G r a d e 6 M a t h e M a t i c s

Grades 5 to 8 Blackline Masters

BLM 5–8.1: Observation Form

Students: Date: Activity:

Observation:

Possible Actions:

Students: Date: Activity:

Observation:

Possible Actions:

Students: Date: Activity:

Observation:

Possible Actions:

BLM

5–

8.2

: C

on

ce

pt

De

scrip

tio

n S

he

et

#1

C

hara

cter

istic

s

Exam

ples

N

on-E

xam

ples

D

iagr

ams/

Pict

ures

BLM 5–8.3: Concept Description Sheet #2

Concept Description

Example

Diagram

Non-Example

Concept Description

Example

Diagram

Non-Example

BLM 5–8.4: How I Worked in My Group Name ______________________________________________ Date ______________________________________________ Task ______________________________________________

Comments

I took turns

I participated

I encouraged others

I shared materials

I stayed with my group

I listened

I accomplished the task

BLM

5–8

.5:

Nu

mb

er C

ard

s

0

1

2

3

4

5

6

7

8

9

BLM 5–8.6: Blank Hundred Squares

BLM

5–

8.7

: P

lace

-V

alu

e C

ha

rt—

Wh

ole

Nu

mb

ers

hund

reds

te

ns

ones

hund

reds

te

ns

ones

hund

reds

te

ns

ones

Mill

ions

Thou

sand

s

One

s

BLM

5–

8.8

: M

en

tal

Ma

th S

tra

teg

ies

The

follo

win

g lis

t com

pile

s m

enta

l mat

h st

rate

gies

as

foun

d in

the

Kind

erga

rten

to G

rade

8 M

athe

mat

ics:

Man

itoba

Cu

rric

ulum

Fra

mew

ork

of O

utco

mes

. Not

e: T

his

reso

urce

is m

eant

for t

each

er in

form

atio

n, n

ot a

s a

list o

f str

ateg

ies

that

st

uden

ts s

houl

d m

emor

ize.

Gra

de 1

G

rade

2

Gra

de 3

G

rade

4

Gra

de 5

G

rade

6

Gra

de 7

1.

N.1

0.

2.

N.8

. 2.

N.1

0.

3.N

.6.

3.N

.7.

3.N

.10.

3.

N.1

1.

3.N

.12.

4.N

.4.

4.N

.5.

4.N

.6.

4.N

.11.

5.N

.2.

5.N

.3.

5.N

.4.

6.N

.8.

7.

N.2

.

Gra

de

Con

cept

St

rate

gy

Mea

ning

Ex

ampl

e 1

Add

ition

C

ount

ing

on

Stud

ents

beg

in w

ith a

num

ber a

nd c

ount

on

to

get t

he s

um. S

tude

nts

shou

ld b

egin

to

reco

gniz

e th

at b

egin

ning

with

the

larg

er o

f the

tw

o ad

dend

s is

gen

eral

ly m

ost e

ffici

ent.

for 3

+ 5

th

ink

5 +

1 +

1 +

1 is

8;

thin

k 5,

6, 7

, 8

1 Su

btra

ctio

n C

ount

ing

back

St

uden

ts b

egin

with

the

min

uend

and

cou

nt

back

to fi

nd th

e di

ffere

nce.

fo

r 6 –

2

thin

k 6

– 1

– 1

is 4;

th

ink

6, 5

, 4

1, 2

A

dditi

on

U

sing

one

m

ore

Star

ting

from

a k

now

n fa

ct a

nd a

ddin

g on

e m

ore.

fo

r 8 +

5 if

you

kno

w

8 +

4 is

12 a

nd o

ne m

ore

is 13

1,

2

Add

ition

Usi

ng o

ne le

ss

Star

ting

from

a k

now

n fa

ct a

nd ta

king

one

aw

ay.

for 8

+ 6

if y

ou k

now

8

+ 7

is 15

and

one

less

is

14

1, 2

, A

dditi

on

Subt

ract

ion

Mak

ing

10

Stud

ents

use

com

bina

tions

that

add

up

to te

n an

d ca

n ex

tend

this

to m

ultip

les

of te

n in

late

r gr

ades

.

