Year 8 multiplicative relationships:...

Interacting with mathematics in Key Stage 3

Year 8 multiplicative relationships: mini-pack

2 | Interacting with mathematics | Year 8 multiplicative relationships: mini-pack © Crown copyright 2002

Contents

Year 8 multiplicative relationships: sample unit 4

Introduction 4

Objectives 5

Differentiation 5

Resources 5

Key mathematical terms and notation 6

Unit plan 8

Supplementary notes 10

Prompts for oral and mental starters 10

Prompts for main activities in phase 1 16

Prompts for main activities in phase 3 19

Problem bank for phase 3 24

© Crown copyright 20023 | Interacting with mathematics | Year 8 multiplicative relationships: mini-pack

Year 8 multiplicative relationships: sample unit

IntroductionThis Year 8 unit has been developed through a flexible use of the Sample medium-termplans for mathematics.

In planning the unit several decisions were made that affect the medium-term plans formathematics.

• The multiplicative relationships unit is additional to those in the Sample medium-termplans.

• The objectives for the unit are from the strand on number, so the list of objectivesaddressed in Number 2, Number 3 and Number 4 is reduced.

• It is taught during the spring term of Year 8.

• Number 2 is covered in the autumn term.

• Number 3 and Number 4 are taught after this unit.

• The objectives for the unit Solving problems are addressed in the sequence Number2, Multiplicative relationships (this unit), Number 3 and Number 4.

Understanding proportionality provides the key to much of the Key Stage 3 mathematicscurriculum. In number, proportionality occurs in work with fractions, decimals,percentages, ratio and rates; in algebra, proportions (y = mx ) are a subset of linearfunctions ( y = mx + c ); in geometry, proportionality occurs in ideas of scale andenlargement; in handling data, it underpins many statistical measures, graphicalrepresentations and probability. Proportional thinking is essential in the solution of manyproblems. This may not be immediately apparent and may not be accessible through asingle technique such as the unitary method. The underlying ideas must therefore betaught systematically.

This unit has been structured into three phases for teaching.

Phase 1 (about four lessons)• Addresses the need to move from additive to multiplicative thinking, introducing the

idea of scaling numbers (including multiplicative inverses) and identifying proportionalsets.

• Works with ‘difficult’ numbers (i.e. numbers that have a remainder when divided),forcing attention to the mathematical relationships and generalities involved.

• Emphasises fraction, decimal and percentage equivalents, treating the latter as aspecial kind of decimal (hundredths).

Phase 2 (one lesson)• Involves identifying practical examples of proportions, leaving further extension to link

with work on linear functions.

Phase 3 (about four lessons)• Examines problems, mainly within the field of number, which are solved by

multiplication or division, or a combination of the two operations.


• Highlights strategic approaches to problems by systematically examining the stagesof a solution. These will include extracting the data, clarifying the relationshipsinvolved, identifying what operations are needed, and considering the meaning andlikely size of numbers at each stage of the solution.

• Emphasises the use of calculators, seeking to automate the process of calculation,using the operational understanding developed in the first phase of the unit.

Objectives

A Understand multiplication and division of integers and decimals; use the laws ofarithmetic and inverse operations; check a result by considering whether it is of theright order of magnitude.

B Use division to convert a fraction to a decimal; calculate fractions of quantities;multiply (and divide) an integer by a fraction.

C Interpret percentage as the operator ‘so many hundredths of’ and express one givennumber as a percentage of another; use the equivalence of fractions, decimalsand percentages to compare proportions; calculate percentages and find theoutcome of a given percentage increase or decrease.

D Consolidate understanding of the relationship between ratio and proportion; reduce aratio to its simplest form, including a ratio expressed in different units, recognisinglinks with fraction notation; divide a quantity into two or more parts in a givenratio; use the unitary method to solve simple word problems involving ratioand direct proportion.

E Identify the necessary information to solve a problem, using the correct notationand appropriate diagrams.

F Solve more complex problems by breaking them into smaller steps, choosing andusing efficient techniques for calculation.

G Suggest extensions to problems, conjecture and generalise; identify exceptional casesor counter-examples.

Differentiation

• The unit references the Framework’s supplement of examples. For each objective thepitch of the work is accessible through Years 7 to 9 and examples can be chosenappropriately.

• Ideas and strategies for progression in oral and mental starters are provided in a setof prompts (see pages 10–15).

• Phase 3 is supported by a bank of Key Stage 3 test questions ranging from level 5 tolevel 7 (see pages 24–29).

• This is only the formative stage of securing understanding of proportionalrelationships. Pupils at all levels should have their thinking challenged by the ‘bigideas’ introduced throughout the unit, and all will need further work to clarify theirthinking and consolidate their skills


Resources

• Calculators

• Pupil resource sheets (included in the school file):

– Parallel number lines: examples

– Parallel number lines

– Multiplicative relationships: key results

• Supplementary notes (pages 10–29 of the mini-pack):

– Prompts for oral and mental starters

– Prompts for main activities in phase 1

– Prompts for main activities in phase 3

– Problem bank for phase 3

Key mathematical terms and notation

scale factor, multiplier, operator

inverse operation, inverse operator, multiplicative inverse

ratio (including notation a : b ), fraction, decimal fraction, percentage (%)

proportion, direct proportion

rate, per, for every, in every

unitary method




Ora

l and

men

tal s

tart

erM

ain

teac

hing

No

tes

Ple

nary

Unit

pla

n

Obj

ectiv

es B

, D(S

ee p

rom

pts

for

lang

uage

and

vis

ual

imag

es t

o us

e; a

lso

Fram

ewor

k p.

