m 2

148 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

In G

rade

4 th

is is

revi

sion

and

con

solid

atio

n of

the

conc

epts

dev

elop

ed in

Ter

m 3

. (Se

e Te

rm 3

not

es)

In T

erm

4 le

ngth

, cap

acity

and

mas

s ca

n be

use

d as

con

text

s fo

r fra

ctio

ns.

Teac

hing

frac

tions

in d

iagr

am fo

rms

shou

ld in

clud

e: re

gion

mod

els

, len

gth

mod

els

(incl

udin

g nu

mbe

r lin

es) a

nd s

et m

odel

s.

Lear

ners

sho

uld

solv

e pr

oble

ms

as w

ell a

s w

ork

with

app

arat

us to

und

erst

and

the

rela

tions

hip

betw

een

fract

ions

and

div

isio

n i.e

. if

you

shar

e 1

amon

gst 3

lear

ners

you

will

be m

akin

g th

irds.

Le

arne

rs m

ust a

ble

to n

ame

fract

ions

. Ter

min

olog

y lik

e “3

ove

r 4” s

houl

d be

avo

ided

as

it te

nds

to e

ncou

rage

lear

ners

to th

ink

abou

t eac

h fra

ctio

n as

two

diffe

rent

num

bers

, rat

her t

han

3 4 bei

ng a

num

ber w

hich

is g

reat

er 1 2 t

han

but l

ess

than

1.

Whe

n na

min

g fra

ctio

n pa

rts it

is u

sefu

l for

lear

ners

to ra

ther

use

the

form

“3 q

uarte

rs”.

Lear

ners

sho

uld,

thro

ugh

wor

k w

ith a

ppar

atus

, dia

gram

s an

d so

lvin

g pr

oble

ms,

lear

n th

e ne

w fr

actio

ns th

at th

ey w

ill de

al w

ith in

G

rade

4

Teac

h fr

actio

ns a

s eq

ual p

arts

of a

who

le

Exam

ple

1 •

Div

ide

a ba

nana

une

venl

y an

d as

k tw

o le

arne

rs w

heth

er th

ey w

ould

be

satis

fied

with

this

div

isio

n. W

hy n

ot?

• Ex

plai

n th

at fr

actio

ns a

re fo

rmed

with

the

divi

ding

of a

who

le in

to E

QU

AL P

ARTS

. Ex

ampl

e 2

Giv

e di

ffere

nt m

athe

mat

ical

sha

pes

to le

arne

rs –

circ

les

/ rec

tang

les

/ squ

ares

/ et

c.

• W

ork

prac

tical

ly –

div

ide

one

(circ

le) e

qual

ly b

etw

een

2 –

then

4 –

then

8 le

arne

rs

• Ex

plai

n w

hat w

e ca

ll th

ese

fract

ions

and

wha

t the

not

atio

n lo

oks

like

1 2 ; 1 4 (

at fi

rst o

nly)

Ex

ampl

e 3

Div

ide

a ch

ocol

ate

(rect

angl

e) e

qual

ly b

etw

een

3 –

then

6 le

arne

rs.

Exam

ple

4 D

ivid

e a

cake

(squ

are)

equ

ally

bet

wee

n 5

– th

en 1

0 le

arne

rs.

Expl

ain

the

nota

tion

ever

y tim

e in

the

abov

e ex

ampl

es.

A w

ritte

n or

ora

l exe

rcis

e ca

n be

don

e at

this

poi

nt. T

he fo

llow

ing

can

be u

sed:

Th

e te

ache

r sho

uld

take

the

lead

and

ask

que

stio

ns. A

sk th

e le

arne

rs to

writ

e th

e fra

ctio

n no

tatio

n on

the

boar

d fo

r the

who

le

clas

s to

see

.

149MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Addi

tion

of fr

actio

ns:

•

Wor

k w

ith c

oncr

ete

obje

cts:

•

Wor

k w

ith d

iagr

ams.

Sha

de o

ne p

art i

n on

e co

lour

. Sha

de a

noth

er p

art i

n a

diffe

rent

col

our.

Whi

ch fr

actio

n is

sha

ded

in

tota

l?

•

Wor

k w

ith fr

actio

n w

all:

(Sta

rt w

ith s

impl

e fra

ctio

ns)

Star

t with

the

sam

e de

nom

inat

or, e

.g.,

1 4 + 1 4

If le

arne

rs c

an d

o th

e ad

ditio

n us

ing

conc

rete

exa

mpl

es, t

hey

can

then

pro

gres

s to

writ

ing

the

fract

ion

nota

tion

belo

w th

e co

ncre

te e

xam

ple.

F

ollo

w th

ese

step

s, e

.g.

Step

1: M

ake

sure

the

botto

m n

umbe

rs (t

he d

enom

inat

ors)

are

the

sam

e St

ep 2

: Add

the

top

num

bers

(the

num

erat

ors)

. Put

the

answ

er o

ver t

he s

ame

deno

min

ator

as

in s

tep

1 St

ep 3

: Sim

plify

the

fract

ion

(if n

eede

d).

DEC

IMAL

S

Impo

rtan

t: R

efer

to th

e Le

sson

Pla

ns o

n D

ecim

al F

ract

ions

in T

erm

2

Con

vert

from

frac

tions

to d

ecim

al fr

actio

ns a

nd to

per

cent

ages

. Onl

y ta

ught

in G

rade

6.

PER

CEN

TAG

ES

Perc

enta

ges

are

a ne

w to

pic

for G

rade

6 le

arne

rs.

Lear

ners

hav

e al

read

y w

orke

d w

ith t

enth

s an

d hu

ndre

dths

in

com

mon

fra

ctio

n fo

rm.

They

sho

uld

star

t by

rew

ritin

g an

d co

nver

ting

tent

hs a

nd h

undr

edth

s in

com

mon

frac

tion

form

to p

erce

ntag

es. W

here

den

omin

ator

s of

oth

er fr

actio

ns a

re fa

ctor

s of

10,

e.g

., 2,

5 o

r fac

tors

of 1

00 e

.g. 2

, 4, 5

, 20,

25,

50

lear

ners

can

con

vert

thes

e to

hun

dred

ths

usin

g w

hat t

hey

know

abo

ut

equi

vale

nce.

Eq

uiva

lenc

e be

twee

n co

mm

on fr

actio

ns a

nd p

erce

ntag

e Le

arne

rs a

re n

ot e

xpec

ted

to b

e ab

le to

con

vert

any

com

mon

frac

tion

into

its

perc

enta

ge fo

rm, m

erel

y to

see

the

rela

tions

hip

betw

een

tent

hs a

nd h

undr

edth

s in

thei

r per

cent

age

form

. Lea

rner

s sh

ould

be

able

to c

onve

rt an

y de

cim

al fr

actio

n in

tent

hs o

r hu

ndre

dths

into

a p

erce

ntag

e.

150 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Cal

cula

tions

Le

arne

rs s

houl

d be

abl

e to

find

per

cent

ages

of w

hole

num

ber,s

e.g

., W

hat i

s 25

% o

f R30

0? H

ere

lear

ners

use

wha

t the

y kn

ow

abou

t bot

h co

nver

ting

betw

een

perc

enta

ge a

nd c

omm

on fr

actio

n fo

rm a

nd a

lso

wha

t the

y kn

ow a

bout

find

ing

fract

ions

of w

hole

nu

mbe

rs e

.g. 2

5% o

f R30

= 1 4

of R

300

= R

75.

LTSM

R

esou

rces

N

umbe

r lin

es: 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,

frac

tion

wal

ls,

fract

ion

circ

les,

diff

eren

t mod

els

of fr

actio

ns, f

ract

ion

strip

s,

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 1

pp.

34

– 39

Act

iviti

es 9

8 - 1

08

WB

2 p

p. 1

28 –

138

Act

iviti

es 1

18 -

123

WB

2 p

p. 1

16 –

132

Ac

tiviti

es 1

12 -

118

WB

2 p

p 13

6 –

148

Activ

ities

115

-120

a&b

DB

E Te

xtbo

ok re

fere

nce

Term

3: U

nit 1

: Com

mon

Fra

ctio

ns, p

. 21

1

Term

3: U

nit 1

: Com

mon

Fra

ctio

ns, p

. 19

9

Term

4: U

nit 3

: Com

mon

Fra

ctio

ns, p

. 30

6 Te

rm 2

: Uni

t 7: D

ecim

als,

p. 1

77

Term

3: U

nit 8

: Per

cent

ages

, p. 2

62

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

151MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LESS

ON

PLA

NS:

TER

M 4

152 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

MU

LTI G

RAD

E LE

SSO

N P

LAN

1

TER

M 4

EDU

CAT

OR

: ……

……

……

……

……

DAT

E: fr

om

……

…to

……

.

T

ime:

3 H

OU

RS

GR

ADE

4 C

APS

p.10

4 (m

etho

dolo

gy e

xpla

ined

on

pp.

37

- 38)

5 C

APS

p.19

4 (m

etho

dolo

gy e

xpla

ined

on

pp.

125

-126

)

6 C

APS

p. 2

76 (m

etho

dolo

gy e

xpla

ined

on

pp.

215

– 2

16)

TOPI

C

Who

le n

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

C

ount

forw

ards

and

bac

kwar

ds in

2s,

3s

, 5s,

10s

, 25s

, 50s

, 100

s be

twee

n 0

and

at le

ast 1

0 00

0

O

rder

, com

pare

and

repr

esen

t

nu

mbe

rs to

at l

east

4 d

igit

num

bers

.

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its in

w

hole

num

bers

to a

t lea

st 4

-dig

it nu

mbe

rs

R

ound

off

to th

e ne

ares

t 10,

100

, 10

00.

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

6-d

igit

num

bers

Rep

rese

nt o

dd a

nd e

ven

num

bers

to

at le

ast 1

000

.

Rec

ogni

ze th

e pl

ace

valu

e of

dig

its in

w

hole

num

bers

to a

t lea

st 6

dig

it nu

mbe

rs.

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00 ,

1 00

0 an

d 10

000

C

ount

forw

ards

and

bac

kwar

ds in

w

hole

num

ber i

nter

vals

up

to a

t lea

st

10 0

00

O

rder

, com

pare

and

repr

esen

t nu

mbe

rs to

at l

east

9-d

igit

num

bers

Rep

rese

nt p

rime

num

bers

to a

t lea

st

100

R

ecog

nizi

ng th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to a

t lea

st 9

-dig

it nu

mbe

rs

R

ound

off

to th

e ne

ares

t 5, 1

0, 1

00,

1 00

0, 1

0 00

0, 1

00 0

00, a

nd

1 00

0 00

0 SU

GG

ESTE

D

MET

HO

DO

LOG

Y R

efer

bac

k to

Ter

m 1

Ex

plai

n fr

om k

now

n (lo

w n

umbe

r ran

ge) t

o un

know

n (h

ighe

r ran

ge o

f num

bers

); w

ork

with

con

cret

e LT

SM

Cou

ntin

g, o

rder

ing,

com

parin

g, re

pres

entin

g of

dig

its

Not

e: A

ccor

ding

to C

APS

num

ber r

ange

per

gra

de p

er te

rm a

s in

con

tent

cla

rific

atio

n

R

ead,

say

and

writ

e up

to a

t lea

st 4

-dig

it nu

mbe

rs (G

rade

4) ,

6-d

igit

(Gra

de 5

) and

9-d

igit

(Gra

de 6

)

153MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

(Lea

rner

s ca

n w

rite

num

bers

with

fing

ers

in th

e ai

r whi

le th

ey a

re re

adin

g / s

ayin

g th

e w

ords

.) e.

g. O

ne h

undr

ed a

nd

twen

ty-n

ine

C

onve

rt fro

m w

ords

to n

umbe

rs a

nd n

umbe

rs to

wor

ds

Ar

rang

e th

e nu

mbe

rs b

elow

from

the

smal

lest

to th

e bi

gges

t:

Exam

ple:

10

111

; 1 1

01; 1

110

; 1 0

11

M

ake

the

bigg

est/s

mal

lest

num

ber y

ou c

an w

ith th

ese

digi

ts: 3

, 2, 5

, 4, 9

. O

dd a

nd e

ven

num

bers

: Ex

plor

e pr

actic

ally

by

wor

king

with

num

bers

to b

e ab

le to

des

crib

e th

e pa

ttern

e.g

. all

odd

num

bers

end

in 1

, 3, 5

, 7 o

r 9

ther

efor

e 2

201

, 2 2

03, 2

205

, 2 2

07, 2

209

are

all

odd

num

bers

.

Prim

e nu

mbe

rs fo

r Gra

de 6

Id

entif

y pr

ime

num

bers

and

com

posi

te n

umbe

rs u

p to

at l

east

100

M

ultip

les:

I c

ount

in m

ultip

les

of 3

, up

to 1

00.

W

ill I c

ount

the

num

ber 1

2? W

hy?

W

ill I c

ount

the

num

ber 4

2? W

hy?

Plac

e va

lue:

Dis

tingu

ish

betw

een

the

num

eric

val

ue a

nd th

e pl

ace

valu

e.

U

se p

lace

val

ue c

ards

to b

uild

up

and

brea

k do

wn

num

bers

R

ound

ing

off:

R

ound

off

as in

dica

ted

per g

rade

(use

bas

e te

n bl

ocks

, num

ber l

ines

, pla

ce v

alue

tabl

e)

154 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

U

se c

once

pt o

f rou

ndin

g of

f, in

pro

blem

sol

ving

con

text

LTSM

R

esou

rces

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

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, mul

tiplic

atio

n ta

ble

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

WB

2 p

. 100

Act

iviti

es 1

05

WB

2 p

. 32

– 38

; p. 9

6 Ac

tiviti

es 7

9 - 8

1; 1

05)

WB

2 p

p. 1

08 –

110

Ac

tiviti

es 1

05a&

b

DB

E Te

xtbo

ok re

fere

nce

Term

4: U

nit 1

: Who

le N

umbe

rs, p

. 291

Term

4: U

nit 1

: Who

le N

umbe

rs, p

. 28

3

Term

4: U

nit 1

: Who

le N

umbe

rs, p

. 29

7

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

test

155MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 2

TE

RM

4

EDU

CAT

OR

: ……

……

……

……

……

DAT

E : f

rom

…

……

to …

……

T

ime:

6 H

OU

RS

GR

ADE

4 C

APS

p.93

(met

hodo

logy

exp

lain

ed o

n pp

. 43

- 45)

5 C

APS

pp 1

82- 1

83 (m

etho

dolo

gy

expl

aine

d on

pp.

132

– 1

35)

6 C

APS

pp.2

62-2

63 (m

etho

dolo

gy

expl

aine

d on

pp.

222

– 2

25)

TOPI

C

Addi

tion

and

subt

ract

ion

of w

hole

num

bers

C

once

pts,

Ski

lls a

nd

know

ledg

e N

umbe

r ran

ge fo

r cal

cula

tions

Ad

ditio

n an

d su

btra

ctio

n of

who

le

num

bers

of a

t lea

st 4

dig

its

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to p

erfo

rm

and

chec

k w

ritte

n an

d m

enta

l ca

lcul

atio

ns o

f who

le n

umbe

rs in

clud

ing

es

timat

ion

bu

ildin

g up

and

bre

akin

g do

wn

num

bers

roun

ding

off

and

com

pens

atin

g

doub

ling

and

halv

ing

us

ing

a nu

mbe

r lin

e

usin

g ad

ditio

n an

d su

btra

ctio

n as

in

vers

e op

erat

ions

Num

ber r

ange

for c

alcu

latio

ns

• Add

ition

and

sub

tract

ion

of w

hole

nu

mbe

rs o

f at l

east

5 d

igits

Cal

cula

tion

tech

niqu

es

Usi

ng a

rang

e of

tech

niqu

es to

per

form

an

d ch

eck

writ

ten

and

men

tal

calc

ulat

ions

with

who

le n

umbe

rs

incl

udin

g

estim

atio

n

addi

ng a

nd s

ubtra

ctin

g in

col

umns

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

us

ing

a nu

mbe

r lin

e

roun

ding

off

and

com

pens

atin

g

• do

ublin

g an

d ha

lvin

g

usin

g ad

ditio

n an

d su

btra

ctio

n as

in

vers

e op

erat

ions

Num

ber

rang

e fo

r cal

cula

tions

•

Addi

tion

and

subt

ract

ion

of w

hole

nu

mbe

rs w

ith a

t lea

st 6

-dig

it nu

mbe

r •

Mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out

brac

kets

C

alcu

latio

n te

chni

ques

U

sing

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal

calc

ulat

ions

with

who

le n

umbe

rs

incl

udin

g

estim

atio

n

addi

ng, s

ubtra

ctin

g in

col

umns

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

ro

undi

ng o

ff an

d co

mpe

nsat

ing

us

ing

addi

tion

and

subt

ract

ion

as

inve

rse

oper

atio

ns

us

ing

a ca

lcul

ator

156 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Prop

ertie

s of

who

le n

umbe

rs

Rec

ogni

ze a

nd u

se th

e co

mm

utat

ive

and

asso

ciat

ive

prop

ertie

s w

ith

who

le n

umbe

rs

Solv

ing

prob

lem

s So

lve

prob

lem

s in

con

text

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

fina

ncia

l co

ntex

ts

Prop

ertie

s of

who

le n

umbe

rs

R

ecog

nize

and

use

the

com

mut

ativ

e,

asso

ciat

ive

and

dist

ribut

ive

prop

ertie

s w

ith w

hole

num

bers

-

0 in

term

s of

its

addi

tive

prop

erty

-

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s So

lve

prob

lem

s in

volv

ing

who

le

num

bers

, inc

ludi

ng:

- fin

anci

al c

onte

xts

- m

easu

rem

ent c

onte

xts

Prop

ertie

s of

who

le n

umbe

rs

R

ecog

nize

and

use

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

-

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs a

nd d

ecim

al fr

actio

ns,

incl

udin

g -

finan

cial

con

text

s -

mea

sure

men

t con

text

s

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

LTSM

R

esou

rces

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

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, mul

tiplic

atio

n ta

ble

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 2

pp.

