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Strand : Number Substrand : NS4.4

Topic : ProbabilitySolves probability problems involving simple events

Language :

Knowledge and Skills Working Mathematically Learning Experiences egister

Students learn about

listing all possible outcomes o! a simple e"ent

using the term #sample space$ to denote all possible

outcomes eg !or tossing a !air die% the sample spaceis &% '% (% )% *% +

assigning probabilities to simple e"ents by

reasoning about e,ually likely outcomes eg theprobability o! a * resulting !rom the throw o! a !air

die is+

&

expressing the probability o! a particular outcome

as a !raction between - and &

assigning a probability o! .ero to e"ents that are

impossible and a probability o! one to e"ents that

are certain recognising that the sum o! the probabilities o! all

possible outcomes o! a simple e"ent is &

identi!ying the complement o! an e"ent

eg #The complement o! drawing a red card !rom adeck o! cards is drawing a black card/$

!inding the probability o! a complementary e"ent

Students learn to

sol"e simple probability problems arising in games

(Applying Strategies)

use language associated with chance e"ents

appropriately (Communicating)

e"aluate media statements in"ol"ing probability

(Applying Strategies, Communicating)

interpret and use probabilities expressed as

percentages or decimals(Applying Strategies,Communicating)

explain the meaning o! a probability o! -% '&

and

& in a gi"en situation (Communicating,Reasoning)

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Strand : Number Substrand : NS5.1.3

Topic : 1robability

Determines relative frequencies and teoretical probabilities

Language :



repeating an experiment a number o! times to

determine the relati"e !re,uency o! an e"ent

estimating the probability o! an e"ent !rom

experimental data using relati"e !re,uencies expressing the probability o! an e"entAgi"en a

!inite number o! e,ually likely outcomes as

23AP 4

n

outco!a"ourableo!number

where nis the total number o! outcomes in thesample space

using the !ormula to calculate probabilities !or

simple e"ents

simulating probability experiments using random

number generators

Students learn to

recognise and explain di!!erences between

relati"e !re,uency and theoretical probability ina simple experiment (Communicating,

Reasoning) apply relati"e !re,uency to predict !uture

experimental outcomes (Applying Strategies)

design a de"ice to produce a speci!ied relati"e

!re,uency eg a !our5coloured circular spinner(Applying Strategies)

recognise that probability estimates become

more stable as the number o! trials increases(Reasoning)

recognise randomness in chance situations

(Communicating)

apply the !ormula !or calculating probabilitiesto problems related to card% dice and othergames (Applying Strategies)

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Strand : Number Substrand : NS5.1.1

Topic : !ational Numbers

"pplies inde# la$s to simplify and evaluate aritmetic

e#pressions and uses scientific notation to $rite large and small

numbers

Language :



describing numbers written in index !orm using

terms such as base% power% index% exponent

e"aluating numbers expressed as powers o!

positi"e whole numbers

establishing the meaning o! the .ero index and

negati"e indices eg by patterns

'(

& -(

&( '(

6 ( &(

&6

&

writing reciprocals o! powers using negati"e

indices

eg0&

&

(

&(

)

)==

translating numbers to index !orm 3integralindices2 and "ice "ersa

de"eloping index laws arithmetically by

expressing each term in expanded !ormeg

+)')'((2((((32((3(( === +

('*'*

((((

((((((( ==

=

( ) ( ) ( ) ( ) ( ) ')' (((((((((( ==

using index laws to simplify expressions

Students learn to

sol"e numerical problems in"ol"ing indices


explain the incorrect use o! index laws

eg why +)' 6(( (Communicating,

Reasoning)

"eri!y the index laws by using a calculator

eg to compare the "alues o! ( ) '* %'

'&

*

and

* (Reasoning)

communicate and interpret technical

in!ormation using scienti!ic notation(Communicating)

explain the di!!erence between numerical

expressions such as )&-'

and )' %particularly with re!erence to calculatordisplays (Communicating, Reasoning)

sol"e problems in"ol"ing scienti!ic notation


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&-


using index laws to de!ine !ractional indices !or

s,uare and cube roots

eg ( ) 66'

= and 66'

'&

=

% hence

'

&

66 =

writing s,uare roots and cube roots in index

!orm

eg '00 ((&

=

recognising the need !or a notation to express

"ery large or "ery small numbers

expressing numbers in scienti!ic notation

entering and reading scienti!ic notation on a

calculator

using index laws to make order o! magnitude

checks !or numbers in scienti!ic notationeg

( ) ( ) &-+) '/&&-&'&-'/)&-&'/( con"erting numbers expressed in scienti!ic

notation to decimal !orm

ordering numbers expressed in scienti!ic

notation

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Strand : Number Substrand : NS5.1.%

Topic : &onsumer "ritmetic

Solves consumer aritmetic problems involving earning andspending money

Language :



calculating earnings !or "arious time periods

!rom di!!erent sources% including:5wage5salary5commission

5piecework5o"ertime5bonuses5holiday loadings5interest on in"estments

calculating income earned in casual and part5

time obs% considering agreed rates and specialrates !or Sundays and public holidays

calculating weekly% !ortnightly% monthly and

yearly incomes

calculating net earnings considering deductions

such as taxation and superannuation calculating simple interest using the !ormula

PRTI = where&--

rR=

whereIis the interest%Pthe principal% theannual interest rate and Tthe number o! years

Students learn to

read and interpret pay slips !rom part5time obs

when ,uestioning the details o! their ownemployment (Questioning, Communicating)

prepare a budget !or a gi"en income%

considering such expenses as rent% !ood%

transport etc(Applying Strategies2

interpret the di!!erent ways o! indicating wages

or salary in newspaper #positions "acant$ad"ertisements eg 8'-K (Communicating)

compare employment conditions !or di!!erent

careers where in!ormation is gathered !rom a"ariety o! mediums including the 9nterneteg employment rates% payment (ApplyingStrategies)

compare simple interest with compound interest

in practical situations eg loans (ApplyingStrategies)

interpret spreadsheets or tables when

comparing simple interest and compoundinterest on an in"estment o"er "arious timeperiods(Applying Strategies, Communicating)

realise the total cost andor hidden costs

in"ol"ed in some types o! purchasearrangements(Applying Strategies)

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&&

&'


applying the simple interest !ormula to problems

related to in"esting money at simple interestrates

calculating compound interest !or two or three

years by repeated multiplication using acalculator eg a rate o! *; per annum leads torepeated multiplication by &/-*

calculating compound interest on in"estments

using a table

calculating and comparing the cost o!

purchasing goods using:5cash5credit card5lay5by5de!erred payment5buying on terms5loans

calculating a #best buy$

make in!ormed decisions related to purchases

eg determining the best mobile phone plan !ora gi"en situation (Applying Strategies)

interpret the

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Strand : Patterns and "lgebra Substrand : P"S5.1.1

Topic : "lgebraic 'ecniques

"pplies te inde# la$s to simplify algebraic e#pressions

Language :



using the index laws pre"iously established !or

numbers to de"elop the index laws in algebraic!orm

eg *('(' '''' == + nmnm aaa +=

('*'*'''' ==

nmnmaaa

=

( ) +(' '' = mnnm aa =23

establishing that &- =a using the index laws

eg -(((( aaaa ==

and &(( =aa

&- =a

simpli!ying algebraic expressions that include

index notationeg

(*'(

)'+

*('

-

+'2(3'

)(&'

+('

0(*

mmmm

aaa

xxx

x

+=+

=

=

=+

Students learn to

"eri!y the index laws using a calculator

eg use a calculator to compare the "alues o!') 2(3 and 0( (Reasoning)

explain why &- =x (Applying Strategies,Reasoning, Communicating)

link use o! indices in =umber with use o!

indices in >lgebra (Reflecting)

explain why a particular algebraic sentence is

incorrect

eg explain why +'( aaa = is incorrect

(Communicating, Reasoning)

examine and discuss the di!!erence between

expressions such as

aa *( '

and aa *( '

+

by substituting "alues !or a(Reasoning, Applying Strategies,

Communicating)

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&)

Strand : Data Substrand : DS5.1.1

Topic : Data !epresentation and "nalysis

(roups data to aid analysis and constructs frequency and

cumulative frequency tables and graps

Language :



constructing a cumulati"e !re,uency table !or

ungrouped data

constructing a cumulati"e !re,uency histogram

and polygon 3ogi"e2 using a cumulati"e !re,uency polygon to !ind the

median

grouping data into class inter"als

constructing a !re,uency table !or grouped data

constructing a histogram !or grouped data

!inding the mean using the class centre

!inding the modal class

Students learn to

construct !re,uency tables and graphs !rom data

obtained !rom di!!erent sources 3eg the 9nternet2and discuss ethical issues that may arise !rom

the data (Applying Strategies, Communicating,Reflecting)

read and interpret in!ormation !rom a

cumulati"e !re,uency table or graph(Communicating)

compare the e!!ects o! di!!erent ways o!

grouping the same data (Reasoning)

use spreadsheets% databases% statistics packages%

or other technology% to analyse collected data%present graphical displays% and discuss ethicalissues that may arise !rom the data

(Applying Strategies, Communicating,Reflecting)

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Strand : )easurement Substrand : )S5.1.1

Topic : Perimeter and "rea

*ses formulae to calculate te area of quadrilaterals and finds

areas and perimeters of simple composite figures

Language :



de"eloping and using !ormulae to !ind the area o!

,uadrilaterals:5 !or a kite or rhombus%Area xy

'

&= where

x andy are the lengths o! the diagonals?

5 !or a trape.ium%Area 23'&

bah += where

h is the perpendicular height and a and b thelengths o! the parallel sides

calculating the area o! simple composite !igures

consisting o! two shapes including ,uadrants andsemicircles

calculating the perimeter o! simple composite!igures consisting o! two shapes including,uadrants and semicircles

Students learn to

identi!y the perpendicular height o! a trape.ium

in di!!erent orientations (Communicating) select and use the appropriate !ormula to

calculate the area o! a ,uadrilateral (ApplyingStrategies)

dissect composite shapes into simpler shapes


sol"e practical problems in"ol"ing area o!

,uadrilaterals and simple composite !igures(Applying Strategies)

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Strand : Patterns and "lgebra Substrand : P"S5.1.%

Topic : &oordinate (eometryDetermines te midpoint+ lengt and gradient of an interval

,oining t$o points on te number plane and graps linear and

simple non-linear relationsips from equations

Language :



i!point, "ength an! #ra!ient determining the midpoint o! an inter"al !rom a

diagram

graphing two points to !orm an inter"al on the

number plane and !orming a right5angled triangleby drawing a "ertical side !rom the higher pointand a hori.ontal side !rom the lower point

using the right5angled triangle drawn between

two points on the number plane and 1ythagoras$theorem to determine the length o! the inter"aloining the two points

using the right5angled triangle drawn between

two points on the number plane and therelationship

run

risegradient =

to !ind the gradient o! the inter"al oining twopoints

determining whether a line has a positi"e or

negati"e slope by !ollowing the line !rom le!t toright @ i! the line goes up it has a positi"e slopeand i! it goes down it has a negati"e slope

Students learn to

describe the meaning o! the midpoint o! an

inter"al and how it can be !ound(Communicating)

describe how the length o! an inter"al oining

two points can be calculated using 1ythagoras$theorem (Communicating, Reasoning)

explain the meaning o! gradient and how it can

be !ound !or a line oining two points(Communicating, Applying Strategies)

distinguish between positi"e and negati"e

gradients !rom a graph (Communicating)

relate the concept o! gradient to the tangent

ratio in trigonometry !or lines with positi"egradients (Reflecting)

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!inding the gradient o! a straight line !rom the

graph by drawing a right5angled triangle a!teroining two points on the line

#raphs of Relationships constructing tables o! "alues and using

coordinates to graph "ertical and hori.ontal linessuch as

(%'

&%(

==

==

yy

xx

identi!ying thex5 andy5intercepts o! graphs identi!ying thex 5axis as the liney4 $

identi!ying they5axis as the line x 4 $

graphing a "ariety o! linear relationships on the

number plane by constructing a table o! "aluesand plotting coordinates using an appropriatescale eg graph the !ollowing:

xy

yx

yx

xy

xy

(

'

'*

'

&

(

=

=

=+

+=

=

graphing simple non5linear relationships

eg

xy

xy

xy

'

''

'

=

=

=

determining whether a point lies on a line by

substituting into the e,uation o! the line

describe hori.ontal and "ertical lines in general

terms (Communicating)

explain why thex5axis has e,uationy 4$

(Reasoning, Communicating)

explain why they5axis has e,uationx4 $

(Reasoning, Communicating)

determine the di!!erence between e,uations o!

lines that ha"e a negati"e gradient and thosethat ha"e a positi"e gradient (Reasoning)

use a graphics calculator and spreadsheet

so!tware to graph% compare and describe arange o! linear and simple non5linearrelationships(Applying Strategies, Communicating)

apply ethical considerations when using

hardware and so!tware (Reflecting)

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Strand : )easurement Substrand : )S5.1.%

Topic : 'rigonometry

"pplies trigonometry to solve problems diagrams given/

including tose involving angles of elevation and depression

Language :



Trigonometric Ratios of Acute Angles

identi!ying the hypotenuse% adacent and opposite

sides with respect to a gi"en angle in a right5angled triangle in any orientation

labelling the side lengths o! a right5angled

triangle in relation to a gi"en angle eg the side cisopposite angle C

recognising that the ratio o! matching sides in

similar right5angled triangles is constant !or e,ualangles

de!ining the sine% cosine and tangent ratios !or

angles in right5angled triangles

using trigonometric notation eg sinA using a calculator to !ind approximations o! the

trigonometric ratios o! a gi"en angle measured indegrees

using a calculator to !ind an angle correct to the

nearest degree% gi"en one o! the trigonometricratios o! the angle

Students learn to

label sides o! right5angled triangles in di!!erent

orientations in relation to a gi"en angle(Applying Strategies, Communicating)

explain why the ratio o! matching sides in

similar right5angle triangles is constant !or e,ualangles (Communicating, Reasoning)

sol"e problems in practical situations in"ol"ing

right5angled triangles eg !inding the pitch o! aroo!(Applying Strategies)

interpret diagrams in ,uestions in"ol"ing angles

o! ele"ation and depression(Communicating)

relate the tangent ratio to gradient o! a line

(Reflecting)

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&6


Trigonometry of Right-Angled Triangles

selecting and using appropriate trigonometric

ratios in right5angled triangles to !ind unknownsides% including the hypotenuse

selecting and using appropriate trigonometric

ratios in right5angled triangles to !ind unknownangles correct to the nearest degree

identi!ying angles o! ele"ation and depression

sol"ing problems in"ol"ing angles o! ele"ation

and depression when gi"en a diagram

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