4 +

____

is 1

0 7

+ __

__ is

10;

so

23

+ __

__ is

30 (c

ontin

ued)

BLM

5–

8.8

: M

en

tal

Ma

th S

tra

teg

ies (

Co

nti

nu

ed

)

Gra

de

Con

cept

St

rate

gy

Mea

ning

Ex

ampl

e 1

Add

ition

Su

btra

ctio

n St

artin

g fr

om

know

n do

uble

s

Stud

ents

nee

d to

wor

k to

kno

w th

eir d

oubl

es

fact

s. 2

+ 2

is 4

and

4 –

2 is

2

1, 2

, 3

Subt

ract

ion

Usi

ng

addi

tion

to

subt

ract

This

is a

form

of p

art-p

art-w

hole

re

pres

enta

tion.

Thi

nkin

g of

add

ition

as:

par

t + p

art =

who

le

Thin

king

of s

ubtr

actio

n as

:

who

le –

par

t = p

art

for 1

2 –

5 th

ink

5 +

____

= 1

2 so

12

– 5

is 7

2 A

dditi

on

Subt

ract

ion

The

zero

pr

oper

ty o

f ad

ditio

n

Kno

win

g th

at a

ddin

g 0

to a

n ad

dend

doe

s no

t ch

ange

its

valu

e, a

nd ta

king

0 fr

om a

min

uend

do

es n

ot c

hang

e th

e va

lue.

0 +

5 =

5;

11 –

0 =

11

2, 3

A

dditi

on

Subt

ract

ion

Usi

ng d

oubl

es

Stud

ents

lear

n do

uble

s, a

nd u

se th

is to

ext

end

fact

s:

usi

ng d

oubl

es

d

oubl

es p

lus

one

(or t

wo)

dou

bles

min

us o

ne (o

r tw

o)

for 5

+ 7

th

ink

6 +

6 is

12;

for 5

+ 7

th

ink

5 +

5 +

2 is

12

for 5

+ 7

th

ink

7 +

7 –

2 is

12

2, 3

A

dditi

on

Subt

ract

ion

Build

ing

on

know

n do

uble

s

Stud

ents

lear

n do

uble

s, a

nd u

se th

is to

ext

end

fact

s. fo

r 7 +

8

thin

k 7

+ 7

is 14

so

7 +

8 is

14

+ 1

is 15

3

Add

ition

Add

ing

from

le

ft to

righ

t U

sing

pla

ce v

alue

und

erst

andi

ng to

add

2-

digi

t num

eral

s. fo

r 25

+ 33

th

ink

20 +

30

and

5 +

3 is

50 +

8 o

r 58

(con

tinue

d)

BLM

5–

8.8

: M

en

tal

Ma

th S

tra

teg

ies (

Co

nti

nu

ed

)

Gra

de

Con

cept

St

rate

gy

Mea

ning

Ex

ampl

e 3

Add

ition

Su

btra

ctio

n M

akin

g 10

St

uden

ts u

se c

ombi

natio

ns th

at a

dd u

p to

ten

to c

alcu

late

oth

er m

ath

fact

s an

d ca

n ex

tend

th

is to

mul

tiple

s of

ten

in la

ter g

rade

s.

for 8

+ 5

th

ink

8 +

2 +

3 is

10 +

3 o

r 13

3 A

dditi

on

Subt

ract

ion

Com

pens

atio

n U

sing

oth

er k

now

n m

ath

fact

s an

d co

mpe

nsat

ing.

For

exa

mpl

e, a

ddin

g 2

to a

n ad

dend

and

taki

ng 2

aw

ay fr

om th

e su

m.

for 2

5 +

33

thin

k 25

+ 3

5 –

2 is

60

– 2

or 5

8 3

Add

ition

C

omm

utat

ive

prop

erty

Sw

itchi

ng th

e or

der o

f the

two

num

bers

bei

ng

adde

d w

ill n

ot a

ffect

the

sum

. 4

+ 3

is th

e sam

e as

3 +

4 3,

4

(dec

imal

s)

Add

ition

Su

btra

ctio

n C

ompa

tible

nu

mbe

rs

Com

patib

le n

umbe

rs a

re fr

iend

ly n

umbe

rs

(ofte

n as

soci

ated

with

com

patib

le n

umbe

rs to

5

or 1

0).

for 4

+ 3

stud

ents

may

th

ink

4 +

1 is

5 an

d 2

mor

e mak

es 7

3

Mul

tiplic

atio

n D

ivis

ion

Arr

ay

Usi

ng a

n or

dere

d ar

rang

emen

t to

show

m

ultip

licat

ion

or d

ivis

ion

(sim

ilar t

o ar

ea).

for 3

x 4

thin

k

fo

r 12

3 th

ink

3

Mul

tiplic

atio

n C

omm

utat

ive

prop

erty

Sw

itchi

ng th

e or

der o

f the

two

num

bers

bei

ng

mul

tiplie

d w

ill n

ot a

ffect

the

prod

uct.

4 x

5 is

the s

ame a

s 5

x 4

3 M

ultip

licat

ion

Sk

ip-c

ount

ing

Usi

ng th

e co

ncep

t of m

ultip

licat

ion

as a

ser

ies

of e

qual

gro

upin

g to

det

erm

ine

a pr

oduc

t. fo

r 4 x

2

thin

k 2,

4, 6

, 8

so 4

x 2

is 8

4

Mul

tiplic

atio

n Ze

ro p

rope

rty

of m

ultip

li-ca

tion

Mul

tiply

ing

a fa

ctor

by

zero

will

alw

ays

resu

lt in

zer

o.

30 x

0 is

0

0 x

15 is

0

(con

tinue

d)

BLM

5–

8.8

: M

en

tal

Ma

th S

tra

teg

ies (

Co

nti

nu

ed

)

Gra

de

Con

cept

St

rate

gy

Mea

ning

Ex

ampl

e 4

Mul

tiplic

atio

n D

ivis

ion

Mul

tiplic

ativ

e id

entit

y M

ultip

lyin

g (d

ivid

ing)

a fa

ctor

(div

iden

d) b

y on

e w

ill n

ot c

hang

e its

val

ue.

1 x

12 is

12

21

1 is

21

4. 5

M

ultip

licat

ion

D

ivis

ion

Skip

-cou

ntin

g fr

om a

kno

wn

fact

Sim

ilar t

o th

e co

untin

g on

str

ateg

y fo

r ad

ditio

n. U

sing

a k

now

n fa

ct a

nd s

kip

coun

ting

forw

ard

or b

ackw

ard

to d

eter

min

e th

e an

swer

.

for 3

x 8

th

ink

3 x

5 is

15 a

nd sk

ip

coun

t by

thre

es 1

5, 1

8,

21, 2

4 4,

5

Mul

tiplic

atio

n D

ivis

ion

Dou

blin

g or

ha

lvin

g U

sing

kno

wn

fact

s an

d do

ublin

g or

hal

ving

th

em to

det

erm

ine

the

answ

er.

for 7

x 4

, thi

nk th

e dou

ble

of 7

x 2

is 2

8 fo

r 48

6, t

hink

the

doub

le of

24

6 is

8

4 M

ultip

licat

ion

Div

isio

n U

sing

the

patte

rn fo

r 9s

Kno

win

g th

e fir

st d

igit

of th

e an

swer

is o

ne

less

than

the

non-

nine

fact

or a

nd th

e su

m o

f th

e pr

oduc

t’s d

igits

is n

ine.

for 7

x 9

thin

k on

e les

s th

an 7

is 6

and

6 p

lus 3

is

nine

, so

7 x

9 is

63

4, 5

M

ultip

licat

ion

Repe

ated

do

ublin

g C

ontin

ually

dou

blin

g to

get

to a

n an

swer

. fo

r 3 x

8, t

hink

3 x

2 is

6,

6 x

2 is

12, 1

2 x

2 is

24

4 D

ivis

ion

Usi

ng

mul

tiplic

atio

n to

div

ide

This

is a

form

of p

art-p

art-w

hole

re

pres

enta

tion.

Thi

nkin

g of

mul

tiplic

atio

n as

:

par

t x p

art =

who

le

Thin

king

of d

ivis

ion

as:

w

hole

p

art =

par

t

for 3

5

7

thin

k 7

x _

___

= 35

so

35

7 is

5

4, 5

M

ultip

licat

ion

Dis

trib

utiv

e pr

oper

ty

In a

rith

met

ic o

r alg

ebra

, whe

n yo

u di

stri

bute

a

fact

or a

cros

s th

e br

acke

ts:

a

x (b

+ c)

= a

x b

+ a

x c

(a

+ b

) x (c

+ d

) = a

c + a

d +

bc +

bd

for 2

x 1

54

thin

k 2

x 10

0 pl

us 2

x 5

0 pl

us 2

x 4

is 2

00 +

100

+

8 or

308

(con

tinue

d)

BLM

5–

8.8

: M

en

tal

Ma

th S

tra

teg

ies (

Co

nti

nu

ed

)

Gra

de

Con

cept

St

rate

gy

Mea

ning

Ex

ampl

e 5

Div

isio

n Re

peat

ed

halv

ing

Con

tinua

lly h

alvi

ng to

get

a n

umbe

r. fo

r 32

4, t

hink

32

2

is 16

and

16

2 is

8 so

32

4

is 8

5

Mul

tiplic

atio

n A

nnex

ing

zero

s W

hen

mul

tiply

ing

by a

fact

or o

f 10

(or a

pow

er

of te

n), t

akin

g of

f the

zer

os to

det

erm

ine

the

prod

uct a

nd a

ddin

g th

em b

ack

on.

for 4

x 7

00, t

hink

4 x

7 is

28

and

add

two

zero

s to

mak

e 280

0 5

Mul

tiplic

atio

n H

alvi

ng a

nd

doub

ling

Hal

ving

one

fact

or a

nd d

oubl

ing

the

othe

r. fo

r 24

x 4,

thin

k 48

x 2

is

96

6, 7

D

ivis

ion

Div

idin

g by

m

ultip

les

of

ten

Whe

n di

vidi

ng b

y 10

, 100

, etc

., th

e di

vide

nd

beco

mes

smal

ler b

y 1,

2, e

tc. p

lace

val

ue

posi

tions

.

for 7

6.3

10

thin

k 76

.3

shou

ld b

ecom

e sm

aller

by

one p

lace

val

ue p

ositi

on

so 7

6.3

10

is 7.

63

BLM 5–8.9: Centimetre Grid Paper

BLM 5–8.10: Base-Ten Grid Paper

BLM 5–8.11: Multiplication Table

× 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81

BLM 5–8.12: Fraction Bars

BLM 5–8.13: Clock Face

BLM 5–8.14: Spinner

BLM

5–

8.1

5:

Th

ou

sa

nd

Grid

BLM

5–

8.1

6:

Pla

ce

-V

alu

e M

at—

De

cim

al

Nu

mb

ers

One

s Te

nths

H

undr

edth

s Th

ousa

ndth

s

BLM

5–

8.1

7:

Nu

mb

er F

an

.

0

1

2

3

4

5

6

BLM

5–

8.1

7:

Nu

mb

er F

an

(C

on

tin

ue

d)

7

8

9

BLM

5–

8.1

8:

KW

L C

ha

rt

K W

L

Wha

t do

you

thin

k yo

u KN

OW a

bout

__

____

____

_?