102

– 1

10; p

.176

Ac

tiviti

es 1

07 –

110

; 142

W

B 2

pp.

98

– 10

8; p

. 186

Ac

tiviti

es

106a

&b -

110;

142

WB

2 p

p. 2

0 –

44

Activ

ities

71a

&b -

78

DB

E Te

xtbo

ok re

fere

nce

Term

3: U

nit 9

: Who

le N

umbe

rs: A

dditi

on

and

subt

ract

ion,

p. 2

68

Term

4: U

nit 2

: Who

le N

umbe

rs: A

dditi

on

and

subt

ract

ion,

p. 2

93

Term

4: U

nit 1

1: W

hole

Num

bers

: Add

ition

an

d su

btra

ctio

n, p

. 343

Term

4: U

nit 2

: Who

le N

umbe

rs:

Addi

tion

and

subt

ract

ion,

p. 2

86

Term

3: U

nit 3

: Who

le N

umbe

rs:

Addi

tion

and

subt

ract

ion,

p. 2

24

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t

157MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

– Te

st

158 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 3

TE

RM

4

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to

…

T

ime:

3 H

OU

RS

GR

ADE

4 M

ultip

licat

ion

- CAP

S pp

.101

5 M

ultip

licat

ion

- CAP

S p.

192

(Ref

er to

p.

166

for m

etho

dolo

gy.)

6 M

ultip

licat

ion

- CAP

S p.

241-

243

TOPI

C

Mul

tiplic

atio

n of

who

le n

umbe

rs

Con

cept

s, S

kills

and

kn

owle

dge

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 2

-di

git b

y 2-

digi

t num

bers

C

alcu

latio

n te

chni

ques

U

se a

rang

e of

tech

niqu

es to

per

form

an

d ch

eck

writ

ten

and

men

tal

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 3

-di

git b

y 2-

digi

t num

bers

C

alcu

latio

n te

chni

ques

U

sing

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

Num

ber

rang

e fo

r cou

ntin

g,

orde

ring,

com

parin

g an

d re

pres

entin

g, a

nd p

lace

val

ue o

f di

gits

•

Ord

er, c

ompa

re a

nd re

pres

ent

num

bers

to a

t lea

st 9

-dig

it nu

mbe

rs

• R

epre

sent

prim

e nu

mbe

rs to

at

leas

t 100

•

Rec

ogni

ze

the

plac

e va

lue

of

digi

ts i

n w

hole

num

bers

to

at

leas

t 9-d

igit

num

bers

•

Rou

nd o

ff to

the

near

est 5

, 10,

10

0 or

1 0

00

Num

ber

rang

e fo

r cal

cula

tions

•

Mul

tiplic

atio

n of

at l

east

who

le 4

-di

git b

y 3-

digi

t num

bers

•

Mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out

brac

kets

C

alcu

latio

n te

chni

ques

incl

ude

• es

timat

ion

• m

ultip

lyin

g in

col

umns

159MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

calc

ulat

ions

with

who

le n

umbe

rs

incl

udin

g •

estim

atio

n •

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• do

ublin

g an

d ha

lvin

g N

umbe

r ra

nge

for m

ultip

les

and

fact

ors

Mul

tiple

s of

1-d

igit

num

bers

to a

t lea

st

100

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e; a

nd

dist

ribut

ive

prop

ertie

s of

who

le

num

bers

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g w

hole

num

bers

, in

clud

ing

finan

cial

con

text

s

men

tal c

alcu

latio

ns o

f who

le

num

bers

incl

udin

g:

• es

timat

ion

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

usin

g a

num

ber l

ine

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• do

ublin

g an

d ha

lvin

g N

umbe

r ra

nge

for m

ultip

les

and

fact

ors

• M

ultip

les

of 2

-dig

its w

hole

nu

mbe

rs to

at l

east

100

•

Fact

ors

of 2

-dig

it w

hole

num

bers

to

at l

east

100

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e an

d di

strib

utiv

e pr

oper

ties

with

who

le

num

bers

o

0 in

term

s of

its

addi

tive

prop

erty

o

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

, inc

ludi

ng

o fin

anci

al

cont

exts

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

usin

g a

calc

ulat

or

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igit

and

3-di

git

num

bers

•

Fact

ors

of 2

-dig

it an

d 3-

digi

t who

le

num

bers

•

Prim

e fa

ctor

s of

num

bers

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e;

dist

ribut

ive

prop

ertie

s of

w

hole

num

bers

o 0

in te

rms

of it

s ad

ditiv

e pr

oper

ty

o 1

in te

rms

of it

s m

ultip

licat

ive

prop

erty

So

lvin

g pr

oble

ms

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs a

nd d

ecim

al

fract

ions

, inc

ludi

ng

o fin

anci

al c

onte

xts

160 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

o

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

o co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

o m

easu

rem

ent

cont

exts

•

Solv

e pr

oble

ms

invo

lvin

g w

hole

nu

mbe

rs, i

nclu

ding

o

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

o m

easu

rem

ent c

onte

xts

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

o

com

parin

g tw

o or

mor

e qu

antit

ies

o of

the

sam

e ki

nd (r

atio

) co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

Topi

c

Div

isio

n of

who

le n

umbe

rs

N

umbe

r ra

nge

for c

alcu

latio

ns

• D

ivis

ion

of a

t lea

st w

hole

3-d

igit

by

1-di

git n

umbe

rs.

Cal

cula

tion

tech

niqu

es

• U

se a

rang

e of

tech

niqu

es to

pe

rform

and

che

ck w

ritte

n an

d m

enta

l cal

cula

tions

of w

hole

nu

mbe

rs in

clud

ing

• es

timat

ion

• bu

ildin

g u

p a

nd

brea

king

do

wn

num

bers

•

roun

ding

off

and

com

pens

atin

g •

doub

ling

and

halv

ing

• us

ing

mul

tiplic

atio

n an

d di

visi

on a

s in

vers

e op

erat

ions

Num

ber

rang

e fo

r cal

cula

tions

D

ivis

ion

of a

t lea

st w

hole

3-d

igit

by

2-di

git n

umbe

rs

Cal

cula

tion

tech

niqu

es

Use

a ra

nge

of te

chni

ques

to

perfo

rm a

nd c

heck

writ

ten

and

men

tal c

alcu

latio

ns w

ith w

hole

nu

mbe

rs in

clud

ing

• es

timat

ion

• bu

ildin

g up

and

bre

akin

g do

wn

num

bers

•

usin

g m

ultip

licat

ion

and

divi

sion

as

inve

rse

oper

atio

ns

Num

ber

rang

e fo

r cal

cula

tions

•

Div

isio

n of

at l

east

who

le 4

-dig

it by

3-

digi

t num

bers

•

mul

tiple

ope

ratio

ns o

n w

hole

nu

mbe

rs w

ith o

r with

out

brac

kets

Cal

cula

tion

tech

niqu

es

• es

timat

ion

• us

ing

the

reci

proc

al re

latio

nshi

p be

twee

n m

ultip

licat

ion

and

divi

sion

•

long

div

isio

n •

build

ing

up a

nd b

reak

ing

dow

n nu

mbe

rs

• ro

undi

ng o

ff an

d co

mpe

nsat

ing

• us

ing

a ca

lcul

ator

161MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

1-d

igit

num

bers

to

at le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e; a

nd

dist

ribut

ive

prop

ertie

s of

who

le

num

bers

So

lvin

g pr

oble

ms

• So

lve

prob

lem

s in

con

text

s in

volv

ing

who

le n

umbe

rs

o fin

anci

al c

onte

xts

o m

easu

rem

ent c

onte

xts

Num

ber

rang

e fo

r cou

ntin

g,

orde

ring

and

repr

esen

ting,

and

pl

ace

valu

e of

dig

its

• Rec

ogni

ze th

e pl

ace

valu

e of

dig

its

in w

hole

num

bers

to

at l

east

6-

digi

t num

bers

. •

Rou

nd o

ff to

the

near

est 1

0, 1

00,

1 00

0 N

umbe

r ra

nge

for m

ultip

les

and

fact

ors

• M

ultip

les

of 2

-dig

it nu

mbe

rs to

at

leas

t 100

•

Fact

ors

of 2

-dig

it w

hole

nu

mbe

rs to

at l

east

100

M

ultip

licat

ion

fact

s •

Uni

ts b

y m

ultip

les

of 1

0 •

Uni

ts b

y m

ultip

les

of 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e; a

nd

dist

ribut

ive

prop

ertie

s of

who

le

num

bers

o

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

in c

onte

xts

invo

lvin

g w

hole

num

bers

, in

clud

ing

finan

cial

con

text

s

Num

ber

rang

e fo

r mul

tiple

s an

d fa

ctor

s •

Mul

tiple

s of

2-d

igit

and

3-di

git

num

bers

•

Fact

ors

of 2

-dig

it an

d 3-

digi

t who

le

num

bers

•

Prim

e fa

ctor

s of

num

bers

up

to a

t le

ast 1

00

Prop

ertie

s of

who

le n

umbe

rs

• R

ecog

nize

and

use

the

com

mut

ativ

e; a

ssoc

iativ

e;

dist

ribut

ive

prop

ertie

s of

w

hole

num

bers

o

0 in

term

s of

its

addi

tive

prop

erty

o

1 in

term

s of

its

mul

tiplic

ativ

e pr

oper

ty

Solv

ing

prob

lem

s •

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

and

dec

imal

fra

ctio

ns, i

nclu

ding

o

finan

cial

con

text

s

162 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

o

grou

ping

and

equ

al s

harin

g

with

rem

aind

ers

o co

mpa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd

(ratio

) o

com

parin

g tw

o qu

antit

ies

of

diffe

rent

kin

ds (r

ate)

•

Solv

e pr

oble

ms

invo

lvin

g w

hole

num

bers

, inc

ludi

ng

o co

mpa

ring

two

or m

ore

quan

titie

s of

the

sam

e ki

nd

(ratio

) o

com

parin

g tw

o qu

antit

ies

of

diffe

rent

kin

ds (r

ate)

o

grou

ping

and

equ

al s

harin

g w

ith

rem

aind

ers

o m

easu

rem

ent c

onte

xts

• So

lve

prob

lem

s in

volv

ing

who

le n

umbe

rs, i

nclu

ding

o

com

parin

g tw

o or

mor

e qu

antit

ies

of th

e sa

me

kind

(ra

tio)

o co

mpa

ring

two

quan

titie

s of

di

ffere

nt k

inds

(rat

e)

o gr

oupi

ng a

nd e

qual

sha

ring

with

rem

aind

ers

SUG

GES

TED

M

ETH

OD

OLO

GY

Ref

er to

Ter

m1

and

Term

2 le

sson

Pla

ns

Expl

ain

from

kno

wn

(low

num

ber r

ange

); w

ork

with

con

cret

e LT

SM.

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

cont

inue

to

• ch

eck

thei

r sol

utio

ns b

y us

ing

mul

tiplic

atio

n w

hen

doin

g di

visi

on a

nd v

ice

vers

a •

judg

e th

e re

ason

able

ness

of t

heir

solu

tions

, by

estim

atin

g be

fore

cal

cula

ting

Mul

tiplic

atio

n (K

now

tim

es ta

bles

– p

ract

ice

regu

larly

up

to 1

0 x

10 fo

r Gra

des

4 an

d 5

and

12 x

12

for G

rade

6).

Follo

w th

ese

step

s in

teac

hing

mul

tiplic

atio

n:

• So

lve

prob

lem

s in

con

text

. U

nder

stan

d th

at: 8

6 +

86 +

86

is e

quiv

alen

t to

86 x

3 o

r 3 x

86

(repe

ated

add

ition

) •

mul

tiplic

atio

n by

1 le

aves

a n

umbe

r unc

hang

ed e

.g. 5

x 1

= 5

; 99

x 1

= 99

•

mul

tiplic

atio

n of

zer

o re

sults

in z

ero

e.g.

7 x

0 =

0; 3

1 x

0 =

0 U

nder

stan

d th

at m

ultip

licat

ion

is th

e in

vers

e of

div

isio

n (m

ultip

licat

ion

reve

rses

div

isio

n an

d vi

ce v

ersa

) and

use

this

to c

heck

re

sults

. e.g

. 15

x 35

= 5

25

525

÷ 2

5 =

15 o

r 525

÷ 1

5 =

25

Not

e: In

Gra

de 5

Ter

m 3

lear

ners

are

intro

duce

d to

col

umn

met

hods

of m

ultip

licat

ion

163MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

DIV

ISIO

N

Men

tal d

ivis

ion

: Tur

ning

the

tabl

es –

pra

ctic

e re

gula

rly (e

.g. 5

x 8

= 4

0 4

0 ÷

8 =

5 / 4

0 ÷

5 =

8) (A

lso

calle

d in

vers

e op

erat

ion

or re

cipr

ocal

ope

ratio

n.)

Mak

e le

arne

rs a

war

e th

at d

ivis

ion

is n

on-c

omm

utat

ive

i.e. 8

1 ÷

9 is

not

the

sam

e as

9 ÷

81.

W

ith th

e co

ncep

t gra

sped

, tea

chin

g di

visi

on w

ill be

com

e m

ore

abou

t gui

ded

prac

tice

to h

elp

the

lear

ners

to b

ecom

e fa

milia

r with

th

e di

visi

on o

pera

tion

(alth

ough

it’s

real

ly g

oing

to b

e a

diffe

rent

type

of m

ultip

licat

ion

prac

tice.

) Sta

rt by

pra

ctis

ing

divi

sion

by

1, 2

an

d 3

and

then

gra

dual

ly m

ove

up to

9.

Not

e: G

rade

4 a

nd 5

focu

s is

on

usin

g th

e cl

ue b

oard

– m

etho

d O

NLY

! Gra

de 6

is th

e in

trodu

ctio

n of

long

div

isio

n in

the

third

te

rm

LTSM

R

esou

rces

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

card

s, B

ase

10 b

lock

s, 1

00 c

harts

, mul

tiplic

atio

n ta

ble

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 2

pp.

78

– 90

Ac

tiviti

es 9

8a-c

, 99a

&b; 1

00-1

01

pp. 1

42 –

144

Ac

tiviti

es 1

24 -

125

WB

2 p

p. 9

0 –

94

Activ

ities

103

a&b;

104

; pp.

138

– 1

52

Activ

ities

120

- 126

a&b

WB

2 p

p. 1

12 –

130

Ac

tiviti

es

106a

&b-1

12a&

b pp

.164

– 1

74

Activ

ities

; 126

-129

a&b

DB

E Te

xtbo

ok re

fere

nce

Term

3: U

nit 1

0: W

hole

Num

bers

: M

ultip

licat

ion,

p. 2

75

Term

4: U

nit 6

: Who

le N

umbe

rs:

Div

isio

n, p

. 313

Term

3: U

nit 1

1: W

hole

Num

bers

: M

ultip

licat

ion,

p. 2

69

Term

4: U

nit 5

: Who

le N

umbe

rs:

Div

isio

n, p

. 310

Term

4: U

nit 2

: Who

le N

umbe

rs:

Mul

tiplic

atio

n, p

. 300

Te

rm 4

: Uni

t 7: W

hole

Num

bers

: D

ivis

ion,

p. 3

36

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

164 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 4

TE

RM

4

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to …

……

T

ime:

6 H

OU

RS

GR

ADE

4 C

APS

pp. 1

11

5 C

APS

pp. 1

98

6 C

APS

pp. 2

81

TOPI

C

Prop

ertie

s of

3-D

sha

pes

Con

cept

s, S

kills

and

kn

owle

dge

Prop

ertie

s of

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms,

sphe

res

cy

linde

rs

py

ram

ids

Cha

ract

eris

tics

of o

bjec

ts

sh

apes

of f

aces

flat a

nd c

urve

d su

rface

s Fu

rthe

r act

iviti

es

M

ake

3-D

mod

els

usin

g cu

t out

po

lygo

ns

Prop

ertie

s of

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms

and

othe

r pr

ism

s

cube

s

cylin

ders

cone

s

pyra

mid

s

sim

ilarit

ies

and

diffe

renc

es

betw

een

cube

s an

d re

ctan

gula

r pr

ism

s C

hara

cter

istic

s of

obj

ects

shap

e of

face

s

num

ber o

f fac

es

fla

t and

cur

ved

surfa

ces

Furt

her a

ctiv

ities

Mak

e 3-

D m

odel

s us

ing

cut o

ut p

olyg

ons

ne

ts

Prop

ertie

s of

3-D

obj

ects

R

ange

of O

bjec

ts:

re

ctan

gula

r pris

ms

cu

bes

te

trahe

dron

s

pyra

mid

s

sim

ilarit

ies

and

diffe

renc

es

betw

een

tetra

hedr

ons

and

othe

r py

ram

ids

Cha

ract

eris

tics

of o

bjec

ts

nu

mbe

r and

sha

pe o

f fac

es

nu

mbe

r of v

ertic

es

nu

mbe

r of e

dges

Fu

rthe

r act

iviti

es

M

ake

3-D

mod

els

usin

g:

dr

inki

ng s

traw

s, to

othp

icks

etc

nets

165MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Ref

er to

Ter

m 2

not

es fo

r rev

isio

n an

d co

nsol

idat

ion.

In T

erm

4 le

arne

rs s

houl

d co

nsol

idat

e w

hat t

hey

lear

nt a

bout

3-D

obj

ects

ear

lier i

n th

e ye

ar. T

his

incl

udes

:

w

orki

ng w

ith a

ll of

the

obje

cts

pres

crib

ed fo

r Gra

de 4

, 5 a

nd 6

.

fo

cusi

ng o

n th

e ki

nd o

f sur

face

and

the

shap

e an

d nu

mbe

r of f

aces

.

bu

ildin

g ob

ject

s us

ing

nets

.

In T

erm

4 le

arne

rs c

an b

uild

ske

leto

n sh

apes

with

stra

ws

or to

othp

icks

. The

y w

ill th

en fo

cus

on th

e ed

ges

and

verti

ces

of th

e

obje

cts.

Thi

s m

eans

that

by

the

end

of th

e ye

ar th

ey w

ill be

abl

e to

des

crib

e 3-

D g

eom

etric

obj

ects

acc

ordi

ng to

the

num

ber

and

shap

e of

face

s an

d th

e nu

mbe

r of e

dges

and

ver

tices

of 3

-D O

bjec

ts. L

earn

ers

need

to w

ork

with

real

obj

ects

. How

ever

,

they

als

o ne

ed to

do

writ

ten

exer

cise

s on

3-D

obj

ects

. Int

erpr

etin

g pi

ctur

es a

bout

3-D

obj

ects

is m

ore

diffi

cult

than

wor

king

with

the

real

obj

ects

. Lea

rner

s sh

ould

pra

ctis

e in

terp

retin

g dr

awin

gs o

f 3-D

obj

ects

. The

y sh

ould

iden

tify

and

nam

e 3-

D

obje

cts

in d

raw

ings

; ide

ntify

eve

ryda

y ob

ject

s th

at lo

ok li

ke g

eom

etric

obj

ects

(e.g

. a m

ilk c

arto

n lo

oks

like

a re

ctan

gula

r

pris

m),

mat

ch n

ets

of o

bjec

ts to

dra

win

g of

obj

ects

, des

crib

e 3-

D o

bjec

ts b

y st

atin

g th

e nu

mbe

r of f

lat a

nd/o

r cur

ved

surfa

ces,

the

num

ber o

f ver

tices

, edg

es, a

nd n

umbe

r and

sha

pe o

f fac

es w

hen

show

n dr

awin

gs o

f 3-D

obj

ects

.

LTSM

R

esou

rces

Va

riety

of 3

D o

bjec

ts fr

om le

arne

r’s c

onte

xt, g

eoso

lids

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 2

pp.

124

– 1

26

Act

iviti

es 1

16 -

117

WB

1 p

p. 1

50 A

ctiv

ity 5

4 W

B 2

pp.

152

Ac

tivity

122

D

BE

Text

book

refe

renc

e Te

rm 4

: Uni

t 4: P

rope

rtie

s of

thre

e di

men

sion

al o

bjec

ts, p

. 306

Te

rm 4

: Uni

t 3: P

rope

rtie

s of

thre

e di

men

sion

al o

bjec

ts, p

. 292

Te

rm 4

: Uni

t 4: P

rope

rtie

s of

thre

e di

men

sion

al o

bjec

ts, p

. 315

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent –

Te

st

166 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 5

TE

RM

4

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to…

……

.…

Tim

e: 8

HO

UR

S G

RAD

E 4

C

APS

p. 1

14

5 C

APS

p. 2

02

6 C

APS

p. 2

82

TOPI

C

Perim

eter

, Are

a an

d Vo

lum

e C

once

pts,

Ski

lls a

nd

know

ledg

e Pe

rimet

er

Mea

sure

per

imet

er u

sing

rule

rs o

r m

easu

ring

tape

s M

easu

rem

ent o

f are

a Fi

nd a

reas

of r

egul

ar a

nd ir

regu

lar

shap

es b

y co

untin

g sq

uare

s on

grid

s in

or

der t

o de

velo

p an

und

erst

andi

ng o

f sq

uare

uni

ts

Mea

sure

men

t of v

olum

e Fi

nd v

olum

e/ca

paci

ty o

f obj

ects

(by

pack

ing

or fi

lling

them

in o

rder

to d

evel

op

an u

nder

stan

ding

of c

ubic

uni

ts

Perim

eter

M

easu

re p

erim

eter

usi

ng ru

lers

or

mea

surin

g ta

pes

Mea

sure

men

t of a

rea

Find

are

as o

f reg

ular

and

irre

gula

r sh

apes

by

coun

ting

squa

res

on g

rids

in

orde

r to

deve

lop

an u

nder

stan

ding

of

squa

re u

nits

M

easu

rem

ent o

f vol

ume

Find

vol

ume/

capa

city

of o

bjec

ts b

y pa

ckin

g or

fillin

g th

em in

ord

er to

de

velo

p an

und

erst

andi

ng o

f cub

ic u

nits

Perim

eter

M

easu

re p

erim

eter

usi

ng ru

lers

or

mea

surin

g ta

pes

Mea

sure

men

t of a

rea

• C

ontin

ue to

find

are

as o

f reg

ular

and

irr

egul

ar s

hape

s by

cou

ntin

g sq

uare

s on

grid

s •

Dev

elop

an

unde

rsta

ndin

g of

why

th

e ar

ea o

f rec

tang

les

can

be

desc

ribed

as

thei

r len

gth

mul

tiplie

d by

thei

r wid

th

Mea

sure

men

t of v

olum

e •

Con

tinue

to fi

nd v

olum

e/ca

paci

ty o

f ob

ject

s (b

y pa

ckin

g or

fillin

g th

em.

• D

evel

op a

n un

ders

tand

ing

of w

hy

the

volu

me

of re

ctan

gula

r pris

ms

can

be d

escr

ibed

as

thei

r len

gth

mul

tiplie

d by

thei

r wid

th m

ultip

lied

by

thei

r hei

ght

Inve

stig

ate

the:

•

rela

tions

hip

betw

een

perim

eter

and

ar

ea o

f rec

tang

les

and

squa

res

• re

latio

nshi

p be

twee

n su

rface

are

a an

d vo

lum

e of

rect

angu

lar p

rism

s

167MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Wha

t is

capa

city

? W

hat i

s vo

lum

e?

Cap

acity

is th

e am

ount

of s

ubst

ance

that

an

obje

ct c

an h

old

or th

e am

ount

of s

pace

insi

de th

e ob

ject

. Vo

lum

e is

the

amou

nt o

f spa

ce th

at a

n ob

ject

occ

upie

s.

A bo

ttle

can

have

a 1

litre

cap

acity

, but

it m

ay n

ot b

e fil

led

to it

s fu

ll ca

paci

ty, i

t cou

ld fo

r exa

mpl

e, o

nly

cont

ain

a vo

lum

e of

25

0ml.

Con

tent

Area

and

vol

ume

are

only

mea

sure

d in

form

ally

in th

e In

term

edia

te P

hase

. Lea

rner

s ar

e no

t req

uire

d to

kno

w o

r app

ly

form

ulae

for t

he p

erim

eter

, are

a or

vol

ume

of a

ny s

hape

or o

bjec

ts.

In

Gra

de 3

lear

ners

onl

y m

easu

red

perim

eter

info

rmal

ly b

y fin

ding

the

dist

ance

aro

und

two-

dim

ensi

onal

sha

pes

usin

g st

ring.

Lea

rner

s in

Gra

de 3

are

not

requ

ired

to s

tate

or w

rite

how

long

a p

erim

eter

is. T

hey

only

sho

w th

e st

ring

leng

th o

r co

mpa

re d

iffer

ent p

erim

eter

s by

com

parin

g st

ring

leng

ths.

In G

rade

4 le

arne

rs m

easu

re th

e pe

rimet

ers

of s

hape

s an

d sp

aces

with

rule

rs a

nd m

easu

ring

tape

s. T

hey

are

requ

ired

to s

tate

and

reco

rd th

is m

easu

rem

ent i

n st

anda

rd u

nits

: mm

, cm

, m. L

earn

ers

are

also

requ

ired

to w

ork

from

dra

win

gs in

whi

ch s

ide

leng

ths

are

spec

ified

in m

m, c

m, m

, km

. Her

e th

ey

add

the

leng

ths.

In G

rade

4 th

ey w

ill al

so c

ount

the

leng

ths

of th

e pe

rimet

ers

by c

ount

ing

the

num

ber o

f sid

es o

f squ

are

grid

s on

whi

ch s

hape

s ar

e dr

awn.

Her

e le

arne

rs n

eed

to k

now

that

the

diag

onal

dis

tanc

es b

etw

een

corn

ers

of a

grid

squ

are

are

long

er th

an th

e ve

rtica

l or h

oriz

onta

l dis

tanc

es b

etw

een

corn

ers

of a

grid

squ

are.

In G

rade

3 le

arne

rs o

nly

inve

stig

ate

area

s us

ing

tilin

g. In

Gra

de 4

are

a m

easu

rem

ents

con

tinue

to b

e in

form

al, b

ut n

ow

lear

ners

use

bot

h til

ing

and

squa

re g

rids.

Lea

rner

s co

unt h

ow m

any

grid

squ

ares

the

shap

e co

vers

. The

are

a is

sta

ted

in

num

ber o

f grid

squ

ares

. Sha

pes

shou

ld in

clud

e re

gula

r sha

pes,

irre

gula

r sha

pes

and

shap

es w

ith c

urve

d si

des.

Lear

ners

do

not w

ork

with

vol

ume

in G

rade

3. I

n G

rade

4 le

arne

rs c

ount

how

man

y cu

bes

or re

ctan

gula

r pris

ms

are

used

to

fill a

con

tain

er. T

he v

olum

e of

the

cont

aine

r is

stat

ed in

num

ber o

f cub

es o

r rec

tang

ular

pris

ms

such

as

boxe

s or

blo

cks.

Le

arne

rs m

ake

stac

ks w

ith c

ubes

or r

ecta

ngul

ar p

rism

s. T

he v

olum

e of

the

stac

k is

sta

ted

in n

umbe

r of c

ubes

or

rect

angu

lar p

rism

s su

ch a

s bo

xes

or b

lock

s.

In

Gra

de 4

, lea

rner

s in

terp

ret p

ictu

res

of s

tack

s m

ade

of c

ubes

or r

ecta

ngul

ar p

rism

s in

ord

er to

sta

te th

e vo

lum

e in

term

s of

the

num

ber o

f cub

es o

r rec

tang

ular

pris

ms.

The

y in

terp

ret p

ictu

res

of c

onta

iner

s fil

led

with

cub

es o

r rec

tang

ular

pris

ms

in

orde

r to

stat

e th

e vo

lum

e in

term

s of

the

num

ber o

f cub

es o

r rec

tang

ular

pris

ms.

In G

rade

5 a

nd 6

lear

ners

pra

ctis

e an

d co

nsol

idat

e w

hat t

hey

have

lear

ned

abou

t per

imet

er, a

rea

and

volu

me

in G

rade

4.

168 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

In

Gra

de 6

they

sho

uld

inve

stig

ate

why

the

area

of a

rect

angl

e ca

n be

sta

ted

as it

s le

ngth

mul

tiplie

d by

its

wid

th. T

hey

are

not r

equi

red

to k

now

this

form

ula

off b

y he

art,

nor a

re th

ey re

quire

d to

app

ly th

is fo

rmul

a in

are

a ca

lcul

atio

ns.

G

rade

6 le

arne

rs in

vest

igat

e th

e re

latio

nshi

p be

twee

n th

e ar

ea a

nd p

erim

eter

of r

ecta

ngle

s an

d sq

uare

s. T

his

inve

stig

atio

n ca

n be

don

e as

an

Asse

ssm

ent T

ask.

The

re a

re tw

o di

ffere

nt in

vest

igat

ions

that

lear

ners

can

do.

1.

If

lear

ners

are

giv

en th

e pe

rimet

er o

f a re

ctan

gle,

they

can

dra

w a

num

ber o

f rec

tang

les

of d

iffer

ing

area

s. D

oes

this

al

so w

ork

with

squ

ares

?

2.

Sim

ilarly

if th

ey a

re g

iven

the

area

of a

squ

are,

ther

e w

ill on

ly b

e on

e po

ssib

ility

for t

he le

ngth

of t

he s

ides

. Is

this

the

sam

e fo

r rec

tang

les?

In

vest

igat

ing

the

rela

tions

hip

betw

een

the

area

s an

d pe

rimet

ers

of s

quar

es a

nd re

ctan

gles

can

be

com

bine

d w

ith th

e sh

ape

and

spac

e re

quire

men

t.

G

rade

6 le

arne

rs a

lso

draw

enl

arge

men

ts a

nd re

duct

ions

of 2

-D s

hape

s us

ing

grid

pap

er to

com

pare

thei

r siz

e an

d sh

ape.

H

ere

lear

ners

can

dra

w a

squ

are

or re

ctan

gle

with

spe

cifie

d si

de le

ngth

s. T

hen

they

can

inve

stig

ate

wha

t hap

pens

to th

e ar

ea o

f the

sha

pe, i

f the

leng

th o

f one

pai

r of o

ppos

ite s

ides

of t

he s

hape

are

dou

bled

or h

alve

d L

TSM

R

esou

rces

Va

riety

of 2

D s

hape

s. R

uler

s, m

easu

ring

tape

s, s

quar

blo

ck p

aper

Gra

de 4

G

rade

5

Gra

de 6

W

orkb

ook

refe

renc

e

WB

2 p

p. 1

46 -

156

Ac

tiviti

es 1

27 -

132

WB

2 p

p. 1

54 -

164

Activ

ities

127

- 13

0

WB

2 p

p. 1

54 –

162

Ac

tiviti

es

123a

&b;1

24;1

25a&

b D

BE

Text

book

refe

renc

e Te

rm 4

: Uni

t 7: P

erim

eter

, are

a an

d vo

lum

e, p

. 318

Te

rm 4

: Uni

t 6: P

erim

eter

, are

a an

d vo

lum

e, p

. 318

Te

rm 4

: Uni

t 5: P

erim

eter

, are

a an

d vo

lum

e, p

. 322

H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

169MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 6

TE

RM

4

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to…

….…

T

ime:

4 H

OU

RS

GR

ADE

4 C

APS

p. 9

5

5 C

APS

p. 1

87 -

188

6 C

APS

p. 2

68 -

269

TOPI

C

Dat

a H

andl

ing

Con

cept

s, S

kills

and

kn

owle

dge

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng

data

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng

data

Ung

roup

ed d

ata

Dat

a ha

ndlin

g

Col

lect

ing

and

orga

nizi

ng d

ata

R

epre

sent

ing

data

Anal

ysin

g, in

terp

retin

g an

d re

porti

ng

data

incl

udin

g do

uble

bar

gra

phs

ce

ntra

l ten

denc

ies

– (m

ode

and

med

ian)

Ung

roup

ed d

ata

SUG

GES

TED

M

ETH

OD

OLO

GY

Teac

hers

in th

is p

hase

sho

uld

ensu

re th

at d

iffer

ent t

opic

s ar

e ch

osen

for d

ata

colle

ctio

n in

eac

h of

the

grad

es. T

he fo

llow

ing

are

new

in G

rade

4

Le

arne

rs re

ad, i

nter

pret

, ana

lyse

and

sum

mar

ise

pie

char

ts, w

here

the

info

rmat

ion

is p

rese

nted

in fr

actio

ns o

nly

Le

arne

rs re

ad, a

naly

se d

ata

repr

esen

ted

in w

ords

i.e.

sho

rt pa

ragr

aphs

– th

e da

ta p

rese

nted

in w

ords

sho

uld

be

repr

esen

ted

in o

ther

form

s an

d th

en a

naly

sed

Anal

ysin

g gr

aphs

An

alys

e gr

aphs

on

envi

ronm

enta

l or s

ocio

-eco

nom

ic c

onte

xts

and

answ

er q

uest

ions

on

grap

hs. B

oth

grap

hs a

nd q

uest

ions

to b

e pr

ovid

ed b

y te

ache

r or t

extb

ook.

Lea

rner

s sh

ould

wor

k w

ith a

t lea

st

1

pie

grap

h w

here

the

info

rmat

ion

is g

iven

in c

omm

on fr

actio

ns a

nd n

ot p

erce

ntag

es

1

bar g

raph

Su

itabl

e to

pics

incl

ude:

quan

titie

s of

mat

eria

ls re

cycl

ed in

the

tow

n, p

rovi

nce,

cou

ntry

quan

titie

s of

recy

clin

g m

ater

ials

col

lect

ed b

y sc

hool

s ar

ound

the

coun

try

so

urce

s of

ligh

ting

and

heat

ing

in S

A

ki

nds

of to

ilets

in S

A ho

mes

kind

s of

hom

es in

SA

170 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

The

follo

win

g ar

e ne

w in

Gra

de 5

orde

ring

data

set

s

anal

yzin

g da

ta n

ot o

nly

acco

rdin

g to

cat

egor

ies

but a

lso

taki

ng in

to a

ccou

nt c

onte

xts

and

sour

ces

of d

ata

an

alyz

ing

ungr

oupe

d nu

mer

ical

dat

a se

ts to

find

the

mod

e

pict

ogra

phs

whi

ch s

how

man

y-to

-one

cor

resp

onde

nce

co

nclu

sion

s an

d pr

edic

tions

whe

n an

alys

ing

and

sum

mar

isin

g da

ta

Dev

elop

crit

ical

ana

lysi

s sk

ills

Lear

ners

com

pare

gra

phs

on th

e sa

me

topi

c, b

ut w

here

dat

a ha

s be

en c

olle

cted

from

diff

eren

t gro

ups

of p

eopl

e, a

t diff

eren

t tim

es, i

n di

ffere

nt p

lace

s or

in d

iffer

ent w

ays.

Her

e le

arne

rs w

ill be

abl

e to

dis

cuss

the

diffe

renc

es b

etw

een

the

grap

hs. T

he a

im

is a

lso

for l

earn

ers

to b

ecom

e aw

are

of fa

ctor

s th

at c

an im

pact

on

the

data

. Lea

rner

s ca

n su

mm

ariz

e th

e fin

ding

s of

thei

r co

mpa

rison

in a

par

agra

ph.

