Maths 2010 Book 2 Teachers' Self Study Guide

8/12/2019 Maths 2010 Book 2 Teachers' Self Study Guide

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Mathematics Teachers Self Study Guideon the National Curriculum Statement

Book 1 of 2

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CONTENTS

Financial Mathematics

Transformation Geometry

Data Handling

inear !rogramming

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FINANCIAL

MATHEMATICS

"


4/42

CONTENT LAYOUT!re#kno$ledge %&'er'ie$ of Grade 1( and 11)*

Sim+le and com+ound a++reciation and de+reciation, Nominal and effecti'e interest rates Hire +urchase

Grade 12 and -eyond*

Time lines

.uture and +resent 'alue annuities Sinking funds &utstanding 'alue of a loan Deferred annuities .i/ed +ayment annuities

EXERCISE 1: DIAGNOSTIC TEST ON GRADE 10 AND 11 CURRICULUM

1, .rom .ocus on Mathematics Grade 11

0n 2((" it $as estimated that 3 million South 4fricans $ere infected $ith H05,0t $as estimated that the num-er of H05 +ositi'e South 4fricans $as increasing ata rate of a++ro/imately 67 +er year,

4ssuming this gro$th rate remains constant ho$ may South 4frican -e infected-y*

1,1 2((6 1,2 2(2(8

2, .rom Shuters Mathematics Grade 11

Calculate the rate of de+reciation of a -us +urchased for 9:;( ((( if its scra+'alue eight years later is 911( ((( and de+reciation is calculated on the*

2,1 straight#line -asishat $ill -e the amount of the monthly re+ayments8

2,2 Sho$ that a -ank loan at 217 +,a, com+ounded monthly on

the reducing -alance is a -etter o+tion,

2," >hat hire +urchase rate $ould -e -etter than the 217 -ankloan8

", 4 loan of 91:( ((( is to -e re+aid o'er 2( years -y means of e=ual monthly+ayments starting one month after the loan is granted, The interest rate is 167+,a, com+ounded monthly,

",1 Calculate the monthly re+ayments,

",2 Calculate the outstanding -alance after 1( years,

D!$!rr!# annui%i!'

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0n many situations the re+ayment of a loan -egins one month or one +eriod after thegranting of the loan, Ho$e'er circumstances arise $here the re+ayment of the loan isdeferred or delayed for an agreed +eriod of time,

E5amp-!

Ntha-i -orro$s 92( ((( to start a small -usiness, She agrees to re+ay the loan o'er a+eriod of fi'e years from the time the loan is granted, The first +ayment ho$e'er $ill-e nine months after the granting of the loan, The interest rate is 1:7 +,a, com+ounded=uarterly, Calculate the =uarterly instalments,

Solution*

4t *(T

2(;"

1:(1,,,

1:(1

1:(1

1:(1(((2(

+++

++

++

+= xxxx

Hence ??313"3Rx=

1


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Solution* Method 2

4t 2(T *

1322(

1:(1,,,

1:(1

1:(1

1:(1(((2(

+++

++

++=

+ xxxx

??313"3

1

1:(1

1:(

1:(1(((2(

1:

2(

Rx =

+

+

= %to the nearest cent)

Solution* Method "*

4t 2T *

2( (((

1:(

1:(11

1:(1

1:

2

+=

+

x%@sing the +resent 'alue formula)

4gain ??"3313Rx=

Fi5!# pa,m!n% annui%i!'

4 loan is sometimes re+aid -y deciding on $hat the -orro$er can afford $hich is often a

+ercentage of hisher income, 0n such circumstances the num-er of +ayments as $ell asthe amount of the final +ayment need to -e calculated,

E5amp-!

