YEAR 8 MATHS FOCUS
NUMBER AND ALGEBRA
COMPUTATION WITH INTEGERS
OUTCOME
A student:
MA44NA:
compares, orders and calculates with integers, applying a range of strategies to aid computation
TEACHING POINTS  To divide two and threedigit numbers by a twodigit number, students may be taught the long division algorithm or, alternatively, to transform the division into a multiplication. So, 356÷52=□ becomes 52×□=356. Knowing that there are two fifties in each 100, students may try 7, obtaining 52 × 7 = 364, which is too large. They may then try 6, obtaining 52 × 6 = 312. The answer is 6 44/52=6 11/13 Students also need to be able to express a division in the following form in order to relate multiplication and division: 356 = 6 × 52 + 44, and then division by 52 gives 356/52=6+44/52=6 11/13 Students should have some understanding of integers, as the concept is introduced in Stage 3 Whole Numbers 2 (year 6). However, operations with integers are introduced in Stage 4. Complex recording formats for integers, such as raised signs, can be confusing. On printed materials, the endash ( – ) should be used to indicate a negative number and the operation of subtraction. The hyphen ( – ) should not be used in either context. The following formats are recommended: −2−3−7+(−4)−2−−3=−5=−7−4=−11=−2+3=1 Brahmagupta (c598–c665), an Indian mathematician and astronomer, is noted for the introduction of zero and negative numbers in arithmetic. 
LANGUAGE  Teachers should model and use a variety of expressions for mathematical operations and should draw students’ attention to the fact that the words used for subtraction and division questions may require the order of the numbers to be reversed when performing the operation. For example, ‘9 take away 3’ and ‘reduce 9 by 3’ require the operation to be performed with the numbers in the same order as they are presented in the question (ie 9 – 3), but ‘take 9 from 3’, ‘subtract 9 from 3’ and ‘9 less than 3’ require the operation to be performed with the numbers in the reverse order to that in which they are stated in the question (ie 3 – 9). Similarly, ‘divide 6 by 2’ and ‘6 divided by 2’ require the operation to be performed with the numbers in the same order as they are presented in the question (ie 6 ÷ 2), but ‘how many 2s in 6?’ requires the operation to be performed with the numbers in the reverse order to that in which they appear in the question (ie 6 ÷ 2). 
PURPOSE RELEVANCE OF SUBSTRAND  The positive integers (1, 2, 3, …) and 0 allow us to answer many questions involving ‘How many?’, ‘How much?’, ‘How far?’, etc, and so carry out a wide range of daily activities. The negative integers (…, –3, –2, –1) are used to represent ‘downwards’, ‘below’, ‘to the left’, etc, and appear in relation to everyday situations such as the weather (eg a temperature of –5° is 5° below zero), altitude (eg a location given as –20 m is 20 m below sea level), and sport (eg a golfer at –6 in a tournament is 6 under par). The Computation with Integers substrand includes the use of mental strategies, written strategies, etc to obtain answers – which are very often integers themselves – to questions or problems through addition, subtraction, multiplication and division. 
Expectations of Attainment
Apply the associative, commutative and distributive laws to aid mental and written computation (ACMNA151)  use an appropriate noncalculator method to divide two and threedigit numbers by a twodigit number 
– compare initial estimates with answers obtained by written methods and check by using a calculator {Problem Solving, Critical and creative thinking}  
show the connection between division and multiplication, including where there is a remainder, e.g. 451÷23=19 14/23 means that 451=19×23+14 {Critical and creative thinking}  
apply a practical understanding of commutativity to aid mental computation, e.g. 3 + 9 = 9 + 3 = 12, 3 × 9 = 9 × 3 = 27 {Critical and creative thinking}  
apply a practical understanding of associativity to aid mental computation, e.g. 3 + 8 + 2 = (3 + 8) + 2 = 3 + (8 + 2) = 13, 2 × 7 × 5 = (2 × 7) × 5 = 2 × (7 × 5) = 70 {Critical and creative thinking}  
– determine by example that associativity holds true for multiplication of three or more numbers but does not apply to calculations involving division, e.g. (80 ÷ 8) ÷ 2 is not equivalent to 80 ÷ (8 ÷ 2) {Communicating, Critical and creative thinking}  
apply a practical understanding of the distributive law to aid mental computation, e.g. to multiply any number by 13, first multiply by 10 and then add 3 times the number {Critical and creative thinking}  
use factors of a number to aid mental computation involving multiplication and division, e.g. to multiply a number by 12, first multiply the number by 6 and then multiply the result by 2 
Compare, order, add and subtract integers (ACMNA280)  recognise and describe the ‘direction’ and ‘magnitude’ of integers 
– construct a directed number sentence to represent a reallife situation {Communicating}  
recognise and place integers on a number line  
compare the relative value of integers, including recording the comparison by using the symbols < and >  
order integers  
interpret different meanings (direction or operation) for the + and – signs, depending on the context  
add and subtract integers using mental and written strategies  
– determine, by developing patterns or using a calculator, that subtracting a negative number is the same as adding a positive number {Reasoning, Critical and creative thinking}  
– apply integers to problems involving money and temperature {Problem Solving, Critical and creative thinking} 
Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (ACMNA183)  multiply and divide integers using mental and written strategies 
– investigate, by developing patterns or using a calculator, the rules associated with multiplying and dividing integers {Reasoning, Critical and creative thinking}  
use a calculator to perform the four operations with integers {Information and communication technology capability}  
– decide whether it is more appropriate to use mental strategies or a calculator when performing certain operations with integers {Communicating, Critical and creative thinking}  
use grouping symbols as an operator with integers  
apply the order of operations to mentally evaluate expressions involving integers, including where an operator is contained within the numerator or denominator of a fraction, e.g. (15+9)/6, (15+9)/(15−3), 5+(18−12)/6, 5+18/6−12, 5×(2−8)  
investigate whether different digital technologies, such as those found in computer software and on mobile devices, apply the order of operations {Problem Solving, Critical and creative thinking, Information and communication technology capability} 
FRACTIONS, DECIMALS & PERCENTAGES
OUTCOME
A student:
MA45NA:
operates with fractions, decimals and percentages
TEACHING POINTS  In Stage 3, the study of fractions is limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100 and calculations involve related denominators only. Students are unlikely to have had any experience with rounding to a given number of decimal places prior to Stage 4. The term ‘decimal place’ may need to be clarified. Students should be aware that rounding is a process of ‘approximating’ and that a rounded number is an ‘approximation’. All recurring decimals are nonterminating decimals, but not all nonterminating decimals are recurring. The earliest evidence of fractions can be traced to the Egyptian papyrus of the scribe Ahmes (about 1650 BC). In the seventh century AD, the method of writing fractions as we write them now was invented in India, but without the fraction bar (vinculum), which was introduced by the Arabs. Fractions were widely in use by the twelfth century. Onecent and twocent coins were withdrawn by the Australian Government in 1990. When an amount of money is calculated, it may have 1, 2, 3 or more decimal places, e.g. when buying petrol or making interest payments. When paying electronically, the final amount is paid correct to the nearest cent. When paying with cash, the final amount is rounded correct to the nearest five cents, eg $25.36, $25.37 round to $25.35 $25.38, $25.39, $25.41, $25.42 round to $25.40 $25.43, $25.44 round to $25.45. 
LANGUAGE  In questions that require calculating a fraction or percentage of a quantity, some students may benefit from first writing an expression using the word ‘of’, before replacing it with the multiplication sign (×). Students may need assistance with the subtleties of the English language when solving word problems. The different processes required by the words ‘to’ and ‘by’ in questions such as ‘find the percentage increase if $2 is increased to $3’ and ‘find the percentage increase if $2 is increased by $3’ should be made explicit. When solving word problems, students should be encouraged to write a few key words on the lefthand side of the equals sign to identify what is being found in each step of their working. The word ‘cent’ is derived from the Latin word centum, meaning ‘one hundred’. ‘Percent’ means ‘out of one hundred’ or ‘hundredths’. When expressing fractions in English, the numerator is said first, followed by the denominator. However, in many Asian languages (eg Chinese, Japanese), the opposite is the case: the denominator is said before the numerator. 
PURPOSE RELEVANCE OF SUBSTRAND  There are many everyday situations where things, amounts or quantities are ‘fractions’ or parts (or ‘portions’) of whole things, whole amounts or whole quantities. Fractions are very important when taking measurements, such as when buying goods (eg threequarters of a metre of cloth) or following a recipe (eg a third of a cup of sugar), when telling the time (eg a quarter past five), when receiving discounts while shopping (eg ‘half price’, ‘half off’), and when sharing a cake or pizza (eg ‘There are five of us, so we’ll get onefifth of the pizza each’). ‘Decimals’ and ‘percentages’ represent different ways of expressing fractions (and whole numbers), and so are other ways of representing a part of a whole. Fractions (and decimals and percentages) are of fundamental importance in calculation, allowing us to calculate with parts of wholes and to express answers that are not whole numbers, e.g. 4÷5=45 (or 0.8 or 80%). 
Expectations of Attainment
Compare fractions using equivalence; locate and represent positive and negative fractions and mixed numerals on a number line (ACMNA152)  determine the highest common factor (HCF) of numbers and the lowest common multiple (LCM) of numbers 
generate equivalent fractions  
write a fraction in its simplest form  
express improper fractions as mixed numerals and vice versa  
place positive and negative fractions, mixed numerals and decimals on a number line to compare their relative values  
– interpret a given scale to determine fractional values represented on a number line {Problem Solving}  
– choose an appropriate scale to display given fractional values on a number line, e.g. when plotting thirds or sixths, a scale of 3 cm for every whole is easier to use than a scale of 1 cm for every whole {Communicating, Reasoning} 
Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)  add and subtract fractions, including mixed numerals and fractions with unrelated denominators, using written and calculator methods 
– recognise and explain incorrect operations with fractions, e.g. explain why 2/3+1/4≠3/7 {Communicating, Reasoning, Literacy, Critical and creative thinking}  
– interpret fractions and mixed numerals on a calculator display {Communicating, Critical and creative thinking}  
subtract a fraction from a whole number using mental, written and calculator methods, e.g. 3−2/3=2+1−2/3=2 1/3 
Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)  determine the effect of multiplying or dividing by a number with magnitude less than one 
multiply and divide decimals by powers of 10  
multiply and divide decimals using written methods, limiting operators to two digits  
– compare initial estimates with answers obtained by written methods and check by using a calculator {Problem Solving, Critical and creative thinking}  
multiply and divide fractions and mixed numerals using written methods  
– demonstrate multiplication of a fraction by another fraction using a diagram to illustrate the process {Communicating, Reasoning, Literacy}  
– explain, using a numerical example, why division by a fraction is equivalent to multiplication by its reciprocal {Communicating, Reasoning, Literacy, Critical and creative thinking}  
multiply and divide fractions and decimals using a calculator  
calculate fractions and decimals of quantities using mental, written and calculator methods  
– choose the appropriate equivalent form for mental computation, e.g. 0.25 of $60 is equivalent to 1/4 of $60, which is equivalent to $60 ÷ 4 {Communicating, Critical and creative thinking} 
Express one quantity as a fraction of another, with and without the use of digital technologies (ACMNA155)  express one quantity as a fraction of another 
– choose appropriate units to compare two quantities as a fraction, e.g. 15 minutes is 15/60=1/4 of an hour {Communicating, Critical and creative thinking} 
Round decimals to a specified number of decimal places (ACMNA156)  round decimals to a given number of decimal places 
use symbols for approximation, e.g. ≑ or ≈ {Literacy} 
Investigate terminating and recurring decimals(ACMNA184)  use the notation for recurring (repeating) decimals, e.g. 0.33333…=0.3˙, 0.345345345…=0.3˙45˙, 0.266666…=0.26˙ {Literacy} 
convert fractions to terminating or recurring decimals as appropriate  
– recognise that calculators may show approximations to recurring decimals, and explain why, e.g. 23 displayed as 0.666666667 {Communicating, Reasoning, Critical and creative thinking} 
Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157)  classify fractions, terminating decimals, recurring decimals and percentages as ‘rational’ numbers, as they can be written in the form ab where a and b are integers and b≠0 {Literacy} 
convert fractions to decimals (terminating and recurring) and percentages  
convert terminating decimals to fractions and percentages  
convert percentages to fractions and decimals (terminating and recurring)  
– evaluate the reasonableness of statements in the media that quote fractions, decimals or percentages, e.g. ‘The number of children in the average family is 2.3’ {Communicating, Problem Solving, Critical and creative thinking}  
order fractions, decimals and percentages 
Investigate the concept of irrational numbers, including π (pie) (ACMNA186)  investigate ‘irrational’ numbers, such as π and √2 {Literacy Critical and creative thinking} 
– describe, informally, the properties of irrational numbers {Communicating, Literacy} 
Find percentages of quantities and express one quantity as a percentage of another, with and without the use of digital technologies (ACMNA158)  calculate percentages of quantities using mental, written and calculator methods 
choose an appropriate equivalent form for mental computation of percentages of quantities, e.g. 20% of $40 is equivalent to 1/5 × $40, which is equivalent to $40 ÷ 5 {Communicating, Critical and creative thinking}  
express one quantity as a percentage of another, using mental, written and calculator methods, e.g. 45 minutes is 75% of an hour 
Solve problems involving the use of percentages, including percentage increases and decreases, with and without the use of digital technologies (ACMNA187)  increase and decrease a quantity by a given percentage, using mental, written and calculator methods 
– recognise equivalences when calculating percentage increases and decreases, e.g. multiplication by 1.05 will increase a number or quantity by 5%, multiplication by 0.87 will decrease a number or quantity by 13% {Reasoning}  
interpret and calculate percentages greater than 100, e.g. an increase from $2 to $5 is an increase of 150%  
solve a variety of reallife problems involving percentages, including percentage composition problems and problems involving money  
– interpret calculator displays in formulating solutions to problems involving percentages by appropriately rounding decimals {Communicating, Information and communication technology capability}  
– use the unitary method to solve problems involving percentages, e.g. find the original value, given the value after an increase of 20% {Problem Solving}  
– interpret and use nutritional information panels on product packaging where percentages are involved {Problem Solving, Literacy}  
– interpret and use media and sport reports involving percentages {Problem Solving, Critical and creative thinking}  
– interpret and use statements about the environment involving percentages, e.g. energy use for different purposes, such as lighting {Problem Solving, Critical and creative thinking, Sustainability} 
FINANCIAL MATHEMATICS
OUTCOME
A student:
MA46NA:
solves financial problems involving purchