YEAR 5 - Norfolk&Suffolk Hub · PDF fileThis project was led by the Educator Solutions Math...

No

rfolk

an

d S

uffo

lk P

rima

ry A

ss

es

sm

en

t Wo

rkin

g P

arty

Th

is p

roje

ct w

as

led

by th

e E

du

ca

tor S

olu

tion

s M

ath

em

atic

s T

ea

m

an

d fu

nd

ed

by th

e N

orfo

lk a

nd

Su

ffolk

Ma

ths

Hu

b.

Gu

ida

nc

e o

n fo

rma

tive

as

se

ss

me

nt m

ate

rials

to e

xe

mp

lify flu

en

cy, re

as

on

ing

an

d p

rob

lem

so

lvin

g

Ye

ar 5

For m

ore

info

rmatio

n a

nd to

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De

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e

Ple

ase

find

atta

ch

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ida

nce

writte

n b

y N

orfo

lk a

nd

Suffo

lk P

rima

ry te

ach

ers

to h

elp

un

pic

k

wh

at flu

en

cy, re

ason

ing a

nd

pro

ble

m s

olv

ing lo

oks lik

e in

ye

ar g

rou

ps 1

-6.

Ra

tion

ale

The

se

mate

rials

we

re p

rod

uce

d b

ecau

se

teach

ers

hig

hlig

hte

d a

ga

p o

n h

ow

to te

ach a

nd

asse

ss th

e P

urp

ose

of S

tud

y a

nd

the

thre

e a

ims o

f the

Prim

ary

ma

the

ma

tics c

urric

ulu

m (D

fE,

20

13

). Pre

vio

us in

ca

rna

tion

s o

f the

Prim

ary

Ma

them

atic

s N

atio

na

l Cu

rricu

lum

ha

ve

alw

ays

inclu

de

d g

uid

an

ce

(and

usua

lly o

bje

ctiv

es) o

n th

is a

rea

, alth

ou

gh

the

y h

ave

be

en k

no

wn

un

de

r

ma

ny d

iffere

nt n

am

es s

uch

as u

sin

g a

nd

ap

ply

ing, w

ork

ing m

ath

em

atic

ally

, pro

ble

m s

olv

ing o

r

inve

stig

atio

ns.

Alth

ou

gh

ea

ch

ye

ar g

rou

p c

on

tain

s o

bje

ctiv

es fo

r the

con

ten

t of th

e n

ew

cu

rricu

lum

(DfE

, 20

13

),

the

re a

re fe

w re

fere

nce

s in

the

bo

dy o

f the N

atio

na

l Cu

rricu

lum

tha

t exe

mp

lify flu

en

cy,

rea

so

nin

g o

r pro

ble

m s

olv

ing, a

nd

ye

t the

se

thre

e a

ims w

ill be

ob

se

rve

d, e

xa

min

ed

an

d te

ste

d.

In a

dd

ition to

the

se

mea

su

res th

ere

are

ma

ny (e

.g. N

RIC

H) w

ho

be

lieve

the

se a

ims a

re

pa

rticu

larly

imp

orta

nt w

ithin

the

lea

rnin

g o

f ma

them

atic

s fo

r all c

hild

ren

.

Org

an

isa

tion

of m

ate

rial

The

ma

teria

ls h

ave

bee

n p

rod

uce

d in

sin

gle

age

ye

ar g

rou

ps.

Tea

ch

ers

loo

ked

at a

nd

iden

tified

the b

ig id

ea

s in

ma

them

atic

s. T

en

big

ide

as w

ere

iden

tified

acro

ss e

ve

ry y

ea

r gro

up

. Th

ese

we

re in

form

ed

by th

e N

atio

na

l Cu

rricu

lum

ob

jectiv

es, th

e N

AH

T

KP

I’s (k

ey p

erfo

rma

nce

ind

icato

rs) a

nd

oth

er s

ou

rce

s s

uch

as N

CE

TM

an

d N

RIC

H. T

he

se

big

ide

as a

re o

nly

su

gge

stio

ns a

nd

co

uld

be

ch

ange

d, d

ele

ted o

r ad

ded

to d

ep

en

din

g o

n s

cho

ol

sp

ecific

crite

ria a

nd

foci.

Un

de

r ea

ch

big

ide

a a

re th

ree

bo

xe

s fo

r fluency, re

aso

nin

g a

nd

pro

ble

m s

olv

ing. T

he

first p

art o

f

ea

ch

bo

x in

clu

de

s s

om

e e

xe

mp

lificatio

n fo

r ea

ch

aim

. Th

ese s

tate

me

nts

are

inte

nde

d to

help

su

ppo

rt the

un

de

rsta

nd

ing o

f ea

ch

aim

with

in th

e b

ig id

ea

. Ho

we

ve

r, as a

bo

ve

, the

y a

re n

ot a

defin

itive

or c

om

ple

te lis

t and

tea

che

rs s

hou

ld c

ha

nge

an

d a

lter th

em

acco

rdin

gly

.

The

se

co

nd p

art o

f the b

ox in

clu

de

s s

om

e p

ossib

le a

ctiv

ities th

at c

ou

ld h

elp

sup

po

rt the

exe

mp

lifica

tion

of e

ach a

im. T

he

se

activ

ities h

ave

be

en s

ele

cte

d b

y th

e te

ache

rs a

nd

are

the

re

to s

up

po

rt the te

ach

ing a

nd le

arn

ing o

f ea

ch

aim

, bu

t are

no

t me

an

t to b

eco

me

a c

he

cklis

t.

Ma

ny o

f the a

ctiv

ities a

re th

e te

ach

er’s

ow

n, b

ut if th

ey b

elo

ng to

a s

ou

rce

this

ha

s b

ee

n

ackn

ow

led

ge

d u

nd

ern

ea

th th

e a

ctiv

ity. H

ow

eve

r, wh

ile th

is s

ectio

n is

usefu

l, the

bo

x w

hic

h

offe

rs p

ossib

le e

xe

mp

lificatio

n fo

r ea

ch

aim

is m

ore

impo

rtan

t in u

nde

rsta

nd

ing th

e p

urp

ose o

f

stu

dy o

f the

ma

them

atic

s c

urric

ulu

m.

For m

ore

info

rmatio

n a

nd to

mak

e a

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ww

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Wo

rkin

g P

arty

Th

is p

roje

ct w

as le

d b

y th

e E

du

ca

tor S

olu

tions M

ath

em

atic

s T

eam

(Alis

on

Bo

rthw

ick) a

nd

fun

de

d b

y th

e N

orfo

lk a

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Suffo

lk M

ath

s H

ub .

Pe

op

le w

ho c

ontrib

ute

d to

the m

ate

rials

Co

pyrig

ht a

nd

us

ag

e o

f the

ma

teria

ls

Re

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d w

ith k

ind

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n o

f NR

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, Un

ive

rsity

of C

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mp

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m T

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aste

ry m

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, text ©

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, illustra

tion

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n ©

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ress 2

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re re

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d w

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e k

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NC

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M

an

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ress. T

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und

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e

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16

With

in th

e p

ossib

le a

ctiv

ities to

exe

mp

lify flu

en

cy, re

aso

nin

g a

nd

pro

ble

m s

olv

ing, te

ach

er’s

ch

ose

activ

ities fro

m a

va

riety

of s

ou

rce

s, in

clu

din

g th

eir o

wn

wh

ich

the

y fe

lt sup

po

rted

this

ma

them

atic

al a

rea. H

ow

eve

r this

do

es n

ot m

ea

n th

at th

ese

activ

ities a

re lim

ited

to th

is s

ectio

n,

an

d w

ou

ld b

e s

uita

ble

for u

se

in e

ach

are

a o

f flue

ncy, re

ason

ing a

nd p

rob

lem

so

lvin

g.

On

be

ha

lf of T

he

No

rfolk

an

d S

uffo

lk P

rima

ry A

sse

ssm

en

t Wo

rkin

g P

arty

Be

st w

ish

es,

Alis

on

Bo

rthw

ick

alis

on

.bo

rthw

ick@

ed

uca

tors

olu

tion

s.o

rg.u

k

David

Bo

ard

(St J

oh

n’s

Prim

ary

, No

rfolk

) L

orn

a D

en

ham

(Saxm

un

dh

am

Prim

ary

, Su

ffolk

)

Alis

on

Bo

rthw

ick (M

ath

em

atic

s A

dvis

er)

Vic

toria

Gate

sh

ill (Harle

sto

n P

rimary

, No

rfolk

)

Liz

Bo

nn

ely

kke (S

tan

ton

Prim

ary

, Su

ffolk

) R

os M

iller (H

eth

ers

ett J

un

ior, N

orfo

lk)

Hele

n C

hatfie

ld (C

aven

dis

h P

rimary

, Su

ffolk

) C

herri M

osele

y (F

reela

nce C

on

su

ltan

t)

Sh

eila

Day (W

ind

mill F

ed

era

tion

, No

rfolk

) H

ele

n N

orris

(Du

ssin

gd

ale

Prim

ary

, No

rfolk

)

Refe

ren

ces

Departm

ent fo

r Educatio

n (D

fE), (2

013), M

ath

em

atic

s

Pro

gra

mm

e o

f Stu

dy K

ey S

tages 1

an

d 2

. Lon

don

: DfE

.

McIn

tosh, J

. (201

5) F

inal R

eport o

f the C

om

mis

sio

n o

n

Assessm

ent W

ithou

t Leve

ls. L

ond

on: C

row

n C

opyrig

ht.

ww

w.N

RIC

H.m

ath

s.o

rg w

ww

.ncetm

.org

.uk

For m

ore

info

rmatio

n a

nd to

mak

e a

bo

okin

g

ww

w.e

du

ca

tors

olu

tion

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ll 01

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Big

ide

as in

Ye

ar 5

1

.

Co

un

t, com

pa

re a

nd

ord

er n

um

be

rs (u

p to

1 0

00

00

0 in

clu

din

g n

ega

tive

nu

mbe

rs).

2.

Re

co

gn

ise

and

use

the p

ositio

na

l, ad

ditiv

e a

nd

mu

ltiplic

ativ

e a

sp

ects

of p

lace

va

lue

(mo

re

tha

n 4

dig

it nu

mb

ers

, de

cim

als

to th

ree

pla

ce

s a

nd n

ega

tive

num

be

rs).

3.

De

ve

lop

num

be

r se

nse

to s

upp

ort m

en

tal c

alc

ula

tion.

4.

Ad

d a

nd s

ubtra

ct n

um

be

rs, re

co

gn

isin

g th

at th

ese

are

inve

rse

op

era

tion

s (in

clu

din

g w

ho

le

nu

mb

ers

an

d n

um

be

rs w

ith d

ecim

al p

lace

s).

5.

Mu

ltiply

an

d d

ivid

e n

um

be

rs, re

co

gn

isin

g th

at th

ese

are

inve

rse

opera

tion

s (fo

r at le

ast th

e

12

x 1

2 tim

es ta

ble

s a

nd

4 d

igit b

y 1

and

2 d

igit).

6.

Use

alg

eb

ra to

exp

ress p

atte

rns a

nd

gen

era

lisa

tion

s w

ithin

ma

them

atic

s.

7.

(a) R

eco

gn

ise

fractio

ns, d

ecim

als

an

d p

erc

en

tage

s o

f sha

pe

s, o

bje

cts

and

qu

antitie

s.

(b) C

alc

ula

te w

ith fra

ctio

ns, d

ecim

als

an

d p

erc

en

tage

s.

8.

Ch

oo

se, u

se a

nd

com

pa

re a

va

riety

of u

nits

of m

ea

su

re to

an

app

rop

riate

leve

l of

accu

racy.

9.

Re

co

gn

ise

and

use

the p

rop

ertie

s o

f sh

ap

es, in

clu

din

g p

ositio

n a

nd d

irectio

n.

10

.

Co

llect, o

rga

nis

e a

nd in

terp

ret d

ata

(dis

cre

te a

nd

co

ntin

uou

s).

Ex

am

ple

s from

Te

ach

ing

for M

aste

ry m

ate

rials, te

xt ©

Cro

wn

Co

py

righ

t 20

15

, illustra

�o

n a

nd

de

sign

© O

xfo

rd U

niv

ersity

Pre

ss

20

15

, are

rep

rod

uce

d w

ith th

e k

ind

pe

rmissio

n o

f the

NC

ET

M a

nd

Ox

ford

Un

ive

rsity P

ress. T

he

Te

ach

ing

for M

aste

ry m

ate

rials

can

be

fou

nd

in fu

ll on

the

NC

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16

Year 5 Big Idea 1: Order, compare and count numbers (up to 1 000 000 including negative numbers)

Fluency Reasoning Problem solving

Exemplification of fluency • Count forwards and backwards in

steps of any single digit number and in steps of powers of 10 across zero

• Order numbers up to at least 1 000 000

• Compare numbers up to at least 1 000 000

• Represent numbers in different ways

Exemplification of reasoning • Complete sequences

• Find errors in sequences and explain answers

• Explain why digits belong in different positions in a number

• Show reasoning when rounding

Exemplification of problem solving • Interpret negative numbers in context

• Solve number problems and practical problems through a systematic and logical process

• Have a range of strategies to solve any problem

For more information and to make a booking

www.educatorsolutions.org.uk or call 01603 307710


Possible activities to exemplify fluency • Finish the sequence: 1000, 2000, 3000, ____, _____ 350, 340, ____, _____, _____ 11800, 11900, _____, _______

• Spot the error: 289636, 299636, 300636, 301636, 302636

• Say 358923 aloud, can you write this number in words?

• How can we describe 580500? It has __ hundred thousands. It has __ ten thousands. It has __ hundreds. It is made of 580000 and ____ together.

• Order the following numbers in ascending order: 362354, 362000, 362453, 359999, 363010

• Round the following numbers to the nearest a) 10 b)100 c) 1000

4821, 69243, 2781

• In 2013, there were 778803 births in the UK. What is this to the nearest 1000? Nearest 10000? Nearest 100000?

Possible activities to exemplify reasoning

• Look at this sequence: 18700, 18800, 18900, 19100 Correct the mistake and explain your working.

• True/False - When I count in 10’s I will say the number 12300. How do you know?

• What are the next 3 numbers in this number sequence?3, 3 ½ , 4, 4 ½ Can you explain the rule?

• Hannah says, ‘Using the digits 0-9 I can make any number up to 1000000’ Is she correct? Convince me.

• Oscar says the number 345050 is three hundred and forty five thousand and five. Can you explain why he is wrong?

• Simon says he can order the following numbers by only looking at the first three digits. Is he correct? Explain your answer. 125161, 128324,

• A number rounded to the nearest 1000 is 54000. What is the largest possible number this could be?

Possible activities to exemplify problem

solving

• Can you count back in 30’s to find the

trail through the grid?

• Using the digits 0-9 make the largest

number possible and the smallest

possible. How do you know these are

the largest and smallest numbers?




• In July 2015, the population of the

UK was estimated to be 64881609. What is this rounded to the nearest million?

• Translate these Roman Numerals: MD 2. MCD 3. CXVI 4. DCLX

• Write the numbers in Roman Numerals: 1. 352. 100 3. 994. 283 5. 570

• Complete these calculations: 1. CD + DC= 2. VI + IV= 3. CX + XC =

• Round the number 259996 to the nearest

1000. Round it to the nearest 10000. What do you notice about the answers? Can you think of 3 more numbers where the same thing would happen?

• True or False? All numbers with a five in the tens column will round up when rounded to the nearest 100 and 1000.

• Count in hundreds and fill in the pattern: C, CC, __, __, D, __, __, __, _, _ Explain what each letter means and write the translation below each letter.

• Arrange the numbers in size order: XXXV, XL, XXX, LX, LV, L, XLV, LXV Explain how you ordered the numbers.

• Complete the calculations. Show how you translated the roman numerals and added them. 1. XI + IX= 2. XL + LX= 3. CM + MC=

• Look at how different numbers are represented i.e. Read and recognise Roman numerals to 1000.

• Roll five dice; make as many 5 digit

numbers as you can from them. Round

each number to the nearest 10, 100,

1000 and 10,000. From your numbers,

how many round to the same 10, 100,

1000 or 10,000?

• Nathan thinks of a number. He says ‘My

number rounded to the nearest 10 is

1150, rounded to the nearest 100 is

1200 and rounded to the nearest 1000

is 1000.’ What could Nathan’s number

be?

• Temperature falls by about 1°C for

every 100 metres height gain. Abigail is

standing on top of a mountain at 900

metres above sea level. The

temperature is – 3°C. Abigail walks

down the mountain to sea level. What

should she expect the temperature to

be?




• Fred is a police officer. He is chasing a

suspect on Floor 5 of an office block.

The suspect jumps into the lift and

presses -1. Fred has to run down the

stairs, how many flights must he run

down?

• What is the longest number between 1

and 1000 when depicted in Roman

Numerals?

• Work out the year of your birth in

Roman Numerals. Work out the current

year in Roman numerals. Can you find

the difference?



Big Idea 2: Recognise and use the positional, additive and multiplicative aspects of place value (more than 4 digits, decimals to three places and negative numbers)


Exemplification of fluency • Count forwards or backwards in

steps of powers of 10 for any given

number up to 1 000 000

• Interpret negative numbers in

context, count forwards and

backwards with positive and

negative whole numbers, including

through zero

• Round any number up to 1 000 000

to the nearest 10, 100, 1000,

10 000 and 100 000

• Identify the positional place value in

large whole numbers and numbers

up to 3 dec places

• Understand that in the number

63472 we can find the multiplicative

place value of each digit by

multiplying each digit by the column

it is in e.g. 6 x 1000

Exemplification of reasoning • Convince a friend of the value of each digit in

integer and decimal numbers

• Explain why 2.85 x 100 – 285 and not 2.8500

• Use positional place value to reason about

numbers between numbers, including

decimals

Exemplification of problem solving

• Solve number problems and practical

problems that involve place value

• Solve number problems and practical

problems that involves negative

numbers

• Represent problems using apparatus to

organise thinking





Exemplification of fluency • Recognise the additive place

values of each digit so that when

the individual values of each digit

are added together, they total the

whole number.

• •




Possible activities to exemplify fluency

• Count on and back in 10s 100s 1000s from: 7; (b) 46; (c) 129

• What temperature does this thermometer show?

• Write these temperatures in order from hottest to coldest. 92°C, 37°C, –12°C, 73°C, 12°C, –2°C

• Look at the number below: 157.382

What does the 3 represent?







• John completes the sequence

1.67, 1.68, 1.69. 1.610

Is the last number correct? How do you know?

• Mary says that the only number which lies between 1.25 and 1.27 is 1.26. Do you agree with her? Explain why.

• Which is correct: -7 < -5 or -7 > -5? Explain

how you know.

Possible activities to exemplify problem solving

• How many pence are there in £4.60? In £5000? How many 10p coins?

• How many millimetres in 2.5cm?

• How many centimetres in 7.5m?

• How many centimetres in 27mm?

• How many metres in 136cm?

• On a cold day, the temperature is -3°C. The temperature rises by 5°. What is the new temperature?



Big Idea 3: Develop number sense to support mental calculation


Exemplification of fluency • Use rounding to estimate the

results of calculations

• Know when to use a mental strategy or jottings to work out answers to calculations.

• Use known facts to work out

unknown facts

Exemplification of reasoning • Use understanding of place value to spot

mistake

• Explain how you know when to use a mental strategy or jottings to work out answers to calculations.

• Give reasons for choices of methods

• Explain why one calculation strategy is more

efficient than another.


• Choose operations and efficient calculation strategies to solve word problems

• Find multiple solutions to a problem and

know when all solutions have been

found or where there are an infinite

number of solutions.



Big Idea 3: Develop number sense to support mental calculation


• 176 × 28 is approximately 200 × 30 = 6000

• 4.67 × 8 is approximately 5 × 8 = 40

• 297 ÷ 61 Estimate: 4 sixties are 240, 5 sixties are 300, so there are between 4 and 5 sixty-ones in 297

• Use mental strategies to add or subtract 19, 29, 199 etc.

• Choose mental or written strategies to calculate: 300 x 5; 346 x 5, 3500 ÷7; 3264÷7

Possible activities to exemplify reasoning • Frankie’s garden is 4.15m by 5.03m. A bag of

grass seed will cover 4 square meters of lawn. He thinks he will need 4 bags of grass seed to make his lawn. Is he right? Explain why?

• 4.27 x 11.6

Which answer is correct? 4.9352; 49.532; 495.32. Explain how you know.

• Billy needs to multiply 240 x 5 – he begins to draw a grid to multiply. Is there a quicker way? Explain it.

• Martin is measuring his room for a new

carpet. It has a width of 2.3m and a length of

5.1m. He rounds his measurements to the

nearest metre. Will he have the right amount

of carpet? Explain your reasoning.


• Billy collects Star Wars stickers. They cost £1.99 per pack. He has £12 birthday money. How many packs can he buy?

• Mrs Hopkins wants 25cm lengths of ribbon for making Christmas decorations. She has a 3m length. How many pieces of ribbon will she be able to cut?

• List all numbers between 1 and 2 – is it

possible to find ALL of them?



Big Idea 4:

Add and subtract numbers, recognising that they are inverse operations (including whole numbers

and numbers with decimal places)


Exemplification of fluency • Add & subtract whole numbers and

numbers with decimals

• Add & subtract numbers mentally

• Follow a set of instructions to calculate a mystery number

• Understand and use commutativity and associativity in addition and subtraction

Exemplification of reasoning • Explain and correct mistakes that have been

made in calculations

• Make conjectures about the relationships between different calculations

• Give reasons for choices of methods and strategies

• Use the vocabulary of addition and subtraction


• Work systematically and logically

• Use information given to find missing information without prompting

• Solve two (or more) step problems that involve addition & subtraction.

• Use different representations to understand and solve problems (e.g. bar model)



Big Idea 4:




• Work out this missing numbers: - 92 = 145

740 + = 1039

= 580 – 401 • Calculate: 1638 + 2517; 4023 –

2918

• “I am thinking of a number – if I

double it and subtract 11, the

answer is 39. What was my

number?”

Possible activities to exemplify reasoning • Rachel has £10. She spends £6.49 at the

shop. Would you use columnar subtraction to work out the answer? Explain why.

• True or False? Are these number sentences true or false? 8.7 + 0.4 = 8.11; 6.1 – 0.9 = 5.2 Give your reasons.

• If 2541 is the answer, what’s the question? Can you create three addition number sentences? Can you create three subtraction number sentences? Did you use a strategy?

• A five digit number and a four digit number have a difference of 4365. Give me three possible pairs of numbers. Explain the strategy you used.

• Which of these number sentences have an answer that is between 0.6 and 0.7?

• 11.48 – 10. 86= ; 53.3 – 52.75=

• Is this true? “When you add up four even numbers, the answer is divisible by four.” Can you explain what you have found?


• Peter bought boxes of crisps when they were on offer. After 12 weeks, his family had eaten 513 packets and there were 714 left. How many did he buy?

• Adam earns £37,566 pounds a year.

His wife, Sarah, earns £22,819 a year. How much do they earn altogether? They have to pay £7887 income tax per year, how much are they left with after this is taken off?

• Kangchenjunga is the third highest mountain in the world at 28,169 feet above sea level. Lhotse is the fourth highest at 27,960 feet above sea level. Find the difference in heights mentally.

• Using 0-9 dice roll three at the same time to create a number. Your partner needs to do the same. Who can add them together correctly first? Who can subtract the smallest from the largest correctly first? Use a calculator to check.



Big Idea 4:



• What numbers go in the boxes? What different answers are there? Convince me.

• True or false. 4999-1999 = 5000-2000 Explain how you know.

• Write a number in each circle so that the number in the square equals the sum of the numbers in the circles. Find different ways of doing it.

•

349

• How many ways have you found? Is that all the possible solutions? Explain your thoughts.

• There are 1231 people on an

aeroplane. 378 people have not ordered an inflight meal. How many people have ordered the inflight meal? Give your answer to the nearest hundred. The inflight meal costs £1.99 per person. The cabin crew have collected £1100 pounds so far. How much more money do they need to collect? Round your answer to the nearest pound.

• Here is a picture of a square drawn on cm² paper. How many other rectangles are there with the same perimeter as the square? Show your working.



Big Idea 5: Multiply and divide numbers, recognising that these are inverse operations (for at least 12x12 times tables up and 4 digit by 1 and 2 digit)


Exemplification of fluency

• Identify factors & multiples of at least 12 x 12; recognising common factors and common multiples

• Multiply and divide mentally using known facts

• Use formal method to multiply and divide up to 4 digits

• Recognise prime and composite

numbers

Exemplification of reasoning

• Decide which operations and methods to use and why

• Explain and convince others how you know a number is a prime number

• Justify why adding 2 odd numbers makes an even number

• Conjecture answers using known facts

• Explain and convince others how you know a number is a prime number

• Justify why adding 2 odd numbers makes an even number

• Conjecture answers using known facts

• Apply to fractions, decimals and percentages

• Use logic to decide how to manage remainders when dividing

Exemplification of problem solving • Work systematically and logically

• Use information given to find missing information

• Solve problems which involve multiplication and division

• Solve problems that involve division





• Write down:

The first 5 multiples of 8.

All the factors of 20.

Find a common factor of 36 and 12

• 8 x 6 = 48. Use this to help you find

the answers to the number

sentences: 48 ÷ 6 = 6 x 80 =

• Write down five multiplication and

division facts that use the number

48.

Complete the table:


• Tom says ‘Factors come in pairs, so all

numbers have an even number of factors.’ Do

you agree? Explain your reasoning.

• Rob and James are talking about multiples

and factors. Rob says ‘0 is a multiple of every

whole number.’ James says ‘0 is a factor of

every whole number.’ Who is correct?

• Explain why 6 is a factor of 24.

• How can you use 10 x 7 to help you find the

9th multiple of 7?

• Do you agree? “A square number has an

even number of factors” Explain your answer.

• Do you agree? Square and Cubed numbers

are always positive. Explain your answer

• What is the connection between the results

for the two and the four times table?


solving

• Clare’s age is a multiple of 7 and 3 less

than a multiple of 8. How old is Clare?

• Sally is thinking of a number. She says

‘my number is a multiple of 3. It is also 3

less than a multiple of 4.’ Find three

different numbers that could be Sally’s

number.

• 40 cupcakes cost £3.60, how much do

20 cupcakes cost? How much do 80

cupcakes cost? How much do 10

cupcakes cost?

• Polly is planting potatoes in her garden.

She has 24 potatoes to plant and she

will arrange them in a rectangular array.

List all the different ways that Polly can

plant potatoes.




• If I know 8 x 36 = 288, I also know

8 x 12 x 3 = 288 and 8 x 6 x 6 =

288. If you know 9 x 24 = 216, what

else do you know?

• Solve: 345 x 10 = ; 345 x 100 =

• Fill the gaps:

3790 x = 379000

3790 ÷ = 379

x 1000 = 497200

• Calculate: 5612 x 4; 654 x 34

• Harry has £20, he wants to save 10

times this amount. How much more

does he need to save?

• Mo Farah runs 135 miles a week.

How far does he run each year?

• To multiply a number by 25 you

multiply by 100 and then divide by

4. Use this strategy to solve 84 x

25; 28 x 25; 5.6 x 25

• Claire says ‘When you multiply a number by

10 you just add a nought and when you

multiply by 100 you add two noughts.’ Do you

agree? Explain your answer.

• Apples weigh about 160g each. How many

apples would you expect to get in a 2kg bag?

Explain your reasoning.

• 10 times a number is 4350, what is 9 times

the same number? Explain your working.

• How do you know whether 16 is even?

• Correct the errors in the calculation below.

266 ÷ 5 = 73.1 Explain the error.

• Andrew says that the answer to 166 ÷ 4 can

be written as ‘41 remainder 2’ or as ’41.5’. Do

you agree? Explain your reasoning.

• Last year my age was a square

number. Next year it will be a cube

number. How old am I? How long must I

wait until my age is both a square

number and a cube?

• If 8 x 24 = 192, how many other pairs of

numbers can you write that have the

product of 192?

• Here are the answers to the questions.

Can you write three different questions

that could make these numbers by

multiplying and dividing by 10, 100 or

1000?

5890, 40, 67000, 2000

• David has £35700 in his bank. He

divides the amount by 100 and takes

that much money out of the bank. Using

the money he has taken out he spends

£268 on furniture for his new house.

How much money does David have left

from the money he took out?




• Calculate: 68 ÷ 4 = ; 1248 ÷ 3 =

• Find the missing numbers:

x 5 = 475

3 x = 726

• I am thinking of a number. When it is

divided by 9, the remainder is 3. When

it is divided by 2, the remainder is 1.

When it is divided by 5, the remainder is

4. What is my number?

• The answer to the division has no

remainders. Find the missing numbers.

• 194 pupils are going on a school trip.

One adult is needed for every 9 pupils.

How many adults are needed?



Big Idea 6:

Use algebra to show patterns and generalisations within mathematics


Exemplification of fluency

• Recognise and describe linear

number sequences, including those

involving fractions and decimals,

and find the term-to-term rule.

• Solve equations with missing

numbers

• Understand what letters represent in algebraic expressions

Exemplification of reasoning

• Explain general rules for sequences e.g. the term-to-term rule

• Verbalise general rules in mathematics

• Give further examples to match one or more criteria

• Generalise and express a rule to find any term

• Use mathematical vocabulary when

generalising

Exemplification of problem solving • Express missing measures algebraically

• Solve missing term problems in sequences

• Solve multi-step problems involving equations, including with more than one missing number

• When solving problems use apparatus to combine terms



Big Idea 6:


Possible activities to exemplify fluency • Find the missing terms in these

sequences: 12, 10, 8, __, __, __

2, 4, 8, 16, __, __, __

½, 2, 3½, 5, 6½, __, __, __

3, 7, __, __, __, 23, 27

• If n = 4 calculate: n + 7

3n – 2

12

n

• Each shape stands for a different number. The totals of each row and column are shown. What number does each shape stand for?

21

28

28

19 19 21 18


• Look at this sequence

3, 6, 9, 12, 15, …

Find the next three terms in the sequence. Work out the tenth and 20

th term.

Tell me how you could find any term in the sequence.

• Jenny has measured the sides on a cube.

She says that to find the surface area you

need to square the length of the side and

then multiply by 6. Is she right or wrong? How

do you know?


7 x = 28 +



Big Idea 6:


• Fill the boxes in these number sentences:

+ 7 = 20

2 x + 6 = 10

9 + 5 = 8



Big Idea 7(a):

Recognise fractions, decimals and percentages of shapes, objects and quantities


Exemplification of fluency • Know common equivalents for

fractions, decimals and percentages

• Identify fractions, decimals and

percentages of shapes, objects and

quantities and move between them

Exemplification of reasoning • Explain why fractions, decimals and

percentages can be compared and how

• Reason about which fraction of something you would prefer



• Solve problems that involve comparing

& ordering fraction, decimals and

percentages



Big Idea 7(a):



• Look at the rectangle below:

shade of the rectangle blue;

shade of the rectangle green;

shade of the rectangle yellow.

What fraction is left unshaded?

Complete this:

½ kg = ____ g

¼ kg = _____ g


• Do you agree with this statement: The greater

the denominator, the bigger the fraction”?

• Do you agree with this statement: The greater

the numerator, the bigger the fraction”?

• This diagram shows four regular hexagons.

Shade in one third of the diagram.

• Which would you rather have 3 cakes or

cakes? Explain why.


solving

• Which is better:½ of £43.00; 75% of

£60.00

• Jamal spent ¼ of his birthday money.

What % is this?

• 4 apples cost 95p. How much does 1

apple cost to the nearest penny?

• How many different ways can you

express the fraction of the grid that is

shaded?



Big Idea 7(a):


• Which has the greater mass:

kg or kg? Explain why.

• Complete the table:

fraction decimal percent-age

75%

0.25

0.2

• Chaz and Caroline each had two sandwiches

of the same size. Chaz ate 1½ of his

sandwiches. Caroline ate of her

sandwiches. Fred said Caroline ate more

because 5 is the biggest number. Tammy

said Chaz ate more because she ate a whole

sandwich. Who do you think ate more? What

do you think of the way Fred and Tammy

have thought about the problem?

• Sam had a toffee bar and ate the amount of

toffee shown shaded in blue. Sam says he

ate 7/8 of a bar of toffee. Jo says Sam ate

14/16 of the toffee. Explain why Sam and Jo

are both correct.

• Graham is serving pizzas at a party.

Each person is given ¾ of a pizza.Fill in

the table below to show how many

pizzas he must buy for each number of

guests.

When will he have pizza left over?

Guests Pizzas

4

6

8

10



Big Idea 7(b):

Calculate with fractions, decimals and percentages


Exemplification of fluency • Convert between mixed numbers

and improper fractions

• Convert between percentages, decimals and fractions

• Compare and order fractions, decimals and percentages using <, >, =, ≠

• Add & subtract different representations of fractions (including different denominators), decimals and percentages by converting them

• Find missing fractions to complete calculations

• Multiply fractions by whole numbers

Exemplification of reasoning • Explain how you can add fractions, decimals

and percentages

• Convince others that an answer is correct by explaining

• Explain your solutions and convince others of

your methods.



• Solve problems that involve comparing & ordering fraction, decimals and percentages.

• Conjecture different patterns that can be seen and why when solving problems

• Solve problems that involve multiplying

fractions by whole numbers



Big Idea 7(b):



• Make each number sentence

correct using =, > or <.

• Mark and label on a number line

where you estimate that and

are positioned.

• Choose numbers for each

numerator to make this number

sentence true.

15 10

• Complete this:

½ kg = _____ g

¼ kg = _____ g


• Russell says 3/8 > 3/4 because 8 > 4. Do you

agree? Explain your reasoning.

• Which is closer to 1: 7/8 or 22/34? Explain

how you know.

• True or false? (explain why)

1·5 kg + 600 g = 2·1 kg + 300 g;

32 cm + 1·05 m = 150 cm – 0·13 m;

3/4 ℓ + 0.05 ℓ = half of 1·6 ℓ.

• A litre of water is approximately a pint and

three quarters. How many pints are

equivalent to 2 litres of water?

Using this approximation, when will the

number of litres and the equivalent number of

pints be whole numbers?


solving

• Chaz and Caroline each had two

sandwiches of the same size. Chaz ate

1 ½ of his sandwiches. Caroline ate 5/4

of her sandwiches. Fred said Caroline

ate more because 5 is the biggest

number. Tammy said Chaz ate more

because she ate a whole sandwich.

Explain why Fred and Tammy are both

wrong.

• Each bar of toffee is the same. On

Monday, Sam ate the amount of toffee

shown shaded in A. On Tuesday, Sam

ate the amount of toffee shown shaded

in B. Sam says he ate 7/8 of a bar of

toffee. Jo says Sam ate 7/16 of the

toffee. Explain why Sam and Jo are

both correct.



Big Idea 7(b):


• Which has the greater mass?1/5 kg

or 1/10 kg. Explain why.

• Order these from smallest to

greatest?

5.21 5.01 5.22 5.02 5.2

• Fill in the box to make this correct:

6.45 = 6 + 0.4 +

• Write the total as a decimal:

4 + 6/10 + 3/100

• Write the answers as fractions:

+ =

– =

+ =

– =

•

• Using the numbers 3, 4, 5 and 6 only once,

make this sum have the smallest possible

answer:

+ =

• Write eight different ways of adding two

numbers to make 1

• Find ways to complete:

% of = 30

• True or false: 3 x = 1 ½ x 3

• Jack and Jill each go out shopping. Jack

spends ¼ of his money. Jill spends 20% of

her money. Frank says Jack spent more

because ¼ is greater than 20%. Alice says

you cannot tell who spent more. Who do you

agree with, Frank or Alice? Explain

• Prove that 3/8 is less than 50%

• Graham is serving pizzas at a party.

Each person is given ¾ of a pizza. Fill

in the table below to show how many

pizzas he must buy for each number of

guests.

When will he have pizza left over?

• 4 apples cost 95p. How much does 1

apple cost to the nearest penny?

• I bought 3 items. 2 of them cost £2.59

each. I got £3.12 change from £10.00.

How much did the third item cost?

• Which is better:½ of £43.00; 75% of

£60.00



Guests Pizzas

4

6

8

10

Big Idea 7(b):


• Calculate:

3 x =

1½ x 3 =

• Find 25% of 300

• Spiralling decimals

Source: NRICH

• Fractions Jigsaw

Source: NRICH

• A football weighs 0·4 kg. Three footballs

weigh the same as eight cricket balls.

How many grams does a cricket ball

weigh?

• A 1·2 m ribbon and a 90 cm ribbon are

joined by overlapping the ends and

gluing them together. The total length of

ribbon needs to be 195 cm long. How

much should the two pieces overlap?

• I have 12.75g of sand and add kg of

pebbles. What did my mixture weigh in

total? If I shared it equally into 2 piles,

how much would there be in each pile?

• How many glasses of milk could I pour

from a 2 litre bottle of milk if the glasses

could hold 0.33l?

• A computer game is reduced in the sale

by 30%. It is now £77. What was its

original price?



Big Idea 8:

Chose, use and compare a variety of units of measure to an appropriate level of accuracy


Exemplification of fluency • Convert between different units of

metric measure (for example, kilometre and metre; centimetre and metre; centimetre and millimetre; gram and kilogram; litre and millilitre)

• Understand and use approximate equivalences between metric units and common imperial units such as inches, pounds and pints

• Choose units and suitable equipment to make measurements of length, mass, capacity

• Tell the time using the 12-hour and 24 hour clock, both digital and analogue

• Read a variety of scales accurately

• Use apparatus to illustrate volume

Exemplification of reasoning • Estimate volume [for example, using 1 cm

3

blocks to build cuboids] and capacity [for example, using water] and justify estimations

Exemplification of problem solving • Solve problems involving measuring

and calculating the perimeter of composite rectilinear shapes in centimetres and metres

• Calculate and compare the area of rectangles (including squares), and including using standard units (cm

2 and

m2) and estimate the area of irregular

shapes

• Solve problems involving converting between units of time

• Use all four operations to solve problems involving measure [for example, length, mass, volume, money] using decimal notation, including scaling.

• Pupils use all four operations in problems involving time and money, including conversions (for example, days to weeks, expressing the answer as weeks and days).



Big Idea 8:

Chose, use and compare a variety of units of measure to an appropriate level of accuracy


• 5 miles is approximately the same as 8 km

Complete the table:

miles km

5 8

10

40

20

30

80

20


• What units and equipment would you use to measure, the length of the classroom, the distance to Kings Lynn, the length of your pencil case, the width of your pencil? Explain your choice.

Possible activities to exemplify problem solving • Calculate the area from scale drawings

using given measurements. • Coconuts cost 78p each.

Bananas cost £1.20 for 1 kilogram. Josh buys one coconut and half a kilogram of bananas. How much does he spend altogether?

• Michael’s watch read 9.05, but it is 15 minutes fast – what is the real time?



Big Idea 9: Recognise and use the properties of shapes


Exemplification of fluency • Measure and draw angles, lines

and shapes accurately using appropriate equipment

• Find missing angles in triangles, quadrilaterals, at a point and on a line

• Reflect shapes across horizontal, vertical & diagonal lines

• Translate a shape from one place to another

Exemplification of reasoning • Explain how to calculate missing angles

• Justify properties of shapes by using known facts and convince others of your findings

• Explain where a shape will end up and why it ends up there

• Understand vocabulary such as perpendicular, parallel, etc.



• Solve problems that involve angles, shapes and/or direction

• Work systematically and logically

• Solve problems that involve reflecting & translating shapes




Possible activities to exemplify fluency:

• Estimate and compare acute,

obtuse and reflex angles

• Draw an angle of 76 degrees.

• Draw a line 53mm long.

• Draw 3 irregular pentagons

• Label the parallel lines on the

following shapes.

Possible activities to exemplify reasoning:

• Do you agree with this statement? Why?

“If you cut a triangle in half the two new

shapes will have angles which add up to

180°”

• Which of these statements are correct?

A square is a rectangle

A rectangle is a square

A rectangle is a parallelogram

A rhombus is a parallelogram

Explain your reasoning.

• Explain how you know the size of the missing

angles:

Possible activities to exemplify problem solving: • In the questions, below all of Harry’s

movement is in a clockwise direction. If Harry is facing North and turns through 180 degrees, in which direction will he be facing? If Harry is facing South and turns through 180 degrees, in which direction will he be facing? What do you notice? If Harry is facing North and wants to face SW how many degrees must he turn? From this position how many degrees must he travel through to face North again?




• Reflect shapes across a diagonal

line

• Reflect across 4 quadrants.

• Look at this shape. Mark acute angles with a; obtuse angles with o; reflex angles with r; right angles with rt.

• Six Places to visit

Source: NRICH

• Transformation Tease

Source: NRICH




Exemplification of fluency:

• Collect and organise information in tables and/or graphs as appropriate

• Answer literal questions involving line graphs & tables, including timetables

Exemplification of reasoning: • Prove that your findings are accurate

• Justify any conjectures you make using the information from the graph or table.

• Reason why a graph might give the results it

does.

Exemplification of problem solving: • Interpret a variety of graphs and table to

solve problems

• Use logic to make up stories to fit

graphs in a variety of forms

Big Idea 10:

Collect, organise and interpret data (discrete and continuous)



Big Idea 10:


Possible activities to exemplify fluency:

What was the highest/lowest temperature? What time did they occur? What is the difference between the highest and lowest temperature?

How long did the temperature stay at freezing point or less?

Possible activities to exemplify reasoning:

How long did it take for the pulse rate to reach the

highest level? Explain using the graph to help.

When do you think the person stopped exercising?

Convince me.

Estimate what the pulse rate was after 2 and a half

minutes. How did you get an accurate estimate?

Possible activities to exemplify problem solving: • Here is a line graph showing a bath

time. Can you write a story to explain what is happening in the graph?



Big Idea 10:


• On the 06:35 bus, how long does it take to get from Shelf Roundabout to Bradford Interchange?Can you travel to Woodside on the 07:43 bus?

• Which journey takes the longest

time between Shelf Village Hall and

Bradford Interchange, the bus that

leaves SVH at 06:46 or the bus that

leaves SVH at 07:23?

• If you needed to travel from Halifax Bus Station to Odsal and had to arrive by 08:20, which would be the best bus to catch? Explain your answer.

• Which journey takes the longest time from Halifax Bus Station to Bradford Interchange?

• Hannah works a 10 minute walk from Bradford Interchange. She has to start work at 08:00. She is on the 07:10 bus from Halifax which is running 5 minutes late. Will she make it to work on time? Explain your reasoning.

• Order the journey times on the timetable from longest to shortest. Can you explain why you think the buses take different lengths of time?

• Can you write a story for the three

graphs below?



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YEAR 5 - Norfolk&Suffolk Hub · PDF fileThis project was led by the Educator Solutions Math...

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Transcript of YEAR 5 - Norfolk&Suffolk Hub · PDF fileThis project was led by the Educator Solutions Math...