Exam

ples

cou

ld in

clud

e:

co

mpa

ring

data

abo

ut c

ars

that

pas

s th

e sc

hool

at d

iffer

ent t

imes

or c

ompa

ring

data

abo

ut c

ars

that

pas

s di

ffere

nt

venu

es (b

usy

and

quie

t are

as, p

oore

r and

rich

er a

reas

etc

)

com

parin

g da

ta c

olle

cted

at y

our s

choo

l to

natio

nal d

ata

from

‘Cen

sus

At S

choo

l e.g

. fav

ourit

e sp

orts

; fav

ourit

e su

bjec

ts;

trans

port

to s

choo

l; tim

e ta

ken

to g

et to

sch

ool;

type

of d

wel

ling;

acc

ess

to g

oods

and

ser

vice

s at

hom

e

com

parin

g da

ta c

olle

cted

from

girl

s an

d bo

ys e

.g. f

avou

rite

spor

ts, f

avou

rite

mov

ies,

favo

urite

sch

ool s

ubje

cts

co

mpa

ring

rain

fall

each

mon

th fo

r a to

wn

in s

umm

er a

nd w

inte

r rai

nfal

l are

as

Lear

ners

sho

uld

do a

t lea

st 1

exa

mpl

e in

whi

ch th

ey c

ompa

re g

raph

s.

Com

plet

e da

ta c

ycle

: con

text

per

sona

l dat

a Th

e co

mpl

ete

data

cyc

le in

clud

es a

skin

g a

ques

tion,

col

lect

ion,

org

anis

ing,

repr

esen

ting,

ana

lyzi

ng a

nd in

terp

retin

g da

ta a

nd

repo

rting

on

the

data

. Cho

ose

a di

ffere

nt to

pic

to T

erm

1. W

ork

thro

ugh

the

who

le d

ata

cycl

e to

mak

e an

indi

vidu

al b

ar g

raph

us

ing

cont

exts

that

rela

te to

them

selv

es, t

heir

clas

s, th

eir s

choo

l or t

heir

fam

ily.

Suita

ble

topi

cs in

clud

e:

fa

vour

ite s

ports

/ fa

vour

ite m

ovie

s / f

avou

rite

mus

ic /

favo

urite

TV

prog

ram

mes

/ fo

ods

or c

ool d

rinks

/ fav

ourit

e co

lour

s,

etc.

heig

hts

of le

arne

rs in

cla

ss

m

ass

of le

arne

rs in

cla

ss

sh

oe s

ize

of le

arne

rs in

cla

ss

av

erag

e tim

e ta

ken

to g

et fr

om h

ome

to s

choo

l

num

ber o

f peo

ple

stay

ing

in h

omes

of l

earn

ers

in th

e cl

ass

171MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Anal

yse

ungr

oupe

d nu

mer

ical

dat

a us

ing

mea

sure

s of

cen

tral

tend

ency

Le

arne

rs d

eter

min

e th

e m

ode

of u

ngro

uped

num

eric

al d

ata

sets

. Su

itabl

e to

pics

incl

ude:

heig

hts

of le

arne

rs in

the

clas

s

mas

s of

lear

ners

in th

e cl

ass

sh

oe s

ize

of le

arne

rs in

the

clas

s

aver

age

time

take

n to

get

from

hom

e to

sch

ool

nu

mbe

r of p

eopl

e st

ayin

g in

the

hom

es o

f lea

rner

s in

the

clas

s

tem

pera

ture

s fo

r a m

onth

Th

e fo

llow

ing

are

new

in G

rade

6

gr

aphs

can

incl

ude

data

exp

ress

ed in

per

cent

ages

. Thi

s is

impo

rtant

in p

ie c

harts

, but

per

cent

ages

can

als

o be

use

d in

ba

r gra

phs

or d

oubl

e ba

r gra

phs

co

llect

ing

data

by

usin

g si

mpl

e qu

estio

nnai

res

do

uble

bar

gra

phs

th

e m

edia

n of

a d

ata

set

Com

plet

e a

data

cyc

le in

clud

ing

draw

ing

a do

uble

bar

gra

ph: c

onte

xt p

erso

nal d

ata

This

is re

com

men

ded

as th

e M

athe

mat

ics

proj

ect i

n G

rade

6

The

com

plet

e da

ta c

ycle

incl

udes

pos

ing

a qu

estio

n, c

olle

ctin

g, o

rgan

isin

g, re

pres

entin

g, a

naly

zing

, int

erpr

etin

g da

ta a

nd

repo

rting

on

the

data

. Lea

rner

s w

ork

thro

ugh

the

who

le d

ata

cycl

e to

mak

e an

indi

vidu

al d

oubl

e ba

r gra

ph u

sing

con

text

s th

at

rela

te to

them

selv

es, t

heir

clas

s, th

eir s

choo

l or t

heir

fam

ily.

Suita

ble

topi

cs in

clud

e:

fa

vour

ite s

ports

/ fa

vour

ite m

ovie

s / f

avou

rite

mus

ic /

favo

urite

TV

prog

ram

mes

/ fo

ods

or c

ool d

rinks

/ fav

ourit

e co

lour

s,

etc.

Incl

ude

boys

ver

sus

girls

heig

hts

of le

arne

rs in

cla

ss. I

nclu

de b

oys

vers

us g

irls

m

ass

of le

arne

rs in

cla

ss. I

nclu

de b

oys

vers

us g

irls

sh

oe s

ize

of le

arne

rs in

cla

ss. I

nclu

de b

oys

vers

us g

irls

Anal

ysin

g un

grou

ped

num

eric

al d

ata

usin

g m

easu

res

of c

entr

al te

nden

cy

Lear

ners

find

the

mod

e an

d m

edia

n of

ung

roup

ed n

umer

ical

dat

a se

ts. S

uita

ble

topi

cs s

ame

as fo

r Gra

de 5

.

Teac

hers

in th

is p

hase

sho

uld

ensu

re th

at d

iffer

ent t

opic

s ar

e ch

osen

for d

ata

colle

ctio

n an

d an

alys

is in

eac

h of

the

grad

es.

172 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LTSM

R

esou

rces

Ex

empl

ers

of p

icto

grap

hs, b

ar g

raph

s, p

ie c

harts

– (c

onte

xt s

ensi

tive)

, IC

T –

Exce

l wor

kshe

ets

with

gra

phs,

vi

deo

clip

s, E

lear

ning

onl

ine

reso

urce

s, M

aths

gam

es,

G

rade

4

Gra

de 5

G

rade

6

Wor

kboo

k re

fere

nce

W

B 2

pp.

52

– 62

Ac

tiviti

es 8

6 - 9

1 W

B 2

pp.

68

– 84

A

ctiv

ities

94

- 100

W

B 2

pp.

76

– 88

Ac

tiviti

es 9

1 - 9

6 D

BE

Text

book

refe

renc

e Te

rm 3

: Uni

t 7: D

ata

Han

dlin

g, p

. 258

Te

rm 4

: Uni

t 12:

Pro

babi

lity,

p. 3

47

Term

3: U

nit 9

: Dat

a H

andl

ing,

p. 2

57

Term

4: U

nit 1

1: P

roba

bilit

y, p

. 346

Te

rm 3

: Uni

t 9: D

ata

Han

dlin

g, p

. 268

Te

rm 4

: Uni

t 11:

Pro

babi

lity,

p. 3

63

HO

MEW

OR

K

AS

SESS

MEN

T E.

g. In

form

al a

sses

smen

t –

Test

173MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 7

TE

RM

4

EDU

CAT

OR

: ……

……

……

……

……

DAT

E : f

rom

…

……

to …

….…

T

ime:

12

HO

UR

S G

RAD

E 4

Com

mon

Fra

ctio

ns

CAP

S p.

91

5 C

omm

on fr

actio

ns

CAP

S pp

.160

-162

, 176

-177

& 1

99

6 C

omm

on fr

actio

ns

C

APS

pp. 2

26-2

27, 2

80

Dec

imal

frac

tions

- C

APS

p. 2

52,

Perc

enta

ges

- CAP

S p.

267

TO

PIC

C

omm

on fr

actio

ns, d

ecim

al fr

actio

ns a

nd p

erce

ntag

es

Con

cept

s, S

kills

and

kn

owle

dge

Des

crib

ing

and

orde

ring

frac

tions

:•

C

ompa

re a

nd o

rder

com

mon

fra

ctio

ns w

ith d

iffer

ent d

enom

inat

ors

(hal

ves;

third

s, q

uarte

rs; f

ifths

; si

xths

; sev

enth

s; e

ight

hs)

D

escr

ibe

and

com

pare

com

mon

fra

ctio

ns in

dia

gram

form

Cal

cula

tions

with

frac

tions

:

Addi

tion

of c

omm

on fr

actio

ns w

ith

the

sam

e de

nom

inat

ors

Rec

ogni

ze, d

escr

ibe

and

use

the

equi

vale

nce

of d

ivis

ion

and

fract

ions

Des

crib

ing

and

orde

ring

frac

tions

: • C

ount

forw

ards

and

bac

kwar

ds in

fra

ctio

ns

• Com

pare

and

ord

er c

omm

on fr

actio

ns

to a

t lea

st tw

elfth

s C

alcu

latio

ns w

ith fr

actio

ns:

Ad

ditio

n an

d su

btra

ctio

n of

com

mon

fra

ctio

ns w

ith th

e sa

me

deno

min

ator

s

Addi

tion

and

subt

ract

ion

of m

ixed

nu

mbe

rs

Fr

actio

ns o

f who

le n

umbe

rs w

hich

re

sult

in w

hole

num

bers

Rec

ogni

ze, d

escr

ibe

and

use

the

equi

vale

nce

of d

ivis

ion

and

fract

ions

Des

crib

ing

and

orde

ring

frac

tions

: • C

ompa

re a

nd o

rder

com

mon

frac

tions

, in

clud

ing

tent

hs a

nd h

undr

edth

s C

alcu

latio

ns w

ith fr

actio

ns:

Ad

ditio

n an

d su

btra

ctio

n of

com

mon

fra

ctio

ns in

whi

ch o

ne d

enom

inat

or

is a

mul

tiple

of a

noth

er

Ad

ditio

n an

d su

btra

ctio

n of

mix

ed

num

bers

Frac

tions

of w

hole

num

bers

174 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Solv

ing

prob

lem

s So

lve

prob

lem

s in

con

text

s in

volv

ing

fract

ions

, inc

ludi

ng g

roup

ing

and

equa

l sh

arin

g Eq

uiva

lent

form

s:

Rec

ogni

ze a

nd u

se e

quiv

alen

t for

ms

of

com

mon

frac

tions

(fra

ctio

ns in

whi

ch

one

deno

min

ator

is a

m

ultip

le o

f ano

ther

)

Solv

ing

prob

lem

s So

lve

prob

lem

s in

con

text

s in

volv

ing

com

mon

frac

tions

, inc

ludi

ng g

roup

ing

and

shar

ing

Equi

vale

nt fo

rms:

R

ecog

nize

and

use

equ

ival

ent f

orm

s of

co

mm

on fr

actio

ns (f

ract

ions

in w

hich

one

de

nom

inat

or is

a

mul

tiple

of a

noth

er)

Solv

ing

prob

lem

s So

lve

prob

lem

s in

con

text

s in

volv

ing

com

mon

frac

tions

, inc

ludi

ng g

roup

ing

and

shar

ing

Perc

enta

ges

• Fin

d pe

rcen

tage

s of

who

le n

umbe

rs

Equi

vale

nt fo

rms:

Rec

ogni

ze a

nd u

se e

quiv

alen

t for

ms

of c

omm

on fr

actio

ns w

ith 1

-dig

it or

2-

digi

t den

omin

ator

s (fr

actio

ns in

w

hich

one

den

omin

ator

is a

mul

tiple

of

ano

ther

)

Rec

ogni

ze e

quiv

alen

ce b

etw

een

com

mon

frac

tion

and

deci

mal

fra

ctio

n fo

rms

of th

e sa

me

num

ber

R

ecog

nize

equ

ival

ence

bet

wee

n co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r R

ecog

nizi

ng, o

rder

ing

and

plac

e va

lue

of d

ecim

al fr

actio

ns

C

ount

forw

ards

and

bac

kwar

ds in

de

cim

al fr

actio

ns to

at l

east

two

deci

mal

pla

ces

C

ompa

re a

nd o

rder

dec

imal

fra

ctio

ns to

at l

east

two

deci

mal

pl

aces

Plac

e va

lue

of d

igits

to a

t lea

st tw

o de

cim

al p

lace

s

175MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Cal

cula

tions

with

dec

imal

frac

tions

Addi

tion

and

subt

ract

ion

of d

ecim

al

fract

ions

with

at l

east

two

deci

mal

pl

aces

Mul

tiply

dec

imal

frac

tions

by

10 a

nd

100

Solv

ing

prob

lem

s So

lve

prob

lem

s in

con

text

invo

lvin

g de

cim

al fr

actio

ns

Cal

cula

tions

Fi

nd p

erce

ntag

es o

f who

le n

umbe

rs.

Equi

vale

nt fo

rms:

Rec

ogni

ze e

quiv

alen

ce b

etw

een

com

mon

frac

tion

and

deci

mal

fra

ctio

n fo

rms

of th

e sa

me

num

ber

R

ecog

nize

equ

ival

ence

bet

wee

n co

mm

on fr

actio

n, d

ecim

al fr

actio

n an

d pe

rcen

tage

form

s of

the

sam

e nu

mbe

r

176 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

SUG

GES

TED

M

ETH

OD

OLO

GY

Ref

er to

the

Less

on p

lans

in T

erm

2

In G

rade

4 th

is is

revi

sion

and

con

solid

atio

n of

the

conc

epts

dev

elop

ed in

Ter

m 3

. (Se

e Te

rm 3

not

es)

In T

erm

4 le

ngth

, cap

acity

and

mas

s ca

n be

use

d as

con

text

s fo

r fra

ctio

ns.

Teac

hing

frac

tions

in d

iagr

am fo

rms

shou

ld in

clud

e: re

gion

mod

els

, len

gth

mod

els

(incl

udin

g nu

mbe

r lin

es) a

nd s

et m

odel

s.

Lear

ners

sho

uld

solv

e pr

oble

ms

as w

ell a

s w

ork

with

app

arat

us to

und

erst

and

the

rela

tions

hip

betw

een

fract

ions

and

div

isio

n i.e

. if

you

shar

e 1

amon

gst 3

lear

ners

you

will

be m

akin

g th

irds.

Le

arne

rs m

ust a

ble

to n

ame

fract

ions

. Ter

min

olog

y lik

e “3

ove

r 4” s

houl

d be

avo

ided

as

it te

nds

to e

ncou

rage

lear

ners

to th

ink

abou

t eac

h fra

ctio

n as

two

diffe

rent

num

bers

, rat

her t

han

3 4 bei

ng a

num

ber w

hich

is g

reat

er 1 2 t

han

but l

ess

than

1.

Whe

n na

min

g fra

ctio

n pa

rts it

is u

sefu

l for

lear

ners

to ra

ther

use

the

form

“3 q

uarte

rs”.

Lear

ners

sho

uld,

thro

ugh

wor

k w

ith a

ppar

atus

, dia

gram

s an

d so

lvin

g pr

oble

ms,

lear

n th

e ne

w fr

actio

ns th

at th

ey w

ill de

al w

ith in

G

rade

4

Teac

h fr

actio

ns a

s eq

ual p

arts

of a

who

le

Exam

ple

1 - D

ivid

e a

bana

na u

neve

nly

and

ask

two

lear

ners

whe

ther

they

wou

ld b

e sa

tisfie

d w

ith th

is d

ivis

ion.

Why

not

? - E

xpla

in th

at fr

actio

ns a

re fo

rmed

with

the

divi

ding

of a

who

le in

to E

QU

AL P

ARTS

. Ex

ampl

e 2

Giv

e di

ffere

nt m

athe

mat

ical

sha

pes

to le

arne

rs –

circ

les

/ rec

tang

les

/ squ

ares

/ et

c.

- Wor

k pr

actic

ally

– d

ivid

e on

e (c

ircle

) equ

ally

bet

wee

n 2

– th

en 4

– th

en 8

lear

ners

- E

xpla

in w

hat w

e ca

ll th

ese

fract

ions

and

wha

t the

not

atio

n lo

oks

like

1 2 ; 1 4 (

at fi

rst o

nly)

Ex

ampl

e 3

Div

ide

a ch

ocol

ate

(rect

angl

e) e

qual

ly b

etw

een

3 –

then

6 le

arne

rs.

177MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Exam

ple

4 D

ivid

e a

cake

(squ

are)

equ

ally

bet

wee

n 5

– th

en 1

0 le

arne

rs.

Exp

lain

the

nota

tion

ever

y tim

e in

the

abov

e ex

ampl

es.

A w

ritte

n or

ora

l exe

rcis

e ca

n be

don

e at

this

poi

nt. T

he fo

llow

ing

can

be u

sed:

Th

e te

ache

r sho

uld

take

the

lead

and

ask

que

stio

ns. A

sk th

e le

arne

rs to

writ

e th

e fra

ctio

n no

tatio

n on

the

boar

d fo

r the

who

le

clas

s to

see

. Ad

ditio

n of

frac

tions

: W

ork

with

con

cret

e ob

ject

s:

Wor

k w

ith d

iagr

ams.

Sha

de o

ne p

art i

n on

e co

lour

. Sha

de a

noth

er p

art i

n a

diffe

rent

col

our.

Whi

ch fr

actio

n is

sha

ded

in to

tal?

W

ork

with

frac

tion

wal

l: (S

tart

with

sim

ple

fract

ions

) St

art w

ith th

e sa

me

deno

min

ator

, e.g

., 1 4 +

1 4 If

lear

ners

can

do

the

addi

tion

usin

g co

ncre

te e

xam

ples

, the

y ca

n th

en p

rogr

ess

to w

ritin

g th

e fra

ctio

n no

tatio

n be

low

the

conc

rete

exa

mpl

e.

Fol

low

thes

e st

eps,

e.g

. St

ep 1

: Mak

e su

re th

e bo

ttom

num

bers

(the

den

omin

ator

s) a

re th

e sa

me

Step

2: A

dd th

e to

p nu

mbe

rs (t

he n

umer

ator

s). P

ut th

e an

swer

ove

r the

sam

e de

nom

inat

or a

s in

ste

p 1

Step

3: S

impl

ify th

e fra

ctio

n (if

nee

ded)

. D

ECIM

ALS

Im

port

ant:

Ref

er to

the

Less

on P

lans

on

Dec

imal

Fra

ctio

ns in

Ter

m 2

Con

vert

from

frac

tions

to d

ecim

al fr

actio

ns a

nd to

per

cent

ages

. Onl

y ta

ught

in G

rade

6.