Wha

t do

you

WAN

T to

kno

w a

bout

__

____

____

__?

Wha

t did

you

LEA

RN a

bout

__

____

____

___?

BLM 5–8.19: Double Number Line

BLM 5–8.20: Algebra Tiles

BLM 5–8.21: Isometric Dot Paper

BLM 5–8.22: Dot Paper

BLM 5–8.23: Understanding Words Chart What does it mean? Word Picture

Example

What does it mean? Word Picture

Example

B

LM

5–8

.24

: N

um

be

r L

ine

BLM

5–8

.25

: M

y S

ucce

ss w

ith

Ma

the

ma

tica

l P

ro

ce

sse

s

Nam

e __

____

____

____

____

____

____

____

____

____

____

____

____

____

Date

___

____

____

____

____

____

____

____

Ta

sk _

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

_

Wha

t are

the

mat

hem

atic

al

proc

esse

s?

How

do

I kno

w th

at I

have

bee

n su

cces

sful

?

How

hav

e I s

how

n m

y su

cces

ses?

Com

mun

icat

ion

I use

mat

hem

atic

al la

ngua

ge a

nd

sym

bols

that

I al

read

y kn

ow a

nd

that

I am

lear

ning

.

I use

real

thin

gs, p

ictu

res,

sy

mbo

ls, t

alki

ng, w

ritin

g, a

nd

thin

king

to c

omm

unic

ate.

Con

nect

ions

I con

nect

the

mat

h I a

m le

arni

ng

to m

ath

I alr

eady

kno

w.

I c

onne

ct th

e m

ath

I am

lear

ning

to

my

life.

Men

tal

Mat

hem

atic

s an

d Es

timat

ion

I c

an q

uick

ly fi

gure

out

the

answ

ers

to q

uest

ions

with

nu

mbe

rs b

y th

inki

ng a

bout

how

nu

mbe

rs w

ork

(and

I do

n’t n

eed

to w

rite

dow

n m

y st

eps)

.

I est

imat

e to

mak

e su

re m

y an

swer

mak

es s

ense

or w

hen

I do

n’t n

eed

an e

xact

ans

wer

or

mea

sure

men

t.

Prob

lem

So

lvin

g

I lis

ten

to o

ther

s, ta

lk w

ith

othe

rs, a

nd tr

y m

any

thin

gs

whe

n I a

m tr

ying

to a

nsw

er a

ki

nd o

f que

stio

n th

at I

have

ne

ver s

een

befo

re.

(con

tinue

d)

BLM

5–8

.25

: M

y S

ucce

ss w

ith

Ma

the

ma

tica

l P

ro

ce

sse

s

(co

nti

nu

ed

)

Wha

t are

the

mat

hem

atic

al

proc

esse

s?

How

do

I kno

w th

at I

have

bee

n su

cces

sful

?

How

hav

e I s

how

n m

y su

cces

ses?

Rea

soni

ng

W

hen

doin

g m

ath,

I se

e pa

ttern

s,

I use

wha

t I k

now

to h

elp

me

figur

e ou

t som

ethi

ng th

at I

don’

t kn

ow, a

nd I

thin

k ab

out m

y an

swer

s.

Tech

nolo

gy

I u

se c

alcu

lato

rs, c

ompu

ters

, and

ot

her t

echn

olog

y to

or

gani

ze a

nd s

how

my

wor

k

fig

ure

out p

atte

rns

chec

k so

met

hing

of w

hich

I am

un

sure

he

lp m

e le

arn

in n

ew w

ays

Vis

ualiz

atio

n

I can

mak

e up

, fig

ure

out,

expl

ain,

an

d lin

k to

geth

er d

iffer

ent p

ictu

res

and

3-di

men

sion

al o

bjec

ts.

W

hen

thin

king

abo

ut n

umbe

rs, I

im

agin

e th

em in

my

head

.

Whe

n m

easu

ring

, I k

now

that

so

met

imes

I ne

ed a

n ex

act n

umbe

r an

d so

met

imes

I ne

ed o

ne th

at is

cl

ose.

BLM 5–8.26: Percent Circle