s8 Measures

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S8 Measures

KS4 Mathematics


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Contents

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S8.1 Converting units

S8 Measures

S8.3 Calculations involving bounds

S8.4 Compound measures

S8.5 earings

S8.2 !ccurac" in measurement


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Converting units

It is important to be able to convert between a variety o

units !uickly and accurately"

#or e$ample% i we are usin& a ormula to ind t'e volume

o an ob(ect we must make sure t'at all t'e len&t's are

written usin& t'e same units beore usin& t'e ormula"

)* #hen using a formula

2* #hen comparing measurements

#or e$ample% suppose one mot'er &ives t'e wei&'t o 'er

baby in pounds and anot'er mot'er &ives t'e wei&'t o 'er

baby in kilo&rams"

+e usually use metric units or calculations"

,ow can we compare t'e babies- wei&'ts.

/o compare t'e wei&'ts we convert t'em to t'e same unit"


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Metric units

/'e metric system o measurement is based on powers o

ten and uses t'e ollowin& prei$es

/'ese prei$es are t'en ollowed by a base unit"

/'e base unit or len&t' is metre"metre.

/'e base unit or mass is &ram"gram.

/'e base unit or capacity is litre"litre.

Kilo$

Centi$

Milli$Micro$

meanin& one t'ousand

meanin& one 'undredt'

meanin& one t'ousandt'meanin& one milliont'


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Metric units of length% mass and capacit"

1ou s'ould know t'e ollowin& metric conversions or

len&t'% mass and capacity li!uid volume*

&ength

&ength

1 'm ( 1))) m

1 m ( 1)) cm

1 m ( 1))) mm

1 cm ( 1) mm

Mass

Mass

1 tonne ( 1))) 'g

1 'g ( 1))) g

1 g ( 1))) mg

Capacit" and

*olume

Capacit" and

*olume

1 litre ( 1))) ml

1 cl ( 1) ml

1 m3 ( 1))) litres

1 cm3 ( 1 ml


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Metric units of area and volume

/'e metric units or area are mm2% cm2% m2 and km2"

) m2 3 )00 cm 4 )00 cm

) m2 3 )0%000 cm2

) m2 3 )0%000 cm2

) cm2 3 )0 mm 4 )0 mm

) cm2 3 )00 mm2

) cm2 3 )00 mm2

/'e metric units or volume are mm% cm% m and km"

) m 3 )00 cm 4 )00 cm 4 )00 cm

) m 3 )%000%000 cm

) m 3 )%000%000 cm

) cm 3 )0 mm 4 )0 mm 4 )0 mm

) cm 3 )000 mm

) cm 3 )000 mm


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Converting metric units

/o convert rom a lar&er metric unit to a smaller

one we need to ------- by )0% )00% or )000"

/o convert rom a lar&er metric unit to a smaller

one we need to ------- by )0% )00% or )000"multipl"

#or e$ample%

0"6 k& 3 0"6 4 )000 &

7onvert 0"6 k& to &rams

3 60 &

/o convert rom a smaller metric unit to a lar&er

one we need to ------- by )0% )00% or )000"

/o convert rom a smaller metric unit to a lar&er

one we need to ------- by )0% )00% or )000"divide

#or e$ample%

"9 cm 3 "9 : )00 m

7onvert "9 cm to metres

3 0"09 m


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Converting metric units


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/rdering units


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0mperial units

Imperial units are still re!uently used and you s'ould be

aware o t'e ollowin& imperial conversions

) oot 3 ;;; inc'es)2

) yard 3 ;;; eet

) pound 3 ;;; ounces)9

) stone 3 ;;; pounds)6

) &allon 3 ;;; pints8



Converting imperial units

+e can convert between metric and imperial units usin& t'e

ollowin& appro$imate conversions

5 miles is about ;;; kilometres8

) oot is about ;;; centimetres0

) inc' is about ;;; centimetres2"5

) kilo&ram is about ;;; pounds2"2

) &allon is about ;;; litres6"5

) litre is about ;;;; pints)"5



Converting imperial units

) kilo&ram is about 2"2 pounds"

About 'ow many kilo&rams are t'ere in ) stone.