4 teacher -orro$s 912 :(( to -uy furniture, He can afford to re+ay 93(( a month, Theinterest rate is 1:7 +,a, com+ounded monthly,

1, Calculate the num-er of months that he $ill -e re=uired to +ay 93((,

2, Determine the amount of the final +ayment%less than 93(() $hich is made one month afterthe last +ayment of 93((,

1;


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Solution*

1,

n

+++

++

++

+=

12

1:(13((,,,

12

1:(13((

12

1:(13((

12

1:(13((:((12

"21

1

1

12

1:(11

12

1:(11

12

1:(13((

:((12

+

+

+

=

n

,,,32;3(12

1:(1 =

+

n

,,,;""21

,,,32;3(log12

1:(1log

=

=

+

n

n

There $ill -e 21 +ayments of 93(( and a 22nd+ayment of less than 93((

2, The final +ayment*

22

1

211

12

1:(1

12

1:(11

12

1:(11

12

1:(13((

:((12

++

+

+

+

= y

;1"3

1(;"(12:((1212

1:(1

22

Ry

y

=

=

+

16


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E5!r+i'! /

1* 4fter mo'ing to a ne$ house you -uy a num-er of household items for a total of9" "((, Aou are a-le to finance the +urchase of these goods under the follo$ingconditions*

the money is re+aid in 2 e=ual monthly +ayments< the first +ayment is made after 6 months< the interest rate is 127+,a, com+ounded monthly,

1,1 Calculate the amount of the monthly re+ayments,

1,2 Ho$ much money $ill you o$e after : +ayments ha'e -een made8

2, .rom 0EB 4dditional Mathematics No'em-er 2((6

4shika takes out a loan of 9;( ((( at an effecti'e interest rate of 17 +,a, in

order to +urchase a to$n house, She re+ays the loan $ith e=ual monthlyinstalments of 93 ;(( starting one month from the granting of the loan, Theinterest is com+ounded monthly,

2,1 Sho$ that the nominal interest rate is a++ro/imately 1"137 +,a,

2,2 Calculate

2,2,1 the time s+an of the loan in months,

2,2,2 the 'alue of the last +ayment %less than 93 ;((),

2," Calculate ho$ much interest is +aid*

2,",1 in the first month

2,",2 in the first year of re+ayment,

", 4 retirement +olicy +ays out 92;( ((( $hen a $orker retires, The $orker in'eststhis amount at an effecti'e 1;7 +,a, 0f the $orker $ishes to dra$ 93 ;(( amonth ho$ long $ill the money last8

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ANS>ERS

E5!r+i'! 1

1,1 ;6 million %to 1 dec, +l,) 1,2 123 million %to 1 dec, +l,)

2,1 1(?7 %to 1 dec, +l,) 2,1 2267 %to 1 dec, +l,)",1 Eff, rate 1167 %to 2 dec, +l,)",2 Nom, rate 1"";7 % to 2 dec, +l,), Monthly +ayments 96 66663 %to nearest cent);, 9 23;6" %to nearest cent)6,1 9";( 6,2 ?7

E5!r+i'! 6

1, ; years %6( months) 2, 2( years", 3 years and ? days %rounded u+),1 years and : months ,2 3 years and 11 months

E5!r+i'! =

1, 96 "12; %to nearest cent) 2, 91 6"";" %to nearest cent)", 92;6 22(1" %to nearest cent) , 926?:: %to nearest cent)

E5!r+i'! LA,

( ) ( ) ( )2


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Ty+e "* glide reflection and 'ertical symmetry,

Ty+e * half turn symmetry or rotational symmetry of order 2,

Ty+e ;* 'ertical symmetry

Ty+e 6* horiKontal symmetry

Ty+e 3* 'ertical and horiKontal symmetry

.ind and make co+ies of actual frieKes from at least 2 different counties including South4frica illustrating each of the se'en frieKe ty+es and classify them according to the ty+esillustrated a-o'e,

2;


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Escher became fascinated by the regular Division of the Plane in 1922. This is here an entire s!ace ista"en u! ith recurring images that tessellate ith each other.

Gra#! 16 2r"!+%: E'+&!r an# Tran'$"rma%i"n G!"m!%r,.Fr"m >!'%!rn Cap! A''!''m!n% ?""k-!% 600

Maurits Cornelis Escher $ho $as -orn in eeu$arden Holland in 1:?: created uni=ueand fascinating $orks of art that e/+lore and e/hi-it a $ide range of mathematical ideas,

During his lifetime he made : lithogra+hs $oodcuts and $ood engra'ings and o'er2((( dra$ings and sketches,

(All M.C. Es!"r #$rks (% &'' T!" M.C. Es!"r C$)*any + ,!" -",!"rlans. All ri/!,sr"s"rv". 0s" by *"r)issi$n.###.)"s!"r.$)%

Ta'k 1

@sing the outline of the fish sho$n -elo$ you need to attem+t to di'ide the +lane $iththis sha+e,

Start $ith a -lank sheet of 4 +a+er andchoose a starting +osition to trace your fishand mark it clearly as the start,

@sing this sha+e alone you must de'elo+ a

+attern using the rules of transformationgeometry, Each time you trace the fish in ane$ +osition you need to descri-e thetransformation you ha'e carried out,

Aou may use translation reflection androtation, 0n your descri+tion remem-er tos+ecify the degrees and direction that youha'e rotated the sha+e or the manner in$hich you ha'e reflected or translated it,

The use of different colours often hel+s tokee+ track of $here you are,

&nce you ha'e finished your design $ithaccom+anying directions a second +ersonshould -e a-le to follo$ your instructionsand come u+ $ith the same design,

26
http://var/www/apps/conversion/tmp/scratch_1/www.mcescher.comhttp://var/www/apps/conversion/tmp/scratch_1/www.mcescher.comhttp://var/www/apps/conversion/tmp/scratch_1/www.mcescher.com


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Ta'k 6

0n the sketch -elo$ Escher has used se'eral different sha+es to di'ide the +lane,

0dentify ho$ many sha+es are used and $hat they are,

Descri-e ho$ they are com-ined and transformed to create the design sho$n,

Make mention of any symmetry that e/ists any reflection rotation or translationthat has taken +lace,

Ta'k =

The a-o'e sketch sho$s a frieKe $here 2 sha+es are tessellating to create a recurring+attern that can -e used as a +age -order,

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Escher as mesmeri#ed by gec"os and did several s"etches using this creature. $elo is one of hismore famous ones. This s"etch uses a com!le% combination of transformations including enlargement&

rotation and translation.

Descri-e the transformation used in detail,Ta'k estern Ca+e 9i'er conducted from 2((; to 2((3, The sur'ey measured the+ollution %total dissol'ed solids) in milligrams +er litre of $ater,

a) Determine the fi'e num-er summary for 2((6,

-) >hich year had the greatest range of results8 E/+lain your ans$er,

c) @sing the data comment on the +ollution le'els in the ri'er o'er the threeyears of monitoring,

, The housing de+artment recently conducted a sur'ey on the num-er of -edroomsin 2(( houses in Saldanha, The follo$ing data $as o-tained*

No, of -edrooms 1 2 " ; 6 3 : ? 1(No, of houses 2; ( 2 2: 22 1: 12 3 2

a) Set u+ a cumulati'e fre=uency ta-le for the gi'en data,-) Hence dra$ an ogi'e %cumulati'e fre=uency gra+h) for this data,c) @se the ogi'e to determine the median lo$er =uartile %P1) and u++er

=uartile %P"),d) Dra$ a -o/ and $hisker diagram for this data,e) >hat is the 'alue of the inter=uartile and semi#inter=uartile range8

"1

2((3

2((6

2((;

;(( 1((( 1;(( 2((( 2;(((TOTAL DISSOLED SOLIDS .m/l

; &n the Game of S@95054 a +iece of

music $as +layed to a grou+ of :(contestants and after$ards they $ere askedto estimate ho$ long the +iece of musichad lasted, The estimates in seconds aregi'en in the cumulati'e fre=uency gra+h-elo$,

@sing the gra+h determine*a)

the median


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6 The ta-le -elo$ gi'es the 'alues of the height !cm of a +lant #$eeks aftergerminating,

# 1 2 " ; 6 3

! (1 (;3 (3: 1(3 16 2(1 236

a) Suggest intuiti'ely $hether a linear =uadratic or e/+onential function$ould -est fit the data, Moti'ate your res+onse,

-) Dra$ a scatter +lot to re+resent the data in the ta-le a-o'e,

c) Estimate the height of the +lant after 1( $eeks, %E/+lain $hether yourans$er is acce+ta-le)

3 4 com+any that rents out cars calculated the a'erage cost +er kilometre ofmaintaining a ne$ car for different distances co'ered during the first year, Thedata gathered is gi'en in the ta-le -elo$,

Annua- mi-!a! .in km ; 1( 1; 2( 2; "(C"'% p!r km .in ran# 2;( 26 22 2"3 2"1 22;

a) Dra$ a scatter diagram to re+resent the a-o'e data,

-) Decide $hich of the follo$ing gra+hs fit the a-o'e#mentioned data -est*straight line +ara-ola or e/+onential,

c) Estimate -y using a suita-le gra+h the a'erage cost +er kilometre ofo+erating a ne$ car if it is dri'en : ((( kilometres during the first year,

: The manager of a $omenOs clothing store $as curious a-out the amount of money$omen of 'arious ages s+ent monthly on clothing items, He o-tained thefollo$ing information from a re+resentati'e sam+le of $omen $ho regularly -uyfrom his store*

>"m!n' a!'.in ,!ar'

1: 21 2" 2; "( "2 "6 ": "? ;

Am"un% 'p!n%.in ran#

""( "(( "(( 2( 2;( 1?( 1:( "1( 1;( 12(

a) 0dentify and estimate any outliers in this data,

-) Dra$ a scatter#+lot to re+resent the a-o'e data on the diagram sheet,

"2

the inter=uartile range

c) the num-er of contestants that o'erestimated the +laying timeof the +iece of music if in fact the music lasted 13 ; minutes,


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c) Dra$ the line of -est fit for the gi'en data on your gra+h,

d) Descri-e the general trend -et$een the age of the $omen and the amount ofmoney s+ent,

e) @se the scatter +lot to +redict the amount that a (#year#old $oman $ills+end,

? The traffic de+artment in'estigated $here it $ould -e most a++ro+riate to ininstall s+eed cameras, 4s +art of their in'estigation a sur'ey $as done of thedifferent s+eeds of 'ehicles on a stretch of a national road, The follo$ing ta-lesho$s the results of the sur'ey*

S2EED

.in kmB&

FREUENCY

.O$ )!&i+-!'

CUMULATIE

FREUENCY( Q 6( ?6( Q :( ?2:( Q 1(( 1"1(( Q 12( 1;:12( Q 1( ?1( Q 16( 1316( Q 1:( 1

a) Ho$ many 'ehicles $ere o-ser'ed in the sur'ey8

-) Com+lete the cumulati'e fre=uency column using the ta-le on the diagramsheet,

c) 9e+resent the information in the ta-le -y dra$ing an ogi'e %cumulati'efre=uency

cur'e) on the grid +ro'ided on the diagram sheet,

d) @se your gra+h to determine the median s+eed, 0ndicate on your gra+h

using the letter T $here you $ould read off your ans$er,

1( The marks of 1; learners in a test for $hich the ma/imum mark is 1(( $ere*

3( < " < ?2 < 3 < ?: < ;6 < ; < 61 < " < ;1 < "? < ?2 < " < 32 < :(

""


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Calculate the follo$ing for this data*

a) the mean

-) the standard de'iation,

"


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LINEAR

2ROGRAMMING

";


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EXERCISES

u!'%i"n 1

@se the 'aria-lesxandyand $rite do$n the e=uation or ine=uality that re+resents eachof the follo$ing situations, State clearly in each case $hatxandyre+resent,

1,1 >hen mi/ing green +aint the amount of yello$ +igment must -e at least dou-lethe amount

of -lue +igment,

1,2 The num-er of hours s+ent on Maths and English home$ork each day should note/ceed t$o,

1," The sum of t$o num-ers must -e greater than 12,

1, The +rofit made on selling an ice cream must -e one#and#a#half times the +rofitmade on

selling a chocolate,

1,; 0n a hotel the rooms can accommodate either t$o +eo+le or three +eo+le, Thetotal num-er

of guests in the hotel must -e less than 2;(,

1,6 M is the sum of t$o num-ers,

1,3 M is the sum of one num-er and dou-le another num-er,

u!'%i"n 6

9e+resent each of the follo$ing systems of ine=ualities gra+hically*

u!'%i"n =

9e+resent each of the follo$ing systems of gra+hs sym-olically in terms of xandy*

"6

f%/)B2((

f%/)B2;

f%/)B#/J"((

/B;(< (QB/Q"((

/B2;(

/JyQ"((


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0n a certain $eek a radio manufacturer makes t$o ty+es of +orta-le radios M%mains)and B%-attery), et x -e the num-er of ty+e M and y -e the num-er of ty+e B, 0n thesketch the shaded area re+resents the feasi-le region,

;,2 0f the +rofit on a ty+e M is 9( and on ty+e B is 9:( $rite do$n the e=uation interms of x and y $hich $ill re+resent the +rofit %!),

;," Dra$ the search line that re+resents the +rofit function on the diagram sheet+ro'ided,

;, @se the gra+h to determine the +air )hat is the ma/imum $eekly +rofit8

;,6 The manager is informed that the $orkersO union +lans a strike for the follo$ing $eek

$hich $ill result in only ;( hours -eing $orked, Ho$ many radios of each ty+eshould no$ -e manufactured for ma/imum +rofit and $hat $ill the ma/imum +rofitno$ -e for the $eek8

":

120

(

160

0 4

00

8

0

1

20

1

60

| | | |

x

80

y

5.1 Write down the constraints to the linearprograing proble! gi"en#

5.1.1 $t ost 60 o% t&pe and 100 o% t&pe canan%actred in a wee*.

5.1.2 $t least 80 radios in total st be prodceda

wee* to co"er costs.

5.1.+ ,t ta*es "2 hor to asseble a t&pe and

horto asseble a t&pe . -he %actor& wor*s aai o% 60 hors er wee*.


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u!'%i"n 4 toy factory makes t$o different ty+es of $ooden toys coloured -locks %x) and mo-iles%y), 4ccording to its $orker contracts the factory guarantees each de+artment aminimum amount of $ork +er day,

The cutting de+artment must ha'e at least :( minutes of $ork +er day, 4 set of coloured-locks takes 2( minutes to cut and a mo-ile takes 1( minutes to cut,

The decorating de+artment must ha'e at least 6(( minutes of $ork +er day, 4 mo-iletakes 2( minutes to decorate and a set of $ooden -locks take 1( minutes to decorate,

The assem-ly de+artment must ha'e at least 1(:( minutes of $ork +er day, 4 set ofcoloured -locks takes 1( minutes to assem-ly and a mo-ile takes 6( minutes to assem+le,

6,1 0fxre+resents the num-er of set of -locks and yre+resents the num-er ofmo-iles +resent the information gi'en a-o'e as a set of ine=ualities,

6,2 &n gra+h +a+er gra+h the set of ine=ualities and indicate the feasi-le region,

6," 0f materials for a set of -locks cost 91; and the materials for a mo-ile cost 9;set u+ an e=uation $hich re+resents the cost %C) of +roducing +aint,

6, !lot the cost e=uation on your gra+h and use it to determine $hich 'erte/ $illresult in the most effecti'e +roduction schedule that minimises costs,

6,; .ind the co#ordinates of this 'erte/ -y sol'ing the a++ro+riate t$o -oundary lines,

u!'%i"n 4n organic farmer gro$s la'ender and 'egeta-les, He has a +lot of land $ith an area of ?hectares, 0n order to fulfil his market contracts the farmer has to +lant at least 2 hectaresof la'ender and 1 hectare of 'egeta-les, 0n order to minimise the damage done -yinsects the area under la'ender should -e at least e=ual to the area under 'egeta-les, 0norder to maintain the nitrogen -alance of the soil the area under la'ender should not -emore than dou-le the area under 'egeta-les,

3,1 Belo$ is a system of e=uations that models the scenario descri-ed, 0fxis thehectares under la'ender andyis the hectares under 'egeta-les e/tract from thete/t the $ords that ha'e resulted in the constraint -eing modelled %i,e, $rite do$nthe $ords from the te/t that -elong to each constraint),

3,1,1 yx 3,1,2 2x 3,1," yx 2

3,1, ?+ yx 3,1,; 1y

3,2 @sing gra+h +a+er dra$ the system of e=uations and shade the feasi-le region,

"?


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3," Set u+ a !rofit e=uation if the +rofit on a hectare of la'ender is 91( ((( and the+rofit on a hectare of 'egeta-les is 9: (((,

3, Gra+h the +rofit e=uation and use it to assist you in determining accurately the

+lanting schedule that $ill result in the farmer ma/imiKing +rofit,3,; 0f the +rofit margins changed so that the +rofit on 'egeta-les and la'ender $ere

e=ual $ould it -e necessary for the farmer to change his +lanting schedule8E/+lain your ans$er,

u!'%i"n

4 certain motorcycle manufacturer +roduces t$o -asic models the Su+er L and theSu+er A, These motorcycles are sold to dealers at a +rofit of 92( ((( +er Su+er Land 91( ((( +er Su+er A,

4 Su+er L re=uires 1;( hours for assem-ly ;( hours for +ainting and finishing and 1(hours for checking and testing, The Su+er A re=uires 6( hours for assem-ly ( hoursfor +ainting and finishing and 2( hours for checking and testing, The total num-er ofhours a'aila-le +er month is*"( ((( in the assem-ly de+artment 1" ((( in the +ainting and finishing de+artmentand ; ((( in the checking and testing de+artment,

The a-o'e information can -e summarised -y the follo$ing ta-le*

et x -e the num-er of Su+er L and y -e the num-er of Su+er A modelsmanufactured +er month,

:,1 >rite do$n the set of constraint ine=ualities,

:,2 @se the gra+h +a+er +ro'ided to re+resent the constraint ine=ualities,

:," Shade the feasi-le region on the gra+h +a+er,

:, >rite do$n the +rofit generated in terms of x and y,

:,; Ho$ many motorcycles of each model must -e +roduced in order to ma/imiKethe monthly +rofit8

:,6 >hat is the ma/imum monthly +rofit8

(

D!par%m!n% H"ur' $"r JSup!r XK H"ur' $"r JSup!r YK Ma5imum &"ur'

a)ai-a(-! p!r m"n%&4ssem-ly 1;( 6( "( (((!ainting and finishing ;( ( 1" (((Checking and testing 1( 2( ; (((


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"

1hour of engine $ork and

;

1hour of -ody$ork is re=uired to +roduce one

ShosholoKa,

The ratio of ShosholoKa mini-uses to Puadrant mini-uses +roduced +er $eek must-e at least "*2,

4 minimum of 2(( Puadrant mini-uses must -e +roduced +er $eek,

et the num-er of Puadrant mini -uses manufactured in a $eek -e x.et the num-er of ShosholoKa mini-uses manufactured in a $eek -e y,

1(,1 >rite do$n the constraints in terms of x and y to re+resent the a-o'e#mentionedinformation,

1(,2 @se gra+h +a+er to re+resent the constraints gra+hically,

1(," Clearly indicate the feasi-le region -y shading it,

1(, 0f the +rofit on one Puadrant mini-us is 912 ((( and the +rofit on one ShosholoKa mini-us i 9 ((( $rite do$n an e=uation that $ill re+resent the +rofit on the

mini-uses,

Documents

Maths 2010 Book 2 Teachers' Self Study Guide