178 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

PER

CEN

TAG

ES

Perc

enta

ges

are

a ne

w to

pic

for G

rade

6 le

arne

rs.

Lear

ners

hav

e al

read

y w

orke

d w

ith t

enth

s an

d hu

ndre

dths

in

com

mon

fra

ctio

n fo

rm.

They

sho

uld

star

t by

rew

ritin

g an

d co

nver

ting

tent

hs a

nd h

undr

edth

s in

com

mon

frac

tion

form

to p

erce

ntag

es. W

here

den

omin

ator

s of

oth

er fr

actio

ns a

re fa

ctor

s of

10,

e.g

., 2,

5 o

r fac

tors

of 1

00 e

.g. 2

, 4, 5

, 20,

25,

50

lear

ners

can

con

vert

thes

e to

hun

dred

ths

usin

g w

hat t

hey

know

abo

ut

equi

vale

nce.

Eq

uiva

lenc

e be

twee

n co

mm

on fr

actio

ns a

nd p

erce

ntag

e Le

arne

rs a

re n

ot e

xpec

ted

to b

e ab

le to

con

vert

any

com

mon

frac

tion

into

its

perc

enta

ge fo

rm, m

erel

y to

see

the

rela

tions

hip

betw

een

tent

hs a

nd h

undr

edth

s in

thei

r per

cent

age

form

. Lea

rner

s sh

ould

be

able

to c

onve

rt an

y de

cim

al fr

actio

n in

tent

hs o

r hu

ndre

dths

into

a p

erce

ntag

e.

Cal

cula

tions

Le

arne

rs s

houl

d be

abl

e to

find

per

cent

ages

of w

hole

num

ber,s

e.g

., W

hat i

s 25

% o

f R30

0? H

ere

lear

ners

use

wha

t the

y kn

ow

abou

t bot

h co

nver

ting

betw

een

perc

enta

ge a

nd c

omm

on fr

actio

n fo

rm a

nd a

lso

wha

t the

y kn

ow a

bout

find

ing

fract

ions

of w

hole

nu

mbe

rs e

.g. 2

5% o

f R30

= 1 4

of R

300

= R

75.

LTSM

R

esou

rces

G

r 4 T

ext b

ooks

, Int

erne

t Web

site

s, N

umbe

r boa

rd,,

Cui

senn

arie

rods

/num

ber r

ods,

Fra

ctio

n ci

rcle

s, 2

D re

latio

nal s

hape

s, B

ase

10 b

lock

s/D

iene

s bl

ocks

, Con

cret

e m

ater

ial,

e.g.

cou

nter

s, N

umbe

r lin

es, F

ract

ion

wal

l cou

nter

s; N

umbe

r lin

es +

em

pty

ones

; H

undr

ed c

hart

G

rade

4

Gra

de 5

G

rdad

e 6

Wor

kboo

k re

fere

nce

W

B 2

pp.

128

– 1

38

Activ

ities

118

- 12

3

WB

2 p

p. 2

– 1

8; 2

8; 1

16

Activ

ities

65

– 7

3, 7

7, 1

12

WB

2 p

p. 1

32 –

148

Ac

tiviti

es 1

13 -

120

DB

E Te

xtbo

ok re

fere

nce

Term

4: U

nit 5

: Com

mon

Fra

ctio

ns, p

. 31

2 Te

rm 4

: Uni

t 4: C

omm

on F

ract

ions

, p.

301

Term

4: U

nit 3

: Com

mon

Fra

ctio

ns, p

. 30

6 H

OM

EWO

RK

ASSE

SSM

ENT

E.g.

Info

rmal

ass

essm

ent

– Te

st

179MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

MAT

HEM

ATIC

S IN

TER

MED

IATE

PH

ASE

M

ULT

I GR

ADE

LESS

ON

PLA

N 8

TE

RM

4

ED

UC

ATO

R: …

……

……

……

……

…

DAT

E : f

rom

…

……

to

……

…

Tim

e: 3

HO

UR

S G

RAD

E 4

CAP

S pp

. 39

– 42

5

CAP

S pp

. 127

-131

6

CAP

S pp

. 217

– 2

22

This

less

on a

cts

as c

onso

lidat

ion

for t

he to

pic

of n

umbe

r sen

tenc

es

TOPI

C

Num

ber s

ente

nces

C

once

pts,

Ski

lls a

nd

know

ledg

e •

Writ

e nu

mbe

r sen

tenc

es to

de

scrib

e pr

oble

m s

ituat

ions

•

Solv

e an

d co

mpl

ete

num

ber

sent

ence

s by

-

insp

ectio

n -

trial

and

impr

ovem

ent

• C

heck

sol

utio

n by

sub

stitu

tion

• W

rite

num

ber s

ente

nces

to

desc

ribe

prob

lem

situ

atio

ns

• So

lve

and

com

plet

e nu

mbe

r se

nten

ces

by

- in

spec

tion

- tri

al a

nd im

prov

emen

t •

Che

ck s

olut

ion

by s

ubst

itutio

n

• W

rite

num

ber s

ente

nces

to

desc

ribe

prob

lem

situ

atio

ns

• So

lve

and

com

plet

e nu

mbe

r se

nten

ces

by

- in

spec

tion

- tri

al a

nd im

prov

emen

t •

Che

ck s

olut

ion

by s

ubst

itutio

n

SUG

GES

TED

M

ETH

OD

OLO

GY

U

se n

umbe

r sen

tenc

es to

des

crib

e pr

oble

m s

ituat

ions

. Ex

ampl

e: P

robl

em s

ituat

ion:

I bo

ught

750

g of

sw

eets

and

div

ided

them

am

ong

25 c

hild

ren.

Num

ber s

ente

nce:

750

÷25

U

se n

umbe

r sen

tenc

es a

s eq

uiva

lent

form

of e

xpre

ssio

n to

sec

tions

of f

low

dia

gram

or t

able

s.

Exam

ple:

Tab

le

1 2

3 4

×3

180 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

Flo

w d

iagr

am

E

quiv

alen

t rep

rese

ntat

ion

of th

e ta

ble

and

the

flow

dia

gram

: ×

3

• U

sing

num

ber

sent

ence

s to

hel

p le

arne

rs u

nder

stan

d an

d us

e th

e fa

ct t

hat

addi

tion

and

subt

ract

ion

are

inve

rse

oper

atio

ns

Exa

mpl

e: 3

7 –

4 +

4 =

∆ •

Use

num

ber s

ente

nces

hel

ps le

arne

rs d

evel

op a

dditi

on a

nd s

ubtra

ctio

n te

chni

ques

Exa

mpl

e: 3

6+13

= th

eref

ore

49 –

13

=

•

Com

mut

ativ

e pr

oper

ty o

f add

ition

Num

bers

can

be

adde

d in

any

ord

er.

Exam

ple:

29

+ 19

= 1

9 +

26

T

here

fore

13

+ 49

= o

r 49

+ 1

3 =

•

Ass

ocia

tive

prop

erty

of a

dditi

on

Th

e as

soci

ativ

e pr

oper

ty a

llow

s nu

mbe

rs t

o be

gro

uped

in

diffe

rent

way

s w

hen

addi

ng m

ore

than

wo

num

bers

, with

out i

t af

fect

ing

the

answ

er.

Exa

mpl

e: (3

1 +

26) +

19=

i

s th

e sa

me

as 3

1 +(

26+

19)

•

Use

num

ber s

ente

nces

to h

elp

lear

ners

see

and

use

pat

tern

s in

add

ition

and

sub

tract

ion

num

ber b

onds

for:

181MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

-

10

-

mul

tiple

s of

10

-

mul

tiple

s of

100

Exa

mpl

es:

-

Ten

3 +

7=

4

+ 6

= 2

+ 8

=

5 +

5 =

- M

ultip

les

of 1

0

1

3 +

7 =

14

+ 6

=

12

+ 8=

15 +

5 =

- M

ultip

les

of 1

00

Sim

ilar e

xam

ples

can

be

give

n fo

r mul

tiple

s of

100

suc

h as

200

; 300

; 400

; 500

; 600

; 700

; 800

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182 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TASKS & MEMORANDA

183MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

EXEMPLAR INVESTIGATION: GRADE 4

TERM 4 Investigating the relationship between perimeter and area 15 marks

1. The side of each square in the grid below is 1unit. Determine the perimeter and the area of each of the shape. Write your answers in the table provided.

B D

C

A

E

Shape A B C D E

Perimeter

Area in number of squares

(10)

2. Does the shape with the longest perimeter also have the greatest area? (1)

3. Which shapes have equal perimeter? (2)

4. Do shapes of equal perimeter have equal area? (1)

5. Does the perimeter tell something about the area of a shape? (1)

184 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

EXEMPLAR INVESTIGATION MEMO: GRADE 4 TERM 4

Investigating the relationship between perimeter and area 15 marks

1. The side of each square in the grid below is 1unit. Determine the perimeter and the area of each of the shape. Write your answers in the table provided.

B D

C

A

E

Shape A B C D E

Perimeter 12 12 14 14

12

Area in number of squares 9 5 6 6 8

(10)

2. Does the shape with the longest perimeter also have the greatest area?

No (1)

3. Which shapes have equal perimeter?

A, B and E C and D (2)

4. Do shapes of equal perimeter have equal area?

No (1)

5. Does the perimeter tell something about the area of a shape?

No (1)

185MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

GRADE 4 EXEMPLAR TEST – TERM 2

INSTRUCTIONS TO LEARNERS:

1. Answer all questions in the spaces provided. 2. The use of calculators is NOT allowed. 3. The test duration is 40 minutes

QUESTION 1 Circle the letter next to the correct answer.

1.1 The multiple of 5 in the following is

A 58

B 90

C 59

B 09 (1)

1.2 The number 967 rounded off to the nearest 10 is

A 900

B 960

C 970

D 1 000 (1)

1.3 Complete: 40 + 35 = 30 + ___

A 35

B 40

C 75

D 45 (1)

(3) QUESTION 2 2.1 Write the following numbers in order from the smallest to the largest.

4 910 9 410 4 190 9 140

_________________________________________________________________

(1)

2.2 What is the biggest number you can make using the digits

7 1 4 8 ?

_________________________________________________________________

(1)

2.3 What is the value of the underlined digit?

8 367

_________________________________________________________________

(1)

2.4 Fill in the missing numbers in the following:

2.4.1 930; 933; 936; __________. (1)

186 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

2.4.2 7 050; 7 000; 6 950; __________. (1)

2.5 Which number is written in expanded notation as

(4 × 1 000) + (5 × 100) + (8 × 10) + (2 × 1)?

_________________________________________________________________

(1)

2.6 Write 6 427 in words.

______________________________________________________________________

____________________________________________________________

(1)

2.7 Draw a circle around all the odd numbers in the group below.

(2)

2.8 Say whether each of the following is TRUE or FALSE.

2.8.1 67 − 45 is equal to 45 − 67 (1)

2.8.2 54 + 29 is equal to 29 + 54 (1)

(11) QUESTION 3 Calculate each of the following.

3.1 3 679 + 3 124 (2)

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

____________________________________________

3.2 5 635 – 4 334 (2)

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

3.3 65 × 24

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

710 477 804 415 502 655 312 666 971

187MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

_____________________________________________ (3)

3.4 584 ÷ 8

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

(10) QUESTION 4 4.1 If 3 153 people live in Phambili and 5 244 peolple live in Sinamuva. How many

people live in Phambili and Sinamuva altogether?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

4.2 Rajen changes his R100 note for R5 coins only. How many coins does he get.

Rajen gets ____________________________ coins.

(3)

4.3 Betty wants to buy a book for R46,99 and a poster for R25,89.

4.3.1 How much will this cost altogether?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

4.3.2 Betty only has R60,00 in her wallet. How much more money does she need?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

4.4 Tshepo planted 11 rows of tomatoes with 6 plants in each row. How many tomato plants

did he plant altogether?

______________________________________________________________________

______________________________________________________________________

188 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(2)

4.5 Calculate the difference between the value of the underlined digits in the numbers 9 008

and 8 190.

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

(3)

(18)

189MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

GRADE 4 EXEMPLAR TEST – MEMORANDUM

QUESTION EXPECTED ANSWER MARKS TOTAL

1.1 B (1) 3 1.2 C (1)

1.3 D (1) 2.1 4 190 4 910 9 140 9 410 (1)

11

2.2 8 741 (1) 2.3 8 000 (1) 2.4.1 939 (1) 2.4.2 6 900 (1) 2.5 4 582 (1) 2.6 Six thousand four hundred and twenty seven (1) 2.7

(2)

2.8.1 False (1) 2.8.2 True (1) 3.1 3 659 + 3 124

= 3 000 + 600 + 50 + 9 + 3 000 + 100 + 20 + 4 = 3 000 + 3 000 + 600 + 100 + 50 + 20 + 9 + 4 = 6 000 + 700 + 70 + 13 = 6 783

(2)

10

3.2 5 635 – 4 334 = 5 000 + 600 + 30 + 5 – (4 000 + 300 + 30 + 4) = (5 000 – 4 000) + (600 – 300) + (30 –30) + (5 – 4) = 1 000 + 300 + 0 + 1 = 1 301

(2) 3.3 65 × 24

= (60 + 5) × (20 + 4) = (60 × 20) + (60 × 4) + (5 × 20) + (5 × 4) = 1 200 + 240 + 100 + 20 = 1 440 + 100 + 20 = 1 540 + 20 = 1 560

(3) 3.4 586 ÷ 8 = 70 + 3 = 73 remainder 2

Multiply Subtract CLUE BOARD 70 × 8 = 560 3 × 8 = 24

586 − 560 = 26 26 – 24 = 2 rem

10 × 8 = 80 20 × 8 = 160 30 × 8 = 240 40 × 8 = 320 50 × 8 = 400 60 × 8 = 480 70 × 8 = 560 80 × 8 = 640

8 × 1 = 8 8 × 2 = 16 8 × 3 = 24

(3) 4.1 3 153 + 5 244

= 3 100 + (5 244 + 53) [Rounding off and compensating] = 3 100 + 5 297 = 8 398 8 398 live in Phambili and Sinamuva,

(3)

18

710 477 804 415 502 655 312 666 971

190 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

4.2 Rajen gets 20 coins.

(3) 4.3.1 R46,99 + R25,89

= R47,00 + R25,88 [Rounding off and compensating] = R40,00 + R32,88 =R72,88 The book and poster cost R72,88

(3)

4.3.2 She needs R72,88 – R60,00 = R 12,88

(3)

4.4 Tshepo planted 11 rows of tomatoes with 6 plants in each row. How many tomato plants did he plant altogether? Tshepo planted 11 × 6 = 66 rows

(3)

4.5 The difference is 9 000 – 90 = 8 910 (3)

GRAND TOTAL 40 MARKS

191MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

EXEMPLAR INVESTIGATION: GRADE 5 TERM 4

Investigating the relationship between perimeter and area 20 marks

QUESTION 1

1.1 Draw the following shapes in the grid provided. Label you’re the shapes. a) A square S1 with a perimeter of 16 units. b) Two different rectangles R1 and R2 also having a perimeter of 16 units

c)

(3)

1.2 Use the shapes that you have drawn to complete the table.

Shape S1 R1 R2 Perimeter Area

(3)

1.3 Is the area of the square greater, smaller or equal to that of the triangles? (1) 1.4 Do the shapes have equal area? (1) 1.5 Is it correct to say that, “quadrilaterals with equal perimeter have equal area”? (1)

192 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

QUESTION 2

2.1 Draw the following shapes in the grid provided; three different rectangles R1, R2, and R3 each with an area of 24 squares. Label your diagrams.

d)

(3)

2.2 Use the shapes that you have drawn to complete the table.

Shape R1 R2 R3 Perimeter Area

(3)

2.3 Do the shapes have equal perimeter? (1) 2.4 Does the shape with a larger perimeter have the biggest area? (1) 2.4 Is it correct to say that, “quadrilaterals with equal area have equal perimeter”? (1) 2.5 Does perimeter of a shape tell us anything about the area of that shape? (1) 2.6 Does area of a shape tell us anything about the perimeter of the shape? (1)

193MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

EXEMPLAR INVESTIGATION MEMO: GRADE 5 TERM 4

Investigating the relationship between perimeter and area 20 marks

QUESTION 1

1.2 Draw the following shapes in the grid provided. Label you’re the shapes. e) A square S1 with a perimeter of 16 units. f) Two different rectangles R1 and R2 also having a perimeter of 16 units

S1

R1

R2

(3)

1.3 Use the shapes that you have drawn to complete the table.

Shape S1 R1 R2 Perimeter 16 16 16 Area 16 15 12

(3)

1.3 Is the area of the square greater, smaller or equal to that of the triangles? Greater

(1) 1.4 Do the shapes have equal area?

No (1) 1.5 Is it correct to say that, “quadrilaterals with equal perimeter have equal area”?

It is incorrect (1)

194 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

QUESTION 2

2.3 Draw the following shapes in the grid provided; three different rectangles R1, R2, and R3 each with an area of 24 squares. Label your diagrams.

g)

R1

R2

R3

[Accept any correct diagrams] (3)

2.4 Use the shapes that you have drawn to complete the table.

Shape R1 R2 R3 Perimeter 20 22 28 Area 24 24 24

(3)

2.3 Do the shapes have equal perimeter? No (1)

2.4 Does the shape with a larger perimeter have the biggest area? No (1)

2.4 Is it correct to say that, “quadrilaterals with equal area have equal perimeter”? No (1) 2.5 Does perimeter of a shape tell us anything about the area of that shape?

It tells us nothing (1) 2.6 Does area of a shape tell us anything about the perimeter of the shape?

It does not tell us anything (1)

195MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

GRADE 5 EXEMPLAR TEST – TERM 2

INSTRUCTIONS TO LEARNERS:

1. Answer all questions in the space provided. 2. The use of calculators is NOT allowed. 3. The test duration is 60 minutes

QUESTION 1 Circle the letter next to the correct answer.

1.1 Which number is 2 000 more than 957 467?

A 972 467

B 959 467

C 957 667

B 977 467 (1)

1.2 The third multiple of 9 is

A 3

B 9

C 18

D 27 (1)

1.3 Which number consists of the following:

6H + 4Th + 8Tth + 7U + 5T

A 64 875

B 57 864

C 75 684

D 84 657 (1)

1.4 A number divided by 5 is 12. What is the number?

A 55

B 60

C 65

D 70 (1)

1.5 The number 33 756 rounded off to the nearest 5 is ____

A 33 750

B 33 755

C 33 760

D 33 765 (1)

[5]

196 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

QUESTION 2 2.1 Write the following numbers in order from the smallest to the largest.

4 910 9 410 4 190 9 140

________________________________________________________________

(1)

2.2 What is the biggest number you can make using the digits

7 1 4 8 5?

________________________________________________________________

(1)

2.3 What is the value of the underlined digit?

128 367

________________________________________________________________

(1)

2.4 Fill in the missing numbers in the following:

2.4.1 5 930 ; 5 933 ; 5 936 ; __________. (1)

2.4.2 27 050 ; 27 000 ; 26 950 ; __________. (1)

2.5 Which number is written in expanded notation as

(6 × 10 000) + (4 × 1 000) + (5 × 100) + (8 × 10) + (2 × 1)?

________________________________________________________________

(1)

2.6 Write 86 427 in words.

_____________________________________________________________________

___________________________________________________________

(1)

2.7 Draw a triangle around all the odd numbers in the group below.

(2)

[9]

QUESTION 3

3.1 Say whether the following are True or False?

3.1.1 51 + 32 = 32 + 51 (1)

3.1.2 34 ÷ 5 = 5 ÷ 34 (1)

3.1.3 3(4 + 5) = (3 × 4) + (3 × 5) (1)

3.1.4 63 ÷ 7 × 7 = 63 (1)

3.1.5 (251 + 27) + 49 has the same answer as 251 + (27 + 49) (1)

1710 3 477 804 6 415 3 502 655 312 666 2 971

197MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

3.2 Write down all the multiples 9 between 40 and 60.

________________________________________________________________

(1)

3.3 3.3.1 Is 21 758 closer to 21 700 or 21 800?

________________________________________________________________

(1)

3.3.2 R214,76 ≈ _________________________ rounded off to the nearest rand. (1)

3.4 3.4.1 Insert the missing factor of 12:

2, _____, 12, 3, 6, 4

(1)

3.4.2 Which 2 whole numbers can I multiply to get 99?

__________ and __________

(2)

[11]

QUESTION 4 Calculate

4.1 73 679 + 25 184

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

__________________________________________________

(2)

4.2 65 635 – 64 384

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(2)

4.3 365 × 24

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

4.4 928 ÷ 12

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

[10]

198 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

QUESTION 5 5.1 There were 41 295 spectators at a soccer match, 23 985 were men,11 378 were women

and the remainder were children.

5.1.1 How many adults attended the soccer match?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

5.2 How many children attended the soccer match?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

__________________________________________________

(3)

5.3 Mr Brown buys a muffin for R2,50 and sells it for R5,50.

5.3.1 How much profit does he make?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

5.3.2 How much profit will he make if he sells 35 muffins?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

5.5 375 learners will use buses during a school excursion. How many buses will they hire if

each bus transposts 95 learners?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

_____________________________________________

(3)

[15]

199MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

GRADE 5 MATHEMATICS TEST MEMORANDUM – TERM 2

General marking notes:

1. Give full marks for answers only, unless otherwise stated. 2. Accept any alternative correct solution that is not included in the memorandum. 3. Apply consistency accuracy [CA], where necessary.

QUESTION

EXPECTED ANSWER MARKS

TOTAL

1.1 B (1)

5

1.2 D (1)

1.3 D (1)

1.4 B (1)

1.5 B (1)

2.1 4 190 4 910 9 140 9 410 (1)

9

2.2 87 541 (1)

2.3 Twenty thousand or 20 000 or (2 × 10 000) (1)

2.4.1 5 930; 5 933; 5 936; 5 939 (1)

2.4.2 27 050; 27 000; 26 950; 26 900 (1)

2.5 64 582 (1)

2.6 Eighty six thousand four hundred and twenty seven (1)

2.7

numbers are 655, 2 971, 3 477 and 6 415

(2)

3.1.1 True (1)

11

3.1.2 False (1)

3.1.3 True (1)

3.1.4 True (1)

3.1.5 True (1)

3.2 45 and 54 (1)

3.3.1 21 758 is closer to 21 800 (1)

3.3.1 R214,76 ≈ R215,00 rounded off to the nearest rand. (1)

3.4.1 1 (1)

3.4.2 9 and 11 (2)

4.1 73 679 + 25 184 98 863

70 000 + 3 000 + 600 + 70 + 9 +20 000 + 5 000 + 100 + 80 + 4 90 000 + 8 000 + 800 + 60 + 3

= 98 863

[Accept any method]

1 mark for 863 and 1

mark for 98.

A mark for correct

hundreds, tens & units

and another mark for

thousands and ten

thousands

(2)

10

4.2 65 635 – 64 384

Any suitable method [1 mark for 251 and 1 (2)

200 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

1 251 mark for 1.

A mark for correct

hundreds, tens & units

and another mark for

thousands]

4.3 365 × 24 = (300 + 60 + 5) × (20 + 4) = (300 × 20) + (60 × 20) + (5 × 20) + (300 × 4) + (60

× 4) + (5 × 4) = 6 000 + 1 200 + 100 + 1 200 + 240 + 20 = 8 400 + 340 + 20 = 8 740 + 20 = 8 760

[Accept any method]

[2 marks for method

1 mark for correct

answer]

(3)

4.4

928 ÷ 12 = 77 rem 4

MULTIPLY SUBTRACT CLUE BOARD 12 × 70 = 840 12 × 7 = 84

928 − 840 = 88 88 − 84 = 4 rem

12 × 10 = 120 12 × 20 = 240 12 × 30 = 360 12 × 40 = 480 12 × 50 = 600 12 × 60 = 720 12 × 70 = 840 12 × 80 = 960

[Accept any method]

[2 marks for metho

1 mark for correct

answer]

(3)

5.1.1 23 985 + 11 378 = 35 363 ∴ 35 363 adults attended the soccer match.

[Accept any method]

[2 marks for method

1 mark for correct

answer]

(3)

15

5.1.2 41 295 + 35 663 5 932 ∴ 5 932 children attended the soccer match.

[Accept any method]

[Apply consistency

accuracy]

[2 marks for method

1 mark for correct

answer]

(3)

5.2.1 R5,50 −R2,50 R3,00 ∴ He makes R3,00 profit.

[Accept any method]

[2 marks for method

1 mark for correct

answer]

(3)

5.2.2 He will make R3,00 × 35 = R105,00 profit. (R3,00 × 30) + (R3,00 × 5) = R90,00 + R15,00 = R105,00

[Accept any method]

[Apply consistency

accuracy]

[2 marks for method

1 mark for correct

answer]

(3)

201MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

5.3 CALCULATION CLUE BOARD [Accept any method]

[Apply consistency

accuracy]

[2 marks for method

1 mark for correct

answer]

(3) 380 is bigger than 375 , so we choose 285. So we can say 3 groups of 95 is 285. We then subtract : 375 – 285 = 90 375 ÷ 95 = 3 rem 90 ∴ The school will hire 4 buses for an

excursion.

1 × 95 = 95 2 × 95 = 190 3 × 95 = 285 4 × 95 = 380

GRAND TOTAL 40

202 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

EXEMPLAR INVESTIGATION: GRADE 6 TERM 4

Investigating the relationship between surface area and volume 15 marks

QUESTION 1

1.1 Determine the Surface area and the volume of each stack of boxes below. Give the surface area in number of squares and the volume in number of cubes.

STACK A B C D E

Surface area in number of squares

VOLUME in number of cubes

(12)

1.2 Which stacks have the same volume? (2)

203MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

1.3 Is the surface area of the stack A equal to that of stack B? (1) 1.4 Is the surface area of the stack D equal to that of stack F? (1) 1.5 Do stacks of equal volume also have equal surface area? (1) 1.6 Which stacks have equal surface area? (1) 1.5 Do stacks of equal surface area have equal volume? (1) 1.6 Does knowing volume of a stack tell us the surface area of the stack? (1) 1.7 What conclusion can be drawn from the above observations? (2)

204 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

EXEMPLAR INVESTIGATION MEMO: GRADE 6 TERM 4

Investigating the relationship between surface area and volume 15 marks

QUESTION 1

1.1 Determine the Surface area and the volume of each stack of boxes below. Give the surface area in number of squares and the volume in number of cubes.

1.2

STACK A B C D E

Surface area in number of squares

26

22

22

28

24

VOLUME in number of cubes

6 6 6 8 8

(10)

205MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

1.2 Which stacks have the same volume? A, B and C D and E (2)

1.3 Is the surface area of the stack A equal to that of stack B?

No (1) 1.4 Is the surface area of the stack D equal to that of stack F?

No (1) 1.5 Do stacks of equal volume also have equal surface area?

Sometimes (1) 1.6 Which stacks have equal surface area?

B and C (1) 1.5 Do stacks of equal surface area have equal volume?

Sometimes (1) 1.6 Does knowing volume of a stack tell us the surface area of the stack?

No (1) 1.7 What conclusion can be drawn from the above observations?

Stacks of equal volume do not necessarily have equal surface area Stacks of equal surface area have equal volume (2)

206 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

GRADE 6 EXEMPLAR TEST – TERM 2

INSTRUCTIONS TO LEARNERS:

1. Answer all questions in the space provided. 2. The use of calculators is NOT allowed. 3. The test duration is 60 minutes

QUESTION 1 Circle the letter next to the correct answer.

1.1 The number, three hunderd and fifty nine thousand eight hundred and three can be

written as

A 359 308

B 395 803

C 359 803

B 359 083 (1)

1.2 Which number is represented by (3 × 100 000) + (40 × 1 000) + (9 000) + (15 tens) +

(7 × 1)?

A 349 570

B 350 507

C 349 082

D 349 157 (1)

1.3 Which number is 3 000 000 more than 567 432 957 ?

A 570 432 957

B 567 732 957

C 597 432 957

D 867 432 957 (1)

1.4 The value of the underlined digit in 65 359 468 is

A 5 × 10 000 000

B 5 × 1 000 000

C 5 × 100 000

D 50 000 000 (1)

1.5 Which is the best estimation of the number of South African soccer fans?

A 50 000

B 1 800

C 1 208 400

D 160 000

(1)

207MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

1.6 The number 133 756 rounded off to the nearest 5 is ____

A 133 750

B 133 755

C 133 760

D 133 765 (1)

1.7 What is the value of B if 51 − 6 = 𝑩𝑩 − 51

A 45

B 67

C 81

D 96 (1)

1.8 Which sum does not give an answer of 49?

A 8 × 5 + 9

B (8 × 5) + 9

C 8 × (5 + 9)

D (8 × 5 + 9) (1)

1.9 Which number is not a factor of 36?

A 3

B 4

C 8

D 18 (1)

10 Which number between 12 and 144 is a multiple of 12?

A 6

B 12

C 240

D 96 (1)

(10) QUESTION 2 2.1 Write the following numbers in descending order.

134 910 139 410 143 190 139 140

_______________________________________________________________________

________

(1)

2.2 What is the biggest number you can make using the digits

7 1 4 8 5 9 4?

_______________________________________________________________________

________

(1)

2.3 What is the value of the underlined digit?

657 128 367

208 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

_______________________________________________________________________

________

(1)

2.4 Fill in the missing numbers in the following:

2.4.1 135 930 ; 135 933 ; 135 936 ; _______________. (1)

2.4.2 1 227 050 ; 1 227 000 ; 1 226 950 ; _______________. (1)

2.5 Write the number 5 103 467 in expanded notation

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

___________________________

(3)

2.6 Write 35 386 427 in words.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

___________________________

(1)

2.7 57; 27; 72; 36; 61; 81; 45

From the numbers above, choose:

2.7.1 a prime number

_______________________________________________________________________

_________

(1)

2.7.2 A number which is the product of two prime numbers

_______________________________________________________________________

________

(1)

(11) QUESTION 3

3.1 Say whether the following statements are True or False?

3.1.1 247 889 > 247 898 ____________________ (1)

3.1.2 25 × 27 ≈ (20 + 5) × (20 + 7) ______________________ (1)

3.1.3 (18 × 4) × 5 = 18 × (5 × 4) ____________________ (1)

3.1.4 When I multiply a number by 1, the answer is always equal to the original number.

____________________

(1)

3.1.5 When we add zero (0) to a number, the number is doubled.

____________________

(1)

209MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

3.2 What is the Lowest Common Multiple (LCM) of 12 and 36.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

____________________________________

(2)

3.3 3.3.1 Is 821 758 closer to 821 700 or 821 800?

_______________________________________________________________________

_________

(1)

3.3.2 R6 214,76 ≈ _________________________ rounded off to the nearest rand. (1)

3.4 3.4.1 Insert the missing factor of 12: 2 , _____ , 12 , 3 , 6 , 4

(1) 3.4.2 List all the factors of 225.

_______________________________________________________________________

_________

(1)

3.4.2 Which 2 whole numbers can I multiply to get to 125?

_______________________________________________________________________

_________ (12)

(1)

QUESTION 4 Calculate

4.1 Add using columns: 3 473 679 ; 725 184 𝑎𝑎𝑎𝑎𝑎𝑎 2 378 487

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_____________________________________________

(3)

4.2 Subtract 1 764 384 from 3 065 635

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_____________________________________________

(3)

210 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

4.3 Multiply 7 365 and 25

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_____________________________________________

(3)

4.4 Calculate the remainder, using long division, if 2 836 is divided by 450

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_

(3)

4.5 Calculate:

4.5.1 (58 + 12) − (33 − 19)

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

___________________________

(3)

4.5.2 2 + 3 × 4 − 6

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

___________________________

(17)

(2)

211MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

QUESTION 5 5.1 A lady buys a specific brand of DVD players for all her stores. She buys 126 789 black,

341 567 white and 344 532 silver DVD players. How many DVD players did she buy

altogether?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

________________________________________

(3)

5.2 Farm workers picked 342 345 pears during the morning. After lunch they picked some

more. By the end of the day, they had 866 589 pears. How many pears did they pick

after lunch?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

________________________________________

(3)

5.3 156 pairs of shoes cost R7 020.

How much will one pair of the same shoes cost?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

________________________________________

(3)

5.4 A company gave 536 boxes of soccer balls to multigrade schools. Each box contained

3 126 soccer balls. How many soccer balls did the company give away?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

________________________________________

(3)

(12)

212 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

GRADE 6 EXEMPLAR TEST MEMORANDUM – TERM 2

General marking notes:

1. Give full marks for answers only, unless otherwise stated. 2. Accept any alternative correct solution that is not included in the memorandum. 3. Apply consistency accuracy [CA], where necessary.

QUESTION EXPECTED ANSWER MARKS

TOTAL

1.1 B (1)

10

1.2 D (1) 1.3 A (1) 1.4 B (1) 1.5 C (1) 1.6 B (1) 1.7 D (1) 1.8 C (1) 1.9 C (1) 1.10 D (1) 2.1 143 190 139 410 139 140 134 910 (1)

8

2.2 9 875 441 (1)

2.3 7 000 000 or 7 million or (7 × 1 000 000) (1)

2.4.1 135 930; 135 933; 135 936; 135 939. (1)

2.4.2 1 227 050; 1 227 000; 1 226 950; 1 226 900. (1)

2.5 Thirty five million three hundred and eighty six thousand four hundred and

twenty seven

(1)

2.6.1 61 (1)

2.6.2 57 [a product of 3 and 19] (1)

3.1.1 False (1)

13

3.1.2 True (1)

3.1.3 True (1)

3.1.4 True (1)

3.1.5 False (1)

3.2 Multiples of 12 = 12; 24; 36; 48; …

Multiples of 36 = 36; 72; 108; …

Lowest Common Multiple = 36

(3)

3.3.1 821 758 is closer to 821 800. (1)

3.3.2 R6 214,76 ≈ R6 215 rounded off to the nearest rand. (1)

3.4.1 The missing factor is 1. (1)

3.4.2 Factors of 225 are: 1; 3; 5; 9;15; 45; 225. (1)

3.4.3 1 and 125 or 5 and 25 (1)

4.1 3 476 679 725 184 + 2 378 487 6 580 350

[2 marks for method 1 mark for correct answer] Correct answer –3 marks

(3) 17

213MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

4.2 3 065 635 −1 764 384 1 301 251

[2 marks for method 1 mark for correct answer] Correct answer –3 marks

(3)

4.3 Calculate the product of 7 365 and 25

7 365 × 25 36 825 + 147 200 184 125

2 marks for method

1 mark for correct answer Correct answer –3 marks

(3)

4.4 2 836 ÷ 255 = 11 remainder 31

11

255 2 836 − 255 286 −255 31 rem

2 marks for method

1 mark for correct answer Correct answer –3 marks

(3)

4.5.1 (58 + 12) − (33 − 19) = 70 − 12 = 58

2 marks for method

1 mark for correct answer

(3)

4.5.2 2 + 3 × 4 − 6 = 2 + 12 − 6 = 14 − 6 = 8

2 marks for method

1 mark for correct answer

(2)

5.1 126 789 341 567 + 344 532 812 888 She bought 812 888 DVD players.

[Accept any

method]

[2 marks for

method

1 mark for correct answer]

(3)

12

5.2 866 589 − 342 345 524 244

Farm workers picked 524 244 pears after lunch.

[Accept any

method]

[2 marks for

method

1 mark for correct answer]

(3)

5.3 7 020 ÷ 156 = 45

∴ Each shoe costs R45

[Accept any

method]

[2 marks for

method

1 mark for correct answer]

(3)

MULTIPLY SUBTRACT CLUE BOARD 156 × 40= 6 240 156 × 5= 780

7 020 − 6 240= 780 780 − 780 = 0

156 × 10 = 1 560 156 × 20 = 3 120 156 × 30 = 4 680 156 × 40 = 6 240 156 × 5 = 780

5.4 3 126 × 536 18 756 93 780

[Accept any

method]

[2 marks for

214 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

+ 1 563 000 1 675 536 The company gave 1 675 536 balls away.

method

1 mark for correct answer]

(3)

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kkin

g to

t tyd

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d

Lees

, sê

en s

kryf

12-

uur

en

24 -

uurty

d op

dig

itale

en

anal

oogi

nstru

men

te in

Ure

, min

en

seko

ndes

Le

es v

an a

lman

akke

Pr

oble

emop

loss

ing

en

bere

keni

nge

met

bet

rekk

ing

tot

tyd

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d

Lees

, sê

en s

kryf

12-

uur

en

24 -

uurty

d op

dig

itale

en

anal

oogi

nstru

men

te in

Ure

, min

en

seko

ndes

Le

es v

an a

lman

akke

Pr

oble

emop

loss

ing

en

bere

keni

nge

met

bet

rekk

ing

tot

tyd

Lees

tyds

ones

op

karte

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d LO

OM

:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

224 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

OPV

OED

ER: …

……

……

……

……

……

….

DAT

UM

: W

EEK

9

11

- 15

MAA

RT2

013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

DAT

AHAN

TER

ING

P.5

9, 1

48, 2

39

BEG

RIP

PE E

N

VAAR

DIG

HED

E Ve

rsam

el e

n or

gani

seer

da

ta

tellin

gsta

belle

] St

el d

ata

voor

Pikt

ogra

mm

e

Staa

fgra

fieke

An

alis

e; in

terp

reta

sie

en

vers

lagd

oeni

ng v

an d

ata

Pi

ktog

ram

me

St

aafg

rafie

ke

Vers

amel

en

orga

nise

er

data

[ te

llinst

abel

leas

ooko

rden

van

kl

eins

te to

t gro

otst

e]

Stel

dat

a vo

or

Pi

ktog

ram

me

St

aafg

rafie

ke

Anal

ise;

inte

rpre

tasi

e en

ve

rsla

gdoe

ning

van

dat

a

Pikt

ogra

mm

e

Staa

fgra

fieke

Sirk

eldi

agra

mm

e w

oord

elik

s

Vers

amel

en

orga

nise

er d

ata

te

llinst

abel

leas

ook

or

den

van

klei

nste

tot g

root

ste

vr

aely

ste

Stel

dat

a vo

or

Pi

ktog

ram

me

St

aafg

rafie

ke

Anal

ise;

inte

rpre

tasi

e en

ver

slag

doen

ing

van

data

Pikt

ogra

mm

e

Staa

fgra

fieke

asoo

kdub

bele

staa

fgra

fieke

Sirk

eldi

agra

mm

e

Woo

rdel

iks

m

odus

LO

OM

:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

225MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

……

……

…..

DAT

UM

: W

EEK

2

18

- 22

MAA

RT

201

3

T

YD: 5

DAE

6 U

RE

GR

AAD

4

5 6

ON

DER

WER

P Ei

ensk

appe

van

2D

-Vor

m K

ABV

Bl.

61 -6

3

KAB

V B

l. 15

0-15

2KAB

V B

l. 23

5 –

238

B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Beno

emvo

rms

dr

ieho

eke

vi

erka

nte

re

ghoe

ke

an

derv

ierh

oeke

pent

agon

e

heks

agon

e

sirk

els

Eien

skap

p va

n vo

rms

ge

boë

/regu

itsye

aant

alsy

e Te

ken

2D -V

orm

s

Beno

emvo

rms

dr

ieho

eke

vi

erka

nte

re

ghoe

ke

an

derv

ierh

oeke

pent

agon

e

heks

agon

e

hept

agon

e

sirk

els

Eien

skap

p va

n vo

rms

ge

boë

/regu

itsye

aant

alsy

e

leng

te v

an s

ye

ho

eke

[regt

ehoe

ke

en k

lein

er/g

rote

r as

regt

ehoe

ke]

Teke

n 2D

-Vor

ms

Ken

en

beno

emre

ëlm

atig

e en

on

reël

mat

igev

ierh

oeke

, dr

ieho

eke,

vie

rkan

te,

regh

oeke

en

para

llelo

gram

me;

pe

ntag

one;

hek

sogo

ne;

hept

agon

e en

okt

agon

e Ei

ensk

appe

van

Vor

ms:

aant

alsy

e

leng

te v

an s

ye

gr

ootte

van

hoe

ke

Teke

n 2D

Vor

ms

; geb

ruik

pa

sser

virs

irkel

s H

erke

n en

ben

oem

hoek

e

LOO

M:

H

AND

BO

EK

VER

WYS

ING

S.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

AS

SESS

ERIN

G

TOET

S 1

226 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

OPV

OED

ER: …

……

……

……

……

……

……

..

DAT

UM

: W

EEK

3 2

8 /0

1 –

01 /0

2 2

013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

Get

alle

sinn

eInl

eidi

ng to

t Alg

ebra

iiseu

itdru

kkin

gs

B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Ek

wiv

alen

teui

tdru

kkin

gs /

geta

lsin

ne

Ko

mm

utat

iew

e-,

asso

siat

iew

e- e

n

Dis

tribu

tiew

eeie

nska

ppe

In

vers

e be

wer

king

s en

dieW

isku

ndig

ewaa

rded

aa

rvan

Ve

rmen

igvu

ldig

ings

eien

skap

van

1

O

ptel

lings

eien

skap

van

0

O

ntw

ikke

lopt

el e

n

aftre

kteg

niek

ed.m

.vge

tals

inn

e

As

sosi

atie

wee

iens

kap

van

opte

lling

G

etal

sinn

e - M

aal e

n

Dee

l met

10,

100

en

1000

Vo

lgor

de b

y

aftre

kkin

g

L

os g

etal

lesi

nne

op

deur

In

spek

sie

Pr

obee

r en

verb

eter

Kont

role

er d

ie

oplo

ssin

gdeu

rver

vang

ing

LOO

M:

Patro

onka

arte

, vlo

eika

arte

,

227MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

228 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

OPV

OED

ER: …

……

……

……

……

…..

DAT

UM

: W

EEK

4

4 –

8 F

EBR

UAR

IE

2013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

Hee

lget

alle

Opt

ellin

g&Af

trekk

ing

B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Get

alge

bied

: 3-s

yfer

get

alle

Te

gnie

ke:

Skat

ting

Opb

ou e

n af

bree

k va

n ge

talle

Ve

rdub

bel e

n ha

lvee

r G

etal

lely

ne

Afro

ndin

g en

kom

pens

erin

g Ei

ensk

appe

van

hee

lget

alle

: Ko

mm

utat

iew

e en

As

sosi

atie

wee

iens

kapp

e Pr

oble

emop

loss

ing:

Los

pr

oble

me

in k

onte

ks o

p;

insl

uite

ndfin

ansi

ëlek

onte

kste

Get

alge

bied

: 5-s

yfer

get

alle

Te

gnie

ke: S

katti

ngO

pbou

en

afbr

eek

van

geta

lle

Verd

ubbe

l en

halv

eeG

etal

lely

neAf

rond

ing

en k

ompe

nser

ing

Inve

rse

bew

erki

ngs

Eien

skap

pe v

an h

eelg

etal

le:

Kom

mut

atie

we;

As

sosi

atie

we

en

dist

rubi

tiew

eeie

nska

ppe

0 se

opt

ellin

gs- e

iens

kap

Prob

leem

oplo

ssin

g: L

os

prob

lem

e in

kon

teks

op;

in

slui

tend

finan

siël

ekon

teks

te

Get

alge

bied

: 6 –

syfe

rget

alle

Ve

elvu

ldig

ebew

erki

nge

met

/s

onde

rhak

ies

Tegn

ieke

: Ska

tting

O

pbou

en

afbr

eek

van

geta

lle

Verd

ubbe

l en

halv

eer

Get

alle

lyne

Af

rond

ing

en k

ompe

nser

in

Inve

rse

bew

erki

ngsS

akre

kena

ar

Eien

skap

pe v

an h

eelg

etal

le:

Kom

mut

atie

we;

Ass

osia

tiew

e en

di

stru

bitie

wee

iens

kapp

e

0 se

op

tellin

gs- e

iens

kap

Prob

leem

oplo

ssin

g: L

os p

robl

eme

in

kont

eks

op;

insl

uite

ndfin

ansi

ëlek

onte

kste

Ve

rgel

yk 2

of m

eerh

oeve

elhe

de v

an

dies

elfe

de s

ort [

verh

oudi

ng]

LO

OM

:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K

OPD

RAG

TE

OPM

ERK

ING

229MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

230 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

..

DAT

UM

: W

EEK

5

11

– 15

FEB

RU

ARIE

201

3

TY

D: 4

DAE

5 U

RE

GR

AAD

4

5 6

ON

DER

WER

P N

umer

iese

Patr

oon

K

ABV

Dok

. B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Ond

erso

ek e

n br

eipa

troon

uit

Ky

kna

die

reel

s

Ree

ks m

et

kons

tant

ever

skil/

verh

oudi

ngs

Besk

ryfre

ëls

in w

oord

e

Leer

der s

e ei

eske

ppin

g Be

paal

Inse

t- ;

uits

etw

aard

es e

n re

ëls

met

vlo

eidi

agra

mm

e Ek

wiv

alen

tevo

rms

Woo

rdel

iks

In

vlo

eidi

agra

mm

e M

et g

etal

lesi

nne

Ond

erso

ek e

n br

eipa

troon

uit

Ky

kna

die

reel

s

Ree

ks m

et

kons

tant

ever

skil/

verh

oud

ings

Besk

ryfre

ëls

in w

oord

e

Leer

der s

e ei

eske

ppin

g Be

paal

Inse

t- ;

uits

etw

aard

es e

n re

ëls

met

vlo

eidi

agra

mm

e Ek

wiv

alen

tevo

rms

W

oord

elik

s

Invl

oeid

iagr

amm

e M

et g

etal

lesi

nne

Ond

erso

ek e

n br

eipa

troon

uit

Ky

kna

die

reel

s

Ree

ks m

et

niek

onst

ante

vers

kil/v

erho

udin

gs

Be

skry

freël

s in

woo

rde

Le

erde

r se

eies

kepp

ing

Bepa

al In

set-

; ui

tset

waa

rdes

en

reël

s m

et v

loei

diag

ram

me

Ekw

ival

ente

vorm

s

Woo

rdel

iks

In

vloe

idia

gram

me

Met

get

alle

sinn

e

LOO

M:

H

AND

BO

EK

VER

WYS

ING

S.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K

OPD

RAG

TE

OPM

ERK

ING

231MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

……

……

……

…..

DAT

UM

: W

EEK

6

– 7

18 F

EBR

UAR

IE -

01

MAA

RT

201

3

T

YD: 1

0 D

AE

12

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

Hee

lget

alle

Verm

enig

vuld

igin

g&D

elin

g

K

ABV

Dok

B

l. 54

– 5

8 B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Get

alge

bied

: Le

erde

rson

twik

kel e

n le

erta

fels

[2x

– 10

x]

Verm

: 2-s

yfer

met

1-s

yfer

D

eel3

-syf

er d

eur 1

-syf

er

Tegn

ieke

:

ska

t O

pbou

en

afbr

eek

Afro

ndin

g en

kom

pens

erin

g Ve

rdub

bel&

halv

eerG

ebru

ik

van

‘n g

etal

lely

n O

mge

keer

debe

wer

king

Ve

elvo

ude

en F

akto

re: 1

-sy

fer g

etal

le to

t 100

Ei

ensk

appe

venH

eelg

etal

le:

Kom

mut

atie

we;

As

sosi

atie

we;

Dis

tribu

tiew

e Pr

oble

emop

loss

ing:

In

kont

eks

Fina

nsië

leKo

ntek

s Ve

rgel

ykin

gs v

an

dies

efde

soor

t [ra

tio]V

erge

lyki

ngs

van

vers

kille

ndes

oorte

G

roep

erin

g en

ge

lyke

verd

elin

g

Get

alge

bied

: Lee

rder

sont

wik

kel

en le

erta

fels

[2x

– 10

x]

Verm

: 3-s

yfer

met

2-s

yfer

D

eel 3

-syf

er d

eur 2

-syf

er

Tegn

ieke

: s

kat

Opb

ou e

n af

bree

k Af

rond

ing

en k

ompe

nser

ing

Verd

ubbe

l&ha

lvee

r G

ebru

ik v

an ‘n

get

alle

lyn

Om

geke

erde

rbew

erki

ng

Veel

voud

e en

Fak

tore

: 2-s

yfer

ge

talle

tot 1

00

Eien

skap

peve

nHee

lget

alle

: Ko

mm

utat

iew

e; A

ssos

iatie

we;

D

istri

butie

we0

en

1 in

term

e va

n sy

verm

enig

vuld

igin

gsei

ensk

ap

Prob

leem

oplo

ssin

g: In

kon

teks

Fi

nans

iële

Kont

eks

Verg

elyk

ings

van

die

selfd

esoo

rt

[ratio

] Ve

rgel

ykin

gs v

an

vers

kille

ndes

oorte

G

roep

erin

g en

ver

delin

g m

et re

ste

Get

alge

bied

: Lee

rder

sont

wik

kel

en le

erta

fels

[11x

; 12x

; 13x

; 15x

; 20

x; 2

5x; 5

0x]

Verm

: 4-s

yfer

met

3-s

yfer

[met

of

sond

erha

kies

]Dee

l 4-s

yfer

deu

r 3-

syfe

r[ m

et o

f son

derh

akie

s]

Tegn

ieke

:

ska

tOpb

ou e

n af

bree

kAfro

ndin

g en

ko

mpe

nser

ingG

ebru

iksa

krek

enaa

r Ko

lom

verm

enig

vuld

igin

g Ve

elvo

ude

en

Fak

tore

: 2-

syfe

r en

3-s

yfer

geta

lle to

t Pr

iem

fakt

ore

van

geta

lle to

t ten

m

inst

e 10

0 Ei

ensk

appe

venH

eelg

etal

le:

Kom

mut

atie

we;

Ass

osia

tiew

e;

Dis

tribu

tiew

e0 e

n 1

in te

rme

van

syve

rmen

igvu

ldig

ings

eien

skap

Pr

oble

emop

loss

ing:

In k

onte

ks

Fina

nsië

leKo

ntek

s Ve

rgel

ykin

gs v

an d

iese

lfdes

oort

[ra

tio]

Verg

elyk

ings

van

ve

rski

llend

esoo

rte

Gro

eper

ing

en v

erde

ling

met

rest

e

LOO

M:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

232 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

TUIS

WER

K

OPD

RAG

TE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

…..

DAT

UM

: W

EEK

8

4- 8

MAA

RT

2013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

TYD

KAB

V P.

57, 1

46, 2

34

BEG

RIP

PE E

N

VAAR

DIG

HED

E Le

es, s

ê en

skr

yf 1

2- u

ur e

n 24

- uu

rtyd

op d

igita

le e

n an

aloo

gins

trum

ente

in

U

re, m

in e

n se

kond

es

Lees

van

alm

anak

ke

Be

reke

naan

tald

aetu

ssen

2

datu

ms

Prob

leem

oplo

ssin

g en

ber

eken

inge

m

et b

etre

kkin

g to

t tyd

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d

Lees

, sê

en s

kryf

12-

uur

en

24 -

uurty

d op

dig

itale

en

anal

oogi

nstru

men

te in

Ure

, min

en

seko

ndes

Le

es v

an a

lman

akke

Pr

oble

emop

loss

ing

en

bere

keni

nge

met

bet

rekk

ing

tot

tyd

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d

Lees

, sê

en s

kryf

12-

uur

en

24 -

uurty

d op

dig

itale

en

anal

oogi

nstru

men

te in

Ure

, min

en

seko

ndes

Le

es v

an a

lman

akke

Pr

oble

emop

loss

ing

en

bere

keni

nge

met

bet

rekk

ing

tot

tyd

Lees

tyds

ones

op

karte

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d

233MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

LOO

M:

H

AND

BO

EK

VER

WYS

ING

S.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

234 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

OPV

OED

ER: …

……

……

……

……

……

….

DAT

UM

: W

EEK

9

11

- 15

MAA

RT2

013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

DAT

AHAN

TER

ING

P.5

9, 1

48, 2

39

BEG

RIP

PE E

N

VAAR

DIG

HED

E Ve

rsam

el e

n or

gani

seer

da

ta

tellin

gsta

belle

] St

el d

ata

voor

Pikt

ogra

mm

e

Staa

fgra

fieke

An

alis

e; in

terp

reta

sie

en

vers

lagd

oeni

ng v

an d

ata

Pi

ktog

ram

me

St

aafg

rafie

ke

Vers

amel

en

orga

nise

er

data

[ te

llinst

abel

leas

ooko

rden

van

kl

eins

te to

t gro

otst

e]

Stel

dat

a vo

or

Pi

ktog

ram

me

St

aafg

rafie

ke

Anal

ise;

inte

rpre

tasi

e en

ve

rsla

gdoe

ning

van

dat

a

Pikt

ogra

mm

e

Staa

fgra

fieke

Sirk

eldi

agra

mm

e w

oord

elik

s

Vers

amel

en

orga

nise

er d

ata

te

llinst

abel

leas

ook

or

den

van

klei

nste

tot g

root

ste

vr

aely

ste

Stel

dat

a vo

or

Pi

ktog

ram

me

St

aafg

rafie

ke

Anal

ise;

inte

rpre

tasi

e en

ver

slag

doen

ing

van

data

Pikt

ogra

mm

e

Staa

fgra

fieke

asoo

kdub

bele

staa

fgra

fieke

Sirk

eldi

agra

mm

e

Woo

rdel

iks

m

odus

LO

OM

:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

235MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

……

……

…..