69)

Say

frac

tion

tabl

es a

loud

(who

lecl

ass

toge

ther

or

taki

ng t

urns

), in

diffe

rent

form

s:1

×1 /

4=

1/ 4

, 2 ×

1 /4

= 1

/ 2, …

1 /4

of 1

= 1

/ 4, 1

/ 4of

2 =

1/ 2

, …H

ow d

o yo

u fin

d 1 /

4of

a n

umbe

r?(L

ink

‘of’

with

×.)

1 /4

of 6

= 6

/ 4=

11 /

2, 2

/ 4of

6 =

3, 3

/ 4of

6 =

41 /

2, 4

/ 4of

6 =

6, …

How

do

you

find

5 /4

of a

num

ber?

(Rel

ate

to ÷

4 ×

5 an

d to

×5 /

4.)

Gen

erat

e pr

opor

tiona

l set

s in

an

orde

red

sequ

ence

: e.g

. mul

tiple

s of

10 w

ith m

ultip

les

of 4

, alte

rnat

ing

betw

een

term

s of

the

tw

o se

ts.

Pha

se 1

(fo

ur le

sso

ns)

Obj

ectiv

es A

, B, C

, D, G

(Fra

mew

ork,

pp

65, 6

7, 7

1)S

calin

g nu

mb

ers

(per

hap

s tw

o le

sson

s)H

ow c

an y

ou g

et fr

om 5

to

8 us

ing

only

mul

tiplic

atio

n an

d di

visi

on?

(5 ÷

5) ×

8, (

5 ×

8) ÷

5 o

r 5

× 8 / 5 (8

/ 5 o

f 5).

Inve

rtin

g th

is, h

ow c

an y

ou s

cale

from

8 t

o 5?

(8 ÷

8) ×

5, (

8 ×

5) ÷

8 o

r 8

× 5 / 8 (5

/ 8 o

f 8).

Illus

trat

e di

ffere

nt m

etho

ds g

raph

ical

ly w

ith li

ne s

egm

ents

on

sets

of p

aral

lel n

umbe

r lin

es.

From

sim

ilar

exam

ples

est

ablis

h a

×b

/ a=

b, a

lso

divi

ding

by

ath

en m

ultip

lyin

g by

bis

equi

vale

nt t

o si

ngle

ope

ratio

n of

mul

tiply

ing

by b

/ a(c

alle

d a

mul

tiplie

r or

sca

lefa

ctor

).E

stab

lish

(mul

tiplic

ativ

e) in

vers

e.

Con

side

r de

cim

al a

nd p

erce

ntag

e fo

rms

of s

cale

fact

ors:

×8 /

5=

×1.

6 =

160

% a

nd t

hein

vers

e ×

5 /8

= ×

0.6

25 =

× 6

2.5%

(and

pro

babl

y ÷

1.6

as w

ell).

Rat

io a

nd p

rop

ortio

nC

onsi

der

rela

tions

hips

bet

wee

n tw

o se

ts o

f num

bers

aan

d b

. Ide

ntify

mul

tiplie

rs fo

r ea

chpa

ir of

ent

ries.

Mul

tiplie

r ca

n al

so b

e ca

lled

the

ratio

b: a

. Whe

re r

atio

is e

quiv

alen

t fo

r ea

chpa

ir of

num

bers

, the

set

s of

num

bers

are

inpr

opor

tion.

Est

ablis

h in

vers

e ra

tio a

: b.

Giv

e m

ore

tabl

es o

f num

bers

: ide

ntify

whi

ch s

ets

of n

umbe

rs a

re in

pro

port

ion

and

whi

char

e no

t (fo

r th

ose

in p

ropo

rtio

n, id

entif

y ra

tios

a: b

and

b: a

). If

time

allo

ws,

dra

w g

raph

s of

pro

port

ions

, not

ing

enla

rgin

g tr

iang

les.

Usi

ng r

atio

and

pro

por

tion

Giv

en s

ets

of n

umbe

rs in

pro

port

ion,

iden

tify

appr

opria

te r

atio

s an

d us

e to

cal

cula

teun

know

n en

trie

s:a

b3

5a

3y

910

.57

xx

= 7

×5 /

3b

520

15

17.5

915

10

.517

.5y

= 2

0 ×

3 /5

Set

s of

num

bers

red

uced

to

two

entr

ies

can

now

be

thou

ght

of a

s ro

ws

or c

olum

ns –

with

unkn

own

in a

ny o

f the

four

pos

ition

s:a

bor

a3

53

5b

x6.

4x

= 3

×6.

4 /5

x6.

4x

= 6

.4 ×

3 /5

Use

pup

il re

sour

ce s

heet

‘Par

alle

l num

ber

lines

’:

Sup

port

: Sta

rt w

ith e

asie

r nu

mbe

rs a

ndsc

ale

fact

ors.

Intr

oduc

e id

ea o

f mul

tiplic

ativ

ein

vers

e bu

t sp

end

less

tim

e on

it.

All

pupi

ls c

over

frac

tion,

dec

imal

and

perc

enta

ge fo

rms,

tre

atin

g pe

rcen

tage

s as

hund

redt

hs.

Ext

ensi

on: P

upils

to

mak

e up

tab

les

for

apa

rtne

r to

exp

lore

.

Link

to

shap

e an

d sp

ace

wor

k on

enla

rgem

ent

(Sha

pe, s

pace

and

mea

sure

s3;

Fra

mew

ork,

pp.

212

–5):

Sup

port

: Kee

p un

know

n in

sec

ond

colu

mn.

Sta

rt w

ith e

asy

scal

e fa

ctor

s.

Ext

ensi

on: G

ive

exam

ples

with

mor

e th

anon

e un

know

n. A

sk p

upils

to

cons

ider

mor

eth

an o

ne w

ay o

f fin

ding

an

entr

y.

Ask

pup

ils t

o de

mon

stra

te e

xam

ples

and

cla

rify

met

hods

.

Link

to

oral

and

men

tal s

tart

ers:

•3 /

4an

d 41

/ 2ar

e in

the

sam

e fra

ctio

n ta

ble,

wha

tco

uld

it be

?