) stone 3 )6 pounds



Metric and imperial conversions



Spider diagram



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S8.5 earings

S8 Measures


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Continuous measurements

/'e continuous nature o measurements means t'at t'ey

can never be e$act" /'ere is always an element oroundin& involved"

I we were measurin& t'e len&t' o a pencil% or e$ample% we

would measure to t'e nearest cm or mm"

/'ese t'ree pencils all measure )6 cm to t'e nearest cm

13 14 15 13 14 15 13 14 15

/'e len&t' o a pencil &iven as )6 cm to t'e nearest cm can

be any len&t' between )"5 cm and )6"5 cm"


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!ccurac" in measurement

>emember% any measurement &iven to t'e nearest w'ole

unit could be up to 'al a unit lon&er or s'orter"

>emember% any measurement &iven to t'e nearest w'ole

unit could be up to 'al a unit lon&er or s'orter"

Suppose we are told t'at a pencil measures )6"2 cm"

7an we assume t'at t'is measurement is e$act.

?o% alt'ou&' t'is measurement 'as been &iven to a 'i&'er

de&ree o accuracy it is not e$act"

+'at is t'e s'ortest and lon&est len&t' it could be.

/'e len&t' l o a pencil &iven as )6"2 cm% to t'e nearest 0")

cm% could be anyw'ere in t'e ran&e

)6")5 cm @ l )6"25 cm)6")5 cm @ l )6"25 cm



pper and loer bounds

+'en we &ive a ran&e or a measurement as in

)6")5 cm @ len&t' )6"25 cm

t'is value is calledt'e lower bound t'is value is calledt'e lower bound

and t'is value iscalled t'e upper bound" and t'is value iscalled t'e upper bound"

/'e len&t' could be e!ual to )6")5 cm so we use a &reater

t'an or e!ual to symbol"

I t'e len&t' was e!ual to )6"25 cm 'owever% it would 'ave

been rounded up to )6" cm" /'e len&t' is t'ereore Cstrictly

less thanD )6"25 cm and so we use t'e symbol"

/'is is an

ineualit"



pper and loer bounds


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S8.5 earings

S8 Measures



!dding measures

/'e ollowin& trian&le 'as sides o len&t' cm% 6 cm and 5 cm"

cm5 cm

6 cm

+'at is t'e ran&e o possible

len&t's or t'e perimeter.

/'e least t'e len&t's could be is 2"5 cm% "5 cm and 6"5 cm"

/'e smallest possible perimeter 3 2"5 E "5 E 6"5 3 )0"5 cm

/'e most t'e len&t's could be is "5 cm% 6"5 cm and 5"5 cm"

/'e lar&est possible perimeter 3 "5 E 6"5 E 5"5 3 )"5 cm



!dding measures

/'e ollowin& trian&le 'as sides o len&t' cm% 6 cm and 5 cm"

cm5 cm

6 cm


len&t's or t'e perimeter.

/'e ran&e o possible values or t'e perimeter is

?otice t'at t'e more len&t's t'at are added to&et'er t'e

&reater t'e error on eit'er side"

)0"5 cm @ perimeter )"5 cm)0"5 cm @ perimeter )"5 cm



!dding measures

+'en a calculation involves adding two or more

measurements to&et'er

/'e lower bound is ound by

addin& t'e lower bounds to&et'er addin& t'e lower bounds to&et'er

/'e upper bound is ound by

addin& t'e upper bounds to&et'er addin& t'e upper bounds to&et'er



Calculations involving bounds

A piece o wood measurin& )0 cm 'as

a piece o len&t' 50 cm cut o o it"

/'ese are &iven to t'e nearest )0 cm"


siFes or t'e remainin& piece.

/'e ori&inal piece o wood could be between )95 cm and

)5 cm"

/'e piece o wood t'at is cut o could be between 65 cmand 55 cm"

/'e smallest possible siFe o t'e remainin& piece is

)95 G 55 3 ))0 cm










)5 cm"


/'e lar&est possible siFe o t'e remainin& piece is

)5 G 65 3 )0 cm


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)5 cm"


/'e ran&e o possible siFes or t'e remainin& piece is

))0 cm @ len&t' )0 cm))0 cm @ len&t' )0 cm


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+'at is t'e ran&e o possible values or t'e an&le a.

/'e smallest a could be is )80H G 5"5H E 8"5H* 3

/'e ran&e o possible values or a is t'ereore%

99H @ a 98H 99H @ a 98H

/'e an&les in t'e ollowin& dia&ram are rounded to t'e

nearest de&ree

99H

5H 8Ha

/'e lar&est a could be is )80H G 6"5H E "5H* 3 98H

/'is an&le could

be between 6"5H

and 5"5H"

/'is an&le could

be between "5H

and 8"5H"


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Subtracting measures

+'en a calculation involves subtracting two measurements


subtractin& t'e upper bound rom t'e lower boundsubtractin& t'e upper bound rom t'e lower bound


subtractin& t'e lower bound rom t'e upper boundsubtractin& t'e lower bound rom t'e upper bound


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/'e dimensions o a small tile are

&iven as )5 mm by )8 mm"

)5 mm

)8 mm

+'at is t'e smallest possible

area t'e tile could 'ave.

/'e smallest values or t'e len&t' and t'e widt' o t'e tile

are )6"5 mm by )"5 mm"

/o calculate t'e smallest possible area we multiply t'ese

values to&et'er"

Smallest possible area 3 )6"5 4 )"5 3 25"5 mm2


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)5 mm

)8 mm

+'at is t'e lar&est possible

area t'e tile could 'ave.

/'e lar&est values or t'e len&t' and t'e widt' o t'e tile are

)5"5 mm by )8"5 mm"

/o calculate t'e lar&est possible area we multiply t'ese

values to&et'er"

Lar&est possible area 3 )5"5 4 )8"5 3 289"5 mm2


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)5 mm

)8 mm


values or t'e area.

/'e smallest value or t'e area is 25"5 mm2 and t'e

lar&est value or t'e area is 289"5 mm2

/'e ran&e o possible values or t'e area is t'ereore%

25"5 mm2 @ area 289"5 mm225"5 mm2 @ area 289"5 mm2


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Multipl"ing measures

+'en a calculation involves multipl"ing two or more

measurements to&et'er


multiplyin& t'e lower bounds to&et'er multiplyin& t'e lower bounds to&et'er


multiplyin& t'e upper bounds to&et'er multiplyin& t'e upper bounds to&et'er


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A boy runs 200 metres in 2"8 seconds"

+'at is 'is &reatest possible

avera&e speed to 2 decimal places.

/'e distance is &iven to t'e nearest metre and t'e time is

&iven to t'e nearest tent' o a second"

/'e distance could be between )5"5 m and 200"5 m"

/'e time taken could be between 2"5 s and 2"85 s"&reatest possible

avera&e speed3

&reatest distance

s'ortest time3

200"5 m

2"5 s3 "2 mJs


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ividing measures

+'en a calculation involves dividing two measurements


dividin& t'e lower bound by t'e upper bounddividin& t'e lower bound by t'e upper bound


dividin& t'e upper bound by t'e lower bounddividin& t'e upper bound by t'e lower bound


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S8.5 earings


S8 Measures


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Compound measures

+'ic' is 'eavier a kilo&ram o cotton wool

or a kilo&ram o rocks.


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Compound measures

K course% a kilo&ram o rocks wei&'s t'e same as a

kilo&ram o cotton wool G t'ey bot' wei&' one kilo&ram

I you t'ou&'t t'at t'e rocks were 'eavier% you were probably

t'inkin& o t'eir density rat'er t'an t'eir wei&'t"

ensity is an e$ample o a compound measure"

It is a measure o t'e mass o an ob(ect per unit volume"

ensity is usually measured in &Jcm or k&Jm" It can also

be measured in k&Jl "

A density o )"2 &Jcm means t'at every ) cm o t'e material

'as a mass o )"2 &rams"

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Compound measures

/'is is a more sensible !uestion" >ock is denser t'an cotton

wool and so ) cm o rock is 'eavier t'an ) cm o cotton wool"

+e can ind t'e density o a &iven material usin& t'e

ollowin& ormula

ensity 3mass

volume

+'ic' is 'eavier ) cm o cotton wool or ) cm o rock.