DAT

UM

: W

EEK

2

18

- 22

MAA

RT

201

3

T

YD: 5

DAE

6 U

RE

GR

AAD

4

5 6

ON

DER

WER

P Ei

ensk

appe

van

2D

-Vor

m K

ABV

Bl.

61 -6

3

KAB

V B

l. 15

0-15

2KAB

V B

l. 23

5 –

238

BEG

RIP

PE E

N

VAAR

DIG

HED

E Be

noem

vorm

s

drie

hoek

e

vier

kant

e

regh

oeke

ande

rvie

rhoe

ke

pe

ntag

one

he

ksag

one

si

rkel

s Ei

ensk

app

van

vorm

s

gebo

ë /re

guits

ye

aa

ntal

sye

Teke

n 2D

-Vor

ms

Beno

emvo

rms

dr

ieho

eke

vi

erka

nte

re

ghoe

ke

an

derv

ierh

oeke

pent

agon

e

heks

agon

e

hept

agon

e

sirk

els

Eien

skap

p va

n vo

rms

ge

boë

/regu

itsye

aant

alsy

e

leng

te v

an s

ye

ho

eke

[regt

ehoe

ke

en k

lein

er/g

rote

r as

regt

ehoe

ke]

Teke

n 2D

-Vor

ms

Ken

en

beno

emre

ëlm

atig

e en

on

reël

mat

igev

ierh

oeke

, dr

ieho

eke,

vie

rkan

te,

regh

oeke

en

para

llelo

gram

me;

pe

ntag

one;

hek

sogo

ne;

hept

agon

e en

okt

agon

e Ei

ensk

appe

van

Vor

ms:

aant

alsy

e

leng

te v

an s

ye

gr

ootte

van

hoe

ke

Teke

n 2D

Vor

ms

; ge

brui

k pa

sser

virs

irkel

s H

erke

n en

ben

oem

hoek

e

LOO

M:

H

AND

BO

EK

VER

WYS

ING

S.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

AS

SESS

ERIN

G

TOET

S 1

236 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

OPV

OED

ER: …

……

……

……

……

……

……

..

DAT

UM

: W

EEK

3 2

8 /0

1 –

01 /0

2 2

013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

Get

alle

sinn

eInl

eidi

ng to

t Alg

ebra

iiseu

itdru

kkin

gs

B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Ek

wiv

alen

teui

tdru

kkin

gs /

geta

lsin

ne

Ko

mm

utat

iew

e-,

asso

siat

iew

e- e

n

Dis

tribu

tiew

eeie

nska

ppe

In

vers

e be

wer

king

s en

dieW

isku

ndig

ewaa

rded

aa

rvan

Ve

rmen

igvu

ldig

ings

eien

skap

van

1

O

ptel

lings

eien

skap

van

0

O

ntw

ikke

lopt

el e

n

aftre

kteg

niek

ed.m

.vge

tals

inn

e

As

sosi

atie

wee

iens

kap

van

opte

lling

G

etal

sinn

e - M

aal e

n

Dee

l met

10,

100

en

1000

Vo

lgor

de b

y

aftre

kkin

g

L

os g

etal

lesi

nne

op

deur

In

spek

sie

Pr

obee

r en

verb

eter

Kont

role

er d

ie

oplo

ssin

gdeu

rver

vang

ing

LOO

M:

Patro

onka

arte

, vlo

eika

arte

,

237MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

238 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

OPV

OED

ER: …

……

……

……

……

…..

DAT

UM

: W

EEK

4

4 –

8 F

EBR

UAR

IE

2013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

Hee

lget

alle

Opt

ellin

g&Af

trekk

ing

B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Get

alge

bied

: 3-s

yfer

get

alle

Te

gnie

ke:

Skat

ting

Opb

ou e

n af

bree

k va

n ge

talle

Ve

rdub

bel e

n ha

lvee

r G

etal

lely

ne

Afro

ndin

g en

kom

pens

erin

g Ei

ensk

appe

van

hee

lget

alle

: Ko

mm

utat

iew

e en

As

sosi

atie

wee

iens

kapp

e Pr

oble

emop

loss

ing:

Los

pr

oble

me

in k

onte

ks o

p;

insl

uite

ndfin

ansi

ëlek

onte

kste

Get

alge

bied

: 5-s

yfer

get

alle

Te

gnie

ke: S

katti

ngO

pbou

en

afbr

eek

van

geta

lle

Verd

ubbe

l en

halv

eeG

etal

lely

neAf

rond

ing

en k

ompe

nser

ing

Inve

rse

bew

erki

ngs

Eien

skap

pe v

an h

eelg

etal

le:

Kom

mut

atie

we;

As

sosi

atie

we

en

dist

rubi

tiew

eeie

nska

ppe

0 se

opt

ellin

gs- e

iens

kap

Prob

leem

oplo

ssin

g: L

os

prob

lem

e in

kon

teks

op;

in

slui

tend

finan

siël

ekon

teks

te

Get

alge

bied

: 6 –

syfe

rget

alle

Ve

elvu

ldig

ebew

erki

nge

met

/s

onde

rhak

ies

Tegn

ieke

: Ska

tting

O

pbou

en

afbr

eek

van

geta

lle

Verd

ubbe

l en

halv

eer

Get

alle

lyne

Af

rond

ing

en k

ompe

nser

in

Inve

rse

bew

erki

ngsS

akre

kena

ar

Eien

skap

pe v

an h

eelg

etal

le:

Kom

mut

atie

we;

Ass

osia

tiew

e en

di

stru

bitie

wee

iens

kapp

e

0 se

op

tellin

gs- e

iens

kap

Prob

leem

oplo

ssin

g: L

os p

robl

eme

in

kont

eks

op;

insl

uite

ndfin

ansi

ëlek

onte

kste

Ve

rgel

yk 2

of m

eerh

oeve

elhe

de v

an

dies

elfe

de s

ort [

verh

oudi

ng]

LO

OM

:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K

OPD

RAG

TE

OPM

ERK

ING

239MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

..

WIS

KU

ND

E IN

TERM

EDIê

RE

FASE

M

ULT

IGR

AAD

KW

ARTA

AL E

EN

O

PVO

EDER

: …

……

……

……

……

……

……

..

DAT

UM

: W

EEK

6

– 7

18 F

EBR

UAR

IE -

01

MAA

RT

201

3

T

YD: 1

0 D

AE

12

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

Hee

lget

alle

Verm

enig

vuld

igin

g&D

elin

g

K

ABV

Dok

B

l. 54

– 5

8

DAT

UM

: W

EEK

5

11

– 15

FEB

RU

ARIE

201

3

TY

D: 4

DAE

5 U

RE

GR

AAD

4

5 6

ON

DER

WER

P N

umer

iese

Patr

oon

K

ABV

Dok

. B

EGR

IPPE

EN

VA

ARD

IGH

EDE

Ond

erso

ek e

n br

eipa

troon

uit

Ky

kna

die

reel

s

Ree

ks m

et

kons

tant

ever

skil/

verh

oudi

ngs

Besk

ryfre

ëls

in w

oord

e

Leer

der s

e ei

eske

ppin

g Be

paal

Inse

t- ;

uits

etw

aard

es e

n re

ëls

met

vlo

eidi

agra

mm

e Ek

wiv

alen

tevo

rms

Woo

rdel

iks

In

vlo

eidi

agra

mm

e M

et g

etal

lesi

nne

Ond

erso

ek e

n br

eipa

troon

uit

Ky

kna

die

reel

s

Ree

ks m

et

kons

tant

ever

skil/

verh

oud

ings

Besk

ryfre

ëls

in w

oord

e

Leer

der s

e ei

eske

ppin

g Be

paal

Inse

t- ;

uits

etw

aard

es e

n re

ëls

met

vlo

eidi

agra

mm

e Ek

wiv

alen

tevo

rms

W

oord

elik

s

Invl

oeid

iagr

amm

e M

et g

etal

lesi

nne

Ond

erso

ek e

n br

eipa

troon

uit

Ky

kna

die

reel

s

Ree

ks m

et

niek

onst

ante

vers

kil/v

erho

udin

gs

Be

skry

freël

s in

woo

rde

Le

erde

r se

eies

kepp

ing

Bepa

al In

set-

; ui

tset

waa

rdes

en

reël

s m

et v

loei

diag

ram

me

Ekw

ival

ente

vorm

s

Woo

rdel

iks

In

vloe

idia

gram

me

Met

get

alle

sinn

e

LOO

M:

H

AND

BO

EK

VER

WYS

ING

S.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K

OPD

RAG

TE

REM

IND

ERS

240 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

BEG

RIP

PE E

N

VAAR

DIG

HED

E G

etal

gebi

ed:

Leer

ders

ontw

ikke

l en

leer

tafe

ls [2

x –

10x]

Ve

rm: 2

-syf

er m

et 1

-syf

er

Dee

l3-s

yfer

deu

r 1-s

yfer

Te

gnie

ke:

s

kat

Opb

ou e

n af

bree

k Af

rond

ing

en k

ompe

nser

ing

Verd

ubbe

l&ha

lvee

rGeb

ruik

va

n ‘n

get

alle

lyn

Om

geke

erde

bew

erki

ng

Veel

voud

e en

Fak

tore

: 1-

syfe

r get

alle

tot 1

00

Eien

skap

peve

nHee

lget

alle

: Ko

mm

utat

iew

e;

Asso

siat

iew

e; D

istri

butie

we

Prob

leem

oplo

ssin

g: In

ko

ntek

s Fi

nans

iële

Kont

eks

Verg

elyk

ings

van

di

esef

deso

ort

[ratio

]Ver

gely

king

s va

n ve

rski

llend

esoo

rte

Gro

eper

ing

en

gely

keve

rdel

ing

Get

alge

bied

: Lee

rder

sont

wik

kel

en le

erta

fels

[2x

– 10

x]

Verm

: 3-s

yfer

met

2-s

yfer

D

eel 3

-syf

er d

eur 2

-syf

er

Tegn

ieke

: s

kat

Opb

ou e

n af

bree

k Af

rond

ing

en k

ompe

nser

ing

Verd

ubbe

l&ha

lvee

r G

ebru

ik v

an ‘n

get

alle

lyn

Om

geke

erde

rbew

erki

ng

Veel

voud

e en

Fak

tore

: 2-s

yfer

ge

talle

tot 1

00

Eien

skap

peve

nHee

lget

alle

: Ko

mm

utat

iew

e; A

ssos

iatie

we;

D

istri

butie

we0

en

1 in

term

e va

n sy

verm

enig

vuld

igin

gsei

ensk

ap

Prob

leem

oplo

ssin

g: In

kon

teks

Fi

nans

iële

Kont

eks

Verg

elyk

ings

van

die

selfd

esoo

rt

[ratio

] Ve

rgel

ykin

gs v

an

vers

kille

ndes

oorte

G

roep

erin

g en

ver

delin

g m

et re

ste

Get

alge

bied

: Lee

rder

sont

wik

kel

en le

erta

fels

[11x

; 12x

; 13x

; 15x

; 20

x; 2

5x; 5

0x]

Verm

: 4-s

yfer

met

3-s

yfer

[met

of

sond

erha

kies

]Dee

l 4-s

yfer

deu

r 3-

syfe

r[ m

et o

f son

derh

akie

s]

Tegn

ieke

:

ska

tOpb

ou e

n af

bree

kAfro

ndin

g en

ko

mpe

nser

ingG

ebru

iksa

krek

enaa

r Ko

lom

verm

enig

vuld

igin

g Ve

elvo

ude

en

Fak

tore

: 2-

syfe

r en

3-s

yfer

geta

lle to

t Pr

iem

fakt

ore

van

geta

lle to

t ten

m

inst

e 10

0 Ei

ensk

appe

venH

eelg

etal

le:

Kom

mut

atie

we;

Ass

osia

tiew

e;

Dis

tribu

tiew

e0 e

n 1

in te

rme

van

syve

rmen

igvu

ldig

ings

eien

skap

Pr

oble

emop

loss

ing:

In k

onte

ks

Fina

nsië

leKo

ntek

s Ve

rgel

ykin

gs v

an d

iese

lfdes

oort

[ra

tio]

Verg

elyk

ings

van

ve

rski

llend

esoo

rte

Gro

eper

ing

en v

erde

ling

met

rest

e

LOO

M:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K

OPD

RAG

TE

241MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

…..

DAT

UM

: W

EEK

8

4- 8

MAA

RT

2013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

TYD

KAB

V P.

57, 1

46, 2

34

BEG

RIP

PE E

N

VAAR

DIG

HED

E Le

es, s

ê en

skr

yf 1

2- u

ur e

n 24

- uu

rtyd

op d

igita

le e

n an

aloo

gins

trum

ente

in

U

re, m

in e

n se

kond

es

Lees

van

alm

anak

ke

Be

reke

naan

tald

aetu

ssen

2

datu

ms

Prob

leem

oplo

ssin

g en

ber

eken

inge

m

et b

etre

kkin

g to

t tyd

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d

Lees

, sê

en s

kryf

12-

uur

en

24 -

uurty

d op

dig

itale

en

anal

oogi

nstru

men

te in

Ure

, min

en

seko

ndes

Le

es v

an a

lman

akke

Pr

oble

emop

loss

ing

en

bere

keni

nge

met

bet

rekk

ing

tot

tyd

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d

Lees

, sê

en s

kryf

12-

uur

en

24 -

uurty

d op

dig

itale

en

anal

oogi

nstru

men

te in

Ure

, min

en

seko

ndes

Le

es v

an a

lman

akke

Pr

oble

emop

loss

ing

en

bere

keni

nge

met

bet

rekk

ing

tot

tyd

Lees

tyds

ones

op

karte

Bere

kent

ydin

terv

alle

G

eski

eden

is v

an ty

d LO

OM

:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

242 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

243MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

OPV

OED

ER: …

……

……

……

……

……

….

DAT

UM

: W

EEK

9

11

- 15

MAA

RT2

013

TYD

: 5 D

AE

6

UR

E G

RAA

D

4 5

6 O

ND

ERW

ERP

DAT

AHAN

TER

ING

P.5

9, 1

48, 2

39

BEG

RIP

PE E

N

VAAR

DIG

HED

E Ve

rsam

el e

n or

gani

seer

da

ta

tellin

gsta

belle

] St

el d

ata

voor

Pikt

ogra

mm

e

Staa

fgra

fieke

An

alis

e; in

terp

reta

sie

en

vers

lagd

oeni

ng v

an d

ata

Pi

ktog

ram

me

St

aafg

rafie

ke

Vers

amel

en

orga

nise

er

data

[ te

llinst

abel

leas

ooko

rden

van

kl

eins

te to

t gro

otst

e]

Stel

dat

a vo

or

Pi

ktog

ram

me

St

aafg

rafie

ke

Anal

ise;

inte

rpre

tasi

e en

ve

rsla

gdoe

ning

van

dat

a

Pikt

ogra

mm

e

Staa

fgra

fieke

Sirk

eldi

agra

mm

e w

oord

elik

s

Vers

amel

en

orga

nise

er d

ata

te

llinst

abel

leas

ook

or

den

van

klei

nste

tot g

root

ste

vr

aely

ste

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dat

a vo

or

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ktog

ram

me

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aafg

rafie

ke

Anal

ise;

inte

rpre

tasi

e en

ver

slag

doen

ing

van

data

Pikt

ogra

mm

e

Staa

fgra

fieke

asoo

kdub

bele

staa

fgra

fieke

Sirk

eldi

agra

mm

e

Woo

rdel

iks

m

odus

LO

OM

:

HAN

DB

OEK

VE

RW

YSIN

GS.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

244 MULTIGRADE TOOLKIT MATHEMATICS INTERMEDIATE PHASE

WIS

KU

ND

E IN

TER

MED

IêR

E FA

SE

MU

LTIG

RAA

D

KW

ARTA

AL E

EN

O

PVO

EDER

: ……

……

……

……

……

…..

DAT

UM

: W

EEK

2

18

- 22

MAA

RT

201

3

T

YD: 5

DAE

6 U

RE

GR

AAD

4

5 6

ON

DER

WER

P Ei

ensk

appe

van

2D

-Vor

m K

ABV

Bl.

61 -6

3

KAB

V B

l. 15

0-15

2KAB

V B

l. 23

5 –

238

BEG

RIP

PE E

N

VAAR

DIG

HED

E Be

noem

vorm

s

drie

hoek

e

vier

kant

e

regh

oeke

ande

rvie

rhoe

ke

pe

ntag

one

he

ksag

one

si

rkel

s Ei

ensk

app

van

vorm

s

gebo

ë /re

guits

ye

aa

ntal

sye

Teke

n 2D

-Vor

ms

Beno

emvo

rms

dr

ieho

eke

vi

erka

nte

re

ghoe

ke

an

derv

ierh

oeke

pent

agon

e

heks

agon

e

hept

agon

e

sirk

els

Eien

skap

p va

n vo

rms

ge

boë

/regu

itsye

aant

alsy

e

leng

te v

an s

ye

ho

eke

[regt

ehoe

ke

en k

lein

er/g

rote

r as

regt

ehoe

ke]

Teke

n 2D

-Vor

ms

Ken

en

beno

emre

ëlm

atig

e en

on

reël

mat

igev

ierh

oeke

, dr

ieho

eke,

vie

rkan

te,

regh

oeke

en

para

llelo

gram

me;

pe

ntag

one;

hek

sogo

ne;

hept

agon

e en

okt

agon

e Ei

ensk

appe

van

Vor

ms:

aant

alsy

e

leng

te v

an s

ye

gr

ootte

van

hoe

ke

Teke

n 2D

Vor

ms

; ge

brui

k pa

sser

virs

irkel

s H

erke

n en

ben

oem

hoek

e

LOO

M:

H

AND

BO

EK

VER

WYS

ING

S.

NAS

ION

ALE

WER

KB

OEK

TUIS

WER

K O

PDR

AGTE

OPM

ERK

ING

AS

SESS

ERIN

G

TOET

S 1