•(2

0,16

) is

a co

rres

pond

ing

pair

in p

ropo

rtio

nal

sets

, wha

t m

ight

the

tw

o pr

evio

us p

airs

be?

Dea

l with

issu

es r

elat

ing

to fr

actio

n, d

ecim

al,

perc

enta

ge c

onve

rsio

n.

Ask

pup

ils if

the

y ca

n ge

nera

lise

resu

lts, p

artic

ular

lyth

at s

cale

fact

or fr

om a

to b

is b

/ aan

d fro

m b

to a

is a

/ b.

Giv

en s

cale

fact

or, p

upils

use

cal

cula

tors

to

gene

rate

tab

les

of n

umbe

rs in

pro

port

ion.

Who

lecl

ass

to c

heck

som

e va

lues

.

Rel

ated

idea

:

Wha

t nu

mbe

rs c

ould

go

in t

he b

oxes

? Is

the

re a

uniq

ue s

et?

Han

d ou

t pu

pil r

esou

rce

shee

t ‘M

ultip

licat

ive

rela

tions

hips

: key

res

ults

’. W

ork

thro

ugh

it, a

skin

gpu

pils

to

give

exa

mpl

es o

f the

ir ow

n.

×8/ 5

÷5

×8

1

5

8

×2/ 3

×5


Ora

l and

men

tal s

tart

erM

ain

teac

hing

No

tes

Ple

nary

Obj

ectiv

e D

Gen

erat

e pr

opor

tiona

l set

s (a

s a

rem

inde

r of

wha

t th

ey a

re).

Obj

ectiv

es B

, C(E

xten

d up

to

15 m

inut

es. S

ee‘P

rom

pts

for

oral

and

men

tal

star

ters

’; al

so F

ram

ewor

k pp

. 61,

65

, 73)

Wor

king

tow

ards

flue

ncy:

•U

sing

cal

cula

tors

with

‘aw

kwar

d’nu

mbe

rs

•R

apid

con

vers

ion

betw

een

ratio

,fra

ctio

n, d

ecim

al a

nd p

erce

ntag

efo

rms

•N

umbe

rs a

nd q

uant

ities

, usi

ngra

tes,

cle

arly

sta

ted

… p

er …

Cov

er t

hese

cal

cula

tions

:

•E

xpre

ssin

g pr

opor

tions

: a/b

• C

ompa

ring

prop

ortio

ns:

a/b

< =

> c

/d

•Fi

ndin

g pr

opor

tions

: a/b

of …

•C

ompa

ring

quan

titie

s: a

/bof

… <

a/b

of ..

. < =

> c

/d o

f …

•U

sing

and

app

lyin

g ra

tes:

e.g

. 4

mac

hine

s ne

ed 1

7 ho

urs

mai

nten

ance

, how

long

for

7m

achi

nes?

(‘ho

urs

per

mac

hine

’,m

ultip

lier

17/4

)

(See

‘Pro

mpt

s fo

r or

al a

nd m

enta

lst

arte

rs’.)

Pha

se 2

(o

ne le

sso

n)O

bjec

tive

DD

escr

ibe

prac

tical

situ

atio

ns a

nd a

sk c

lass

whe

ther

the

y ar

e in

pro

port

ion.

Why

? W

hy n

ot?

Cov

er t

hese

poi

nts:

•S

ugge

st u

nits

in w

hich

qua

ntiti

es m

ight

be

mea

sure

d.

•D

iscu

ss c

once

pt o

f ‘ra

te’,

linki

ng u

nits

by

‘per

’ or

sym

bol /

.

•Ta

bula

te p

ossi

ble

sets

of v

alue

s an

d, if

tim

e, d

raw

gra

phs.

Pha

se 3

(fo

ur le

sso

ns)

Obj

ectiv

es C

, D, E

, F, G

(Fra

mew

ork

pp. 3

, 5, 6

1, 7

1–83

)S

trat

egie

s fo

r so

lvin

g pr

oble

ms

invo

lvin

g m

ultip

licat

ion,

div

isio

n, r

atio

and

pro

port

ion

Dra

w o

n pr

oble

m b

ank,

incl

udin

g so

me

from

sha

pe a

nd s

pace

and

han

dlin

g da

ta.

Act

iviti

es (m

ay in

clud

e m

ini-p

lena

ries)

:

•C

hoos

e on

e pr

oble

m: d

iscu

ss a

ltern

ativ

e st

rate

gies

for

solv

ing;

cha

nge

num

bers

(e.g

.m

ake

them

mor

e di

fficu

lt) a

nd c

onsi

der

how

met

hods

can

be

adap

ted;

ask

diff

eren

tor

sup

plem

enta

ry q

uest

ions

from

sam

e co

ntex

t. (S

ee ‘p

rom

pts

for

mai

n ac

tiviti

es in

phas

e 3’

)

•C

hoos

e sm

all s

et o

f pro

blem

s: c

once

ntra

te o

n ex

trac

ting

and

orga

nisi

ng d

ata

(e.g

.pu

ttin

g in

to t

abul

ar fo

rm) b

efor

e de

cidi

ng o

n po

ssib

le m

etho

ds o

f sol

utio

n, r

athe

rth

an w

orki

ng p

robl

ems

thro

ugh

to a

n an

swer

. (S

ee ‘p

rom

pts

prom

pts

for

mai

nac

tiviti

es in

Pha

se 3

’)

•A

sk p

upils

to

mak

eup

sim

ilar

prob

lem

s fo

r a

part

ner

to s

olve

.

•G

ive

part

solu

tions

and

ask

pup

ils t

o co

ntin

ue a

nd c

ompl

ete

solu

tion

or g

ive

aco

mpl

ete

solu

tion

and

ask

pupi

ls t

o ev

alua

te e

ffici

ency

of s

trat

egy

chos

en a

nd t

oid

entif

y er

rors

.

Pro

blem

-sol

ving

str

ateg

ies

to b

e ta

ught

:

•Tr

ansl

ate

prob

lem

into

a fo

rm t

hat

help

s w

ith t

he s

olut

ion:

e.g

. ext

ract

app

ropr

iate

data

and

put

in t

abul

ar fo

rm

•E

stim

ate

answ

er: a

sk ‘W

ill it

be b

igge

r or

sm

alle

r?’,

‘Will

it be

gre

ater

or

less

tha

n1?

’, et

c.; u

se k

now

ledg

e of

effe

ct o

f mul

tiply

ing

or d

ivid

ing

by n

umbe

rs g

reat

er t

han

or le

ss t

han

1.

•C

onsi

der

scal

ing

met

hods

by

findi

ng a

mul

tiplie

r.

•W

hen

usin

g un

itary

met

hod,

invo

lvin

g di

visi

on, c

larif

y ra

tes

expr

esse

d pa

rt w

ay t

o a

solu

tion:

e.g

. is

it eu

ros

per

poun

d or

pou

nds

per

euro

?

Type

s of

exa

mpl

e:D

ista

nce

/tim

e at

con

stan

t sp

eed

•W

eigh

t/co

st a

t gi

ven

unit

pric

e

•H

eigh

t/w

eigh

t of

a g

roup

of p

eopl

e

•M

ass

atta

ched

/ s

tret

ch in

ela

stic

•A

mou

nt o

f mea

t /

size

of b

urge

r

Link

to

alge

bra

of li

near

func

tions

(Alg

ebra

3 a

nd A

lgeb

ra 5

; Fra

mew

ork

pp.

164–

7,17

2–7)

.

Sel

ect

suita

ble

prob

lem

s, r

angi

ng fr

omle

vel 5

to

leve

l 7.

Sup

port

: Inc

lude

pro

blem

s w

ith‘c

onve

nien

t’ or

eas

y nu

mbe

rs, m

akin

gin

form

al/m

enta

l met

hods

app

ropr

iate

.

Ext

ensi

on: R

epla

ce w

ith ‘a

wkw

ard’

num

bers

, to

forc

e at

tent

ion

on g

ener

alm

etho

ds o

r to

mak

e pr

oble

m m

ore

diffi

cult.

Ask

for

one

or t

wo

exam

ples

iden

tifie

d as

prop

ortio

ns a

nd d

iscu

ss c

ircum

stan

ces

unde

rw

hich

the

y m

ight

not

be

so: e

.g. t

rave

l at

vary

ing

spee

d, e

xcha

nges

of c

urre

ncy

mad

e on

diff

eren

tda

ys, e

tc.

(Idea

s fo

r sh

ort

plen

arie

s, t

o be

use

d as

appr

opria

te.)

Pup

ils t

o sp

ecify

uni

ts o

n ca

lcul

ated

rat

es:

Pup

ils t

o su

gges

t us

es o

f str

ateg

ies

for

solv

ing

mul

tiplic

atio

n an

d di

visi

on p

robl

ems

in o

ther

are

asof

cur

ricul

um. (

Pos

sibl

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Supplementary notes

Prompts for oral and mental starters

Phase 1

Chant fraction tables in different forms, whole class together or taking turns. For visualsupport, write equations on the board, or reproduce extended area diagrams of the formdescribed in the booklet What is a fraction? (page 6).*

Multiples of a fraction

1 × 1/4 = 1/4 , 2 × 1/4 = 1/2 , . . .

Language to use: ‘One times a quarter is a quarter, two times a quarter is ahalf, three times a quarter is three quarters, four times a quarter is one, fivetimes a quarter is one and a quarter, . . .’

Or: ‘One quarter is a quarter, two quarters are a half, three quarters are three quarters,four quarters are one, five quarters are one and a quarter, . . .’

Count in multiples of other fractions, for example thirds or fifths.

Same fraction of different numbers1/4 of 1 = 1/4 , 1/4 of 2 = 1/2 , . . .

Language to use: ‘One quarter of one is a quarter, a quarter of two is a half, a quarter ofthree is three quarters, a quarter of four is one, a quarter of five is one and a quarter, . . .’

Key question: How do you find 1/4 of a number?

Find other fractions of the sequence of integers, for example thirds or fifths.

Different fractions of the same number1/4 of 6 = 6/4 = 11/2 , 2/4 of 6 = 3 , 3/4 of 6 = 41/2 , . . .

Language to use: ‘One quarter of six is one and a half, two quarters of six is three, threequarters of six is four and a half, four quarters of six is six, five quarters of six is seven anda half, . . .’

Key questions: How do you find 1/4 of a number? 5/4 of a number? 9/4 of a number?

Find quarters of another number.

Start with a different unitary fraction of a number, e.g. thirds or fifths.

*The booklet What is a fraction? was provided to teachers attending the additional support courses Planning

and teaching mathematics and Leading developments in mathematics.


Phase 2

Generating proportional sets

Generate two sets of multiples simultaneously, alternating between the terms of the twosets, for example multiples of 10 and multiples of 4.†

Point to markers on the top and bottom of a counting stick; top marker referencesmultiples of 10, bottom marker references multiples of 4. Say the numbers aloud as aclass:

Ten, four, twenty, eight, thirty, twelve, forty, sixteen, . . .

Extend beyond the limit of the counting stick.

Check equality of ratios between the pairs of numbers (use a calculator if appropriate).

Repeat with other pairs of multiples.

† See the video of Walt’s Y7 lesson on ratio and proportion. The video was included in the school materials

(purple box) for the Key Stage 3 conference on the National Numeracy Strategy in the summer term 2000.


0 10 20 30 40 50 60 70 80 90 100

0 4 8 12 16 20 24 28 32 36 40

Phase 3

Expressing proportions

• Expect rapid conversion between fractions in lowest terms, decimals, percentages

• Expect rates, clearly stated as … per …

• Expect use of calculator; Numbers all 'awkward'

Numbers, first smaller than second

• What is 5 as a fraction of 85?

• What is 5 as a percentage of 85?

• What is 5 as a proportion of 85?

Numbers, first larger than second

• What is 15 as a fraction of 8?

• What is 15 as a percentage of 8?


Quantities, first smaller than second (Answer is not in units, e.g. £.)

• What is £3 as a fraction of £17?

• What is £3 as a percentage of £17?

• What is £3 as a proportion of £17?

Quantities, first smaller than second, units mixed (Answer is not in units, e.g. £ or p.)

• What is 60p as a fraction of £2?

• What is 60p as a percentage of £2?

• What is 60p as a proportion of £2?

Quantities, first larger than second (Answer is not in units, e.g. kg.)

• What is 23 kg as a fraction of 8 kg?

• What is 23 kg as a percentage of 8 kg?

• What is 23 kg as a proportion of 8 kg?

Reversing proportions (Deal with various forms – ‘fractions/decimals/percentages’ and‘numbers/quantities’.)



Rates to be written as ‘this’ per ‘that’

• 367 miles travelled at a constant speed for 4 hours

• £87 split equally among 6 children

• 14 hours of work to be covered by 9 clerks


Reversing rates (What is the meaning of ‘that’ per ‘this’? When is it useful?)

• hours per mile

• children per pound

• clerks per hour

Comparing proportions




Numbers, first smaller than second

• What is greater: 5 as a proportion of 85 or 7 as a proportion of 90?

Numbers, first larger than second


Quantities, first smaller than second (Answer is not in units, e.g. minutes.)

• What is greater: 5 minutes as a proportion of 18 minutes or 3 minutes as a proportionof 10 minutes?

Quantities, first smaller than second, units mixed (Answer is not in units, e.g. £ or p.)

• What is greater: 60p as a proportion of £2 or 90p as a proportion of £2.47?

Quantities, first larger than second (Answer is not in units, e.g. cm.)

• What is greater: 7 cm as a proportion of 18 cm or 6 cm as a proportion of 16 cm?

Reversing proportions (Deal with various forms – ‘fractions/decimals/percentages’ and‘numbers/quantities’.)


• Given the above, what is greater: 28 as a proportion of 9 or 40 as a proportion of 13?

Rates to be written as ‘this’ per ‘that’

• What is the greater rate: 367 miles travelled in 4 hours at a constant speed or 640miles travelled in 7 hours at a constant speed?

Finding proportions




Numbers, proportions less than 1, some terminating decimals and some not

• What is three fifths of 83?

• What is 0.6 of 83?

• What is 60% of 83?



Numbers, proportions greater than 1, some terminating decimals and some not

• What is ten sixths of 74?

Quantities, proportions less than 1, some terminating decimals and some not (Answer isin units, e.g. £.)

• What is three sevenths of £17?

Quantities, proportions greater than 1, some terminating decimals and some not (Answeris in units, e.g. litres.)

• What is four thirds of 14 litres?

Comparing quantities




Numbers, proportions less than 1, some terminating decimals and some not

• Which is greater: 16% of £25 or 25% of £16?

Numbers, proportions greater than 1, some terminating decimals and some not

• Which is greater: five thirds of 35 or seven quarters of 33?

Quantities, proportions less than 1, some terminating decimals and some not (Answer isin units, e.g. £.)

• Which is greater: three sevenths of £17 or two fifths of £14?

Quantities, proportions greater than 1, some terminating decimals and some not (Answeris in units, e.g. litres.)

• Which is greater: ten thirds of 14 litres or ten sevenths of 20 litres?

Using and applying rates




What is the rate we use in this calculation? (Expect, not a numerical answer, but a rate –e.g. ‘hours per machine’.)

• 4 machines need 17 hours of maintenance. How many hours for 7 machines?

What is the rate we use in this calculation? (Expect, not a numerical answer, but a rate –e.g. ‘machines per hour’.)

• 4 machines need 17 hours of maintenance. How many machines in 5 hours?

What is the calculation? (Expect, not a numerical answer, but a calculation – e.g. 17/4 × 7.)



What is the calculation? (Expect, not a numerical answer, but a calculation – e.g. 4/17 × 5.)


What is the answer?


What is the answer?


Prompts for main activities in phase 1This sequence of work is planned to be challenging for many pupils. It sets highexpectations in order to build on and take advantage of the National Numeracy Strategyin primary schools, which has aimed to improve pupils’ fluency in mental calculation.Pupils should come into secondary schools not only with better knowledge of basicmultiplication and division facts, but also improved understanding of the relationshipsbetween them.

For example, pupils should see

3 × 6 = 18, 6 × 3 = 18, 18 ÷ 6 = 3, 18 ÷ 3 = 6

as a related set of facts – if one is known then the others follow. So, when asked ‘Whatdoes 3 have to be multiplied by to give 18?’ (3 × ? = 18), as well as knowing the answer(6) they should recognise its relationship to the given numbers (6 = 18 ÷ 3). Pupils whohave this level of understanding should be ready for the work here.

General teaching points:

• Use calculators to pursue multiplicative relationships between numbers that would bedifficult to deal with mentally. This forces pupils to attend to general strategies fordealing with numbers.

• Support pupils, where necessary, by discussing examples with easier numbers, sothat they can make connections with what they already know and understand frommental work. They should then return to using calculators with more difficult numbers.

Scaling numbers

How can you get from the first number (referred to as a) to the second number (referredto as b) using only multiplication?

2 → 5

4 → 5

5 → 8

16 → 5

2 → 3

4 → 3

6.4 → 22.4

Teaching points:

• Line segments drawn on sets of three parallel lines (see the pupil resource sheet‘Parallel number lines’) give a strong visual image to support understanding of howtwo operations can be combined into one: ÷ a × b is replaced by the single operator× b/a , b/a being regarded as a single fraction. It is useful to note that a and b will notalways be integers.

• By reversing the direction of the scalings, pupils can be led to see that × a/b is theinverse of × b/a – treat it as a straightforward matter, not a complicated new idea! Thisis likely to be the first time they have met the concept of a multiplicative inverse:preserving the operation (multiplication) and seeking an inverse operator (a/b). Themore familiar way, particularly if b/a is expressed as a single decimal (e.g. 0.8), is topreserve the operator 0.8 and invert the operation (multiplication) – that is, ‘multiply by0.8’ is inverted to ‘divide by 0.8’. A possible approach, so as not to introduce too


Pupils might start with a and bas whole numbers, but with theratio b/a not a whole number.


many complications, is to discuss divisors only where the multiplier is in tenths. Theidea of a multiplicative inverse is central to phase 1 of the unit and should be givendue emphasis.

• Fractions can be converted easily to decimals (b/a = b ÷ a) and percentages (treatpercentages as hundredths – of which you can have more than one hundred!). Pupilsshould have met these conversions before and they need to become fluent withthem. Retaining ‘of’ for × can aid understanding when expressing fractions orpercentages of numbers.

• Your department will need to agree on how to deal with recurring decimals – forexample, round to 2 decimal places.

There is a lot to cover here, so this material could easily make two one-hour lessons.

Ratio and proportion

a b

7 10.5

0.8 1.2

11/2 21/4

24 36

1.4 2.1

Teaching points:

• Look at both examples and counter-examples, such as those considered earlier, toestablish the concept of proportion as an equality of ratios. Pupils meet lots of tablesof numbers that are well ordered. Avoiding such order here keeps attention on theessential idea behind the concept.

• The terms ‘ratio’ and ‘proportion’ are often used together, and their meanings are notalways clearly distinguished. For example, in everyday language it is common to referto a part of a whole as ‘a proportion’. It is perhaps wise not to dwell on thesedistinctions but to allow the context to take care of the meaning. In fact, manyquestions of a practical nature do not use either term explicitly.

• Pupils need to be familiar with ratio notation b : a (read as ‘the ratio of b to a’) andknow that it is equivalent to b/a .

• Some graphical work is worthwhile, making a link to the algebra of linear functionsand to ideas of enlargement. Time may be short in this unit, however, and it may bebetter to draw out the links in the next appropriate unit.

Using ratio and proportion

Look at this table of numbers in proportion. What must x be?

6 9

3 x

For all pairs of values, b/a = 1.5 and a/b = 0.67(to 2 decimal places)

Using simple numbers can encourage different ways of looking at a problem. In this case,x can be found either by halving 9 or by multiplying 3 by 11/2. In practical problems thenumbers are often simple enough to be dealt with mentally – for example, a recipe forfour can be adapted for six by multiplying the amounts by 11/2. However, as suggested inthe unit plan, this phase is concerned purely with relationships between sets of numbers.Practical contexts are left until phases 2 and 3.

Teaching points:

• Pupils need to understand that the unknown number is found by applying a scalefactor to one of the known numbers, the scale factor being obtained from a knownpair.

• Asking the question ‘Will the answer be bigger or smaller?’ is a useful check onwhether the correct scale factor is being applied.

• This abstract work can be simplified for some pupils by keeping the unknown in onestandard position – for example, always the bottom right-hand entry in the table. Thisis a better strategy than merely simplifying the numbers as it forces pupils to use theideas of ratio and proportion rather than their knowledge of numbers.


Prompts for main activities in phase 3Choose one problem: discuss alternative strategies for solving the problem;change the numbers in the problem (e.g. make them more difficult) and considerhow the methods can be adapted; ask different or supplementary questions fromthe same context.

Consider possible strategies for part (a). A similar approach can be taken to part (b).

Strategy 1: Mental scaling method

3 litres of red paint plus 7 litres of blue paint makes 10 litres of purple.

Double up for 20 litres.

For which kinds of numbers does this strategy lend itself?

red blue purple

3 7 10

How does this help us to calculate for 5 litres of purple?


How does this help us to calculate for 3.85 litres of purple?

How does it help if the components of red and blue change– for example, 3 parts red and 8 parts blue?

Could I use this strategy to see quickly the percentage of any mix that is made up of red paint?


This is the most likelymethod to come frompupils.

Put this table on theboard and have thepupils complete theentries.

Other strategies maystart to emerge.

You can make different colours of paint by mixing red, blue and yellow in differentproportions.

For example, you can make green by mixing 1 part blue to 1 part yellow.

(a) To make purple, you mix 3 parts red to 7 parts blue.

How much of each colour do you need to make 20 litres of purple paint?

Give your answer in litres.

. . . . . litres of red and . . . . . litres of blue

(b) To make orange, you mix 13 parts yellow to 7 parts red.

How much of each colour do you need to make 10 litres of orange paint?

Give your answer in litres.

. . . . . litres of yellow and . . . . . litres of red

From 1998 Key Stage 3 Paper 2 question 10

Strategy 2: Unitary method

Initial stage to calculate what is required for one unit of the mix.

purple red blue

10 3 7

1 3 ÷ 10 7 ÷ 10

20 (3 ÷ 10) × 20 (7 ÷ 10 × 20




How does it help if the components of red and blue change – for example, 3 parts red and 8 parts blue?


Strategy 3: Scale factor method

Finding the proportion of the mix made up by each component part – that is, the factor by which the total is multiplied to calculate each part.

purple red blue

10 3 7

20 20 ÷ 10 × 3 (20 ÷ 10) × 7




How does it help if the components of red and blue change – for example, 3 parts red and 8 parts blue?