#or e$ample% i a block o metal 'as a mass o 90 & and avolume o 80 cm

ensity 390

803 "5 &Jcm

C d


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Compound measures

+e use compound measures w'en we are comparin& 'ow

one measurement c'an&es wit' anot'er"

+'en one measurement c'an&es in direct proportion wit'

anot'er it is said to c'an&e at a constant rate"

#or e$ample% suppose a man is runnin& around a race track"

/'e total distance 'e 'as run c'an&es wit' time"

/'e rate at w'ic' 'e runs is called 'is speed"

Speed is usually measured in kmJ'% mJs or mp'"

Speed 3 distance travelledtime taken

! d


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!verage speed

In many situations t'e speed is not constant"

#or e$ample% t'e man runnin& around t'e track will probably

speed up or slow down as 'e runs"

+e can still calculate 'is avera&e speed usin& t'e ollowin&

ormula

Avera&e speed 3total distance travelled

total time taken

#or e$ample% i t'e man runs )590 metres in 00 seconds

Avera&e speed 3)590

003 5"2 mJs

C d


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Common compound measures

7ommonly used compound measures include

#uel consumptionMeasured in kmJl or

mp&"

NressureMeasured in ?Jm2 or

?Jcm2"

SpeedMeasured in mJs% kmJ'

or mp'"

ensityMeasured in &Jcm%

k&Jm or k&Jl "

mass

volume

distance

time

orce

surace area

distance

volume

i d


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airs 6 compound measures

C l l ti d iti


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Calculating densities

C ti d it


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Converting compound units

+'at is 65 mp' in mJs.

+'en we convert compound units we usually 'ave to do it in

several steps% especially i bot' units are bein& c'an&ed"

65 mp' stands or 65 miles per 'our"

+e 'ave to c'an&e t'e 65 miles into metres and ) 'our into

seconds" 5 miles 3 8 km

So ) mile 3 )"9 km65 miles = 65 4 )"9 km

3 2 km3 2%000 m

) 'our 3 90 minutes

3 900 seconds

C ti d it


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Converting compound units

+'at is 65 mp' in mJs.

/ravellin& 65 miles in ) 'our is e!uivalent to travellin& 2 000 m

in 900 seconds"

/'ereore% 65 mp' =2%000

900mJs

3 20 mJs

,ow could we use t'is answer to convert any

speed &iven in mp' to a speed in mJs.

65 mp' = 20 mJs

ividin& by 5 we 'ave% mp' = 6 mJs

+e can divide t'e speed in mp' by and multiply it by 6"+e can divide t'e speed in mp' by and multiply it by 6"

Contents


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S8.5 earings





S8 Measures

earings


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earings

earings are a measure o direction taken rom ?ort'"

I you were travellin& ?ort' you would be travellin& on a

bearin& o 000H"

I you were travellin& rom t'e point N in t'e direction

s'own by t'e arrow t'en you would be travellin& on abearin& o 000H"

I you were travellin& rom t'e point N in t'e direction

s'own by t'e arrow t'en you would be travellin& on abearin& o ),57.

?

5H

N

Bearin&s are always measured

clockwise rom ?ort' and arewritten as t'ree i&ures"

Compass points


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Compass points

N

E W

S

NE

SE SW

NW

000H

065H

00H

)5H

)80H

225H

20H

)5H

Measuring bearings


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Measuring bearings

ac' bearings


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ac' bearings

/'e bearin& rom point A to point B is )05O"

+'at is t'e bearin& rom point B to point A.

A

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)05O

B

?

.

/'e an&le rom B to A is

)05O E )80O 3 285

/'is is called a bac'

bearing")05O

)80H

0nvestigating bac' bearings


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0nvestigating bac' bearings

s8 Measures

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