Choose a small set of problems: concentrate on extracting and organising thedata (e.g. putting into tabular form) before deciding on possible methods ofsolution, rather than working the problems through to an answer.


Ask: What calculationtakes me from 10purple to 1 purple?

Ask: What calculationtakes me from 10purple to 3 red? Whatis this as a singlemultiplier?

÷ 10 × 3

÷ 10 × 7

× 0.7

× 0.3

Translate the data from each question into a useful form.

Orange juice

20% or 0.2 every 1 litre

20% or 0.2 2.5 litres

What calculation do we perform to find 20% of anything? Roughly how big will the answer be?

Market stall

Bananas £350 1/3

Apples 1/3

Peaches 1/4

Oranges ? ?

Will the answer be more than £350? Do we need oranges as a fraction of the whole or asa fraction of bananas?


• There is 20% orange juice in every litre of a fruit drink. How much orange juice isthere in 2.5 litres of fruit drink? How much fruit drink can be made from 1 litre oforange juice?

• This chart shows the income that a market stall-holder got last week from sellingdifferent kinds of fruit.

The stall-holder got £350 from selling bananas. Estimate how much she got from selling oranges.

• 6 out of every 300 paper clips produced by a machine are rejected. What is this as a percentage?

• Rena put £150 in her savings account. After one year, her interest was £12. John put £110 in his savings account. After one year, his interest was £12. Who had the better rate of interest, Rena or John? Explain you answer.

From page 75 of the Supplement of examples in the Framework for teaching mathematics: Years 7, 8 and 9

applespeaches

oranges

bananas

Paper clips

Produced 300 100

Rejected 6 ?

What relationship is it useful to identify here?

Savings

Rena £150 £12

John £110 £12

(a) The label on yoghurt A shows this information.

How many grams of protein does 100 g of yoghurt provide?

Show your working.

(b) The label on yoghurt B shows different information.

A boy eats the same amount of yoghurt A and yoghurt B.

Which yoghurt provides him with more carbohydrate?

Show your working.


Translate the data from each question into a useful form.

Part (a)

Yoghurt A 125 g 1 g 25 g 100 g

Protein 4.5 g ? ? ?

Which of the entries is useful to calculate?

Part (b)

Yoghurt A 125 g 1 g 25 g 100 g 150 g

Carbohydrates 11.1 g ? ? ? ?

Yoghurt B 150 g 1 g 25 g 100 g 150 g

Carbohydrates 13.1 g ? ? ? ?

Which of the entries is useful to calculate?


300 ÷ 3 = 100

Draw out gutreaction justifiedwith key words.

Yoghurt A 125 g

Each 125 g provides

Energy 430 kJProtein 4.5 gCarbohydrate 11.1 gFat 4.5 g

Yoghurt B 150 g

Each 150 g provides

Energy 339 kJProtein 6.6 gCarbohydrate 13.1 gFat 0.2 g


Problem bank for phase 3These problems support phase 3 of the multiplicative relationships unit. They have beenselected from previous Key Stage 3 test papers. Questions 1–3 are targeted at NC level5, questions 4–8 at level 6, questions 9 and 10 at level 7 and question 11 at level 8. Allquestions are taken from the calculator paper. Additional problems can be found onpages 5, 75 and 79 of the Framework’s supplement of examples.

1 Paint

You can make different colours of paint by mixing red, blue and yellow in differentproportions. For example, you can make green by mixing 1 part blue to 1 part yellow.

(a) To make purple, you mix 3 parts red to 7 parts blue.

How much of each colour do you need to make 20 litres of purple paint? Give youranswer in litres.

. . . . . litres of red and . . . . . litres of blue

(b) To make orange, you mix 13 parts yellow to 7 parts red.

How much of each colour do you need to make 10 litres of orange paint? Give youranswer in litres.

. . . . . litres of yellow and . . . . . litres of red


2 Ratios

(a) Nigel pours 1 carton of apple juice and 3 cartons of orange juice into a big jug.

What is the ratio of apple juice to orange juice in Nigel’s jug?

apple juice : orange juice = . . . . . : . . . . .

(b) Lesley pours 1 carton of apple juice and 11/2 cartons of orange juice into another big jug.

What is the ratio of apple juice to orange juice in Lesley’s jug?

apple juice : orange juice = . . . . . : . . . . .

(c) Tandi pours 1 carton of apple juice and 1 carton of orange juice into another big jug.She wants only half as much apple juice as orange juice in her jug.

What should Tandi pour into her jug now?


3 Ages

These pie charts show some information about the ages of people in Greece and inIreland.

There are about 10 million people in Greece, and there are about 3.5 million people inIreland.

(a) Roughly what percentage of people in Greece are aged 40–59?

(b) There are about 10 million people in Greece. Use your percentage from part (a) towork out roughly how many people in Greece are aged 40–59.

(c) Dewi says: ‘The charts show that there are more people under 15 in Ireland than inGreece.’

Dewi is wrong. Explain why the charts do not show this.

(d) There are about 60 million people in the UK.

The table shows roughly what percentage of people in the UK are of different ages.

under 15 15–39 40–59 over 59

20% 35% 25% 20%

Draw a pie chart below to show the information in the table. Label each section of yourpie chart clearly with the ages.



4 Birds

(a) One morning last summer Ravi carried out a survey of the birds in the school garden.He saw 5 pigeons, 20 crows, 25 seagulls and 45 sparrows.

Complete the line below to show the ratios.

1 : . . . . . : . . . . . : . . . . .

(b) What percentage of all the birds Ravi saw were sparrows?

(c) One morning this spring Ravi carried out a second survey. This time he saw:

the same number of pigeons

25% fewer crows

60% more seagulls

two thirds of the number of sparrows

Pigeons : Crows : Seagulls : Sparrows

1 : . . . . . : . . . . . : . . . . .


5 Salt

(a) What is the volume of this standard size box of salt?

(b) What is the volume of this special offer box of salt, which is 20% bigger?

(c) The standard size box contains enough salt to fill up 10 salt pots.

How many salt pots may be filled up from the special offer box of salt?



6 Population

Emlyn is doing a project on world population. He has found some data about thepopulation of the regions of the world in 1950 and 1990.

(a) In 1950, what percentage of the world’s population lived in Asia? Show each step inyour working.

(b) In 1990, for every person who lived in North America how many people lived in Asia?Show your working.

(c) For every person who lived in Africa in 1950 how many people lived in Africa in 1990?Show your working.

(d) Emlyn thinks that from 1950 to 1990 the population of Oceania went up by 100%.

Is Emlyn right? Tick the correct box.

Yes No Cannot tell

Explain your answer.


7 Continents

The table shows the land area of each of the world’s continents.


Regions of Population Populationthe world (in millions) (in millions)

in 1950 in 1990

Africa 222 642

Asia 1 558 3 402

Europe 393 498

Latin America 166 448

North America 166 276

Oceania 13 26

World 2 518 5 292

Continent Land area (in 1 000 km2)

Africa 30 264

Antarctica 13 209

Asia 44 250

Europe 9 907

North America 24 398

Oceania 8 534

South America 17 793

World 148 355

(a) Which continent is approximately 12% of the world’s land area?

(b) What percentage of the world’s land area is Antarctica? Show your working.

(c) About 30% of the world’s area is land. The rest is water. The amount of land in theworld is about 150 million km2.

Work out the approximate total area (land and water) of the world. Show yourworking.


8 Currency

(a) Use £1 = 9.60 francs to work out how much 45p is in francs. Show your working.

(b) Use 240 pesetas = £1 to work out how much 408 pesetas is in pounds. Show yourworking.

(c) Use £1 = 9.60 francs and £1 = 240 pesetas to work out how much 1 franc is inpesetas. Show your working.


9 Roof frames

Timpkins Builders make wooden frames for roofs on new houses.

In the diagram of the wooden frame shown below, PQ is parallel to BC.

(a) Calculate length PQ using similar triangles. Show your working.

In the diagram of the wooden frame shown below, angle ABC = angle LMC, and angleACB = angle KNB.

(b) Calculate length LM using similar triangles. Show your working.


10 Pupils

The table shows some information about pupils in a school.

Left-handed Right-handed

Girls 32 180

Boys 28 168

There are 408 pupils in the school.


(a) What percentage of the pupils are boys? Show your working.

(b) What is the ratio of left-handed pupils to right-handed pupils?

Write your ratio in the form 1 : . . . . .

Show your working.

(c) One pupil is chosen at random from the whole school.

What is the probability that the pupil chosen is a girl who is right-handed?


11 Births

Look at the table:

1961 1994

England 17.6

Wales 17 12.2

(a) In England, from 1961 to 1994, the birth rate fell by 26.1%. What was the birth rate inEngland in 1994?

Show your working.

(b) In Wales, the birth rate also fell. Calculate the percentage fall from 1961 to 1994.

Show your working.

(c) From 1961 to 1994, the birth rates in Scotland and Northern Ireland fell by the sameamount.

The percentage fall in Scotland was greater than the percentage fall in NorthernIreland.

Put a tick by the statement below which is true.

In 1961, the birth rate in Scotland was higher than the birth rate in Northern Ireland. . . . . .

In 1961, the birth rate in Scotland was the same as the birth rate in Northern Ireland. . . . . .

In 1961, the birth rate in Scotland was lower than the birth rate in Northern Ireland. . . . . .

From the information given, you cannot tell whether Scotland or Northern Ireland had the higher birth rate in 1961. . . . . .



Pupil resource sheet

Parallel number lines: examples

× 5/3

÷3

× 5

1

3

5

× 9/2

÷2

× 9

12

9

× 5/3

÷3

× 5

15

3

5

× 9/2

÷2

× 9

18

2

9

1 | Interacting with mathematics | Pupil resource sheets © Crown copyright 2002



Parallel number lines

© Crown copyright 20023 | Interacting with mathematics | Pupil resource sheets



Multiplicative relationships: key results

Fractions, decimals and percentages

When using a calculator, it is easiest to deal with numbers expressed in decimal form.

To convert a fraction to a decimal: a/b = a ÷ b.

Example: 5/8 = 5 ÷ 8 = 0.625

To convert a percentage to a decimal, or vice versa, remember that percentages arehundredths:

Examples: 89% = 0.89, 2.75 = 275%

Scale factors

To scale a to b, multiply by b/a.

To scale b to a, multiply by a/b.

Example:

5 4 4 5

scale factor = 0.8 scale factor = 1.25

5 × 0.8 = 4 4 × 1.25 = 5

Proportions

Two pairs of numbers are in proportion if the scale factors for the pairs of numbers areequal.

Example:

a b

5 4

10 8

35 28 b/a = 4/5 = 8/10 = 28/35 = 0.8

Bigger or smaller?

When solving problems involving multiplication and division it is useful to ask yourself ‘Will the answer be smaller or will it be bigger?’

Starting with any positive number n:

• Multiplying n by a number bigger than 1 increases n.

Example: 0.25 × 5 = 1.25 (bigger than 0.25)

• Multiplying n by a number less than 1 decreases n.

Example: 0.25 × 0.2 = 0.05 (smaller than 0.25)

• Dividing n by a number bigger than 1 decreases n.

Example: 0.25 ÷ 5 = 0.05 (smaller than 0.25)

• Dividing n by a number smaller than 1 increases n.

Example: 0.25 ÷ 0.2 = 1.25 (bigger than 0.25)

© Crown copyright 20025 | Interacting with mathematics | Pupil resource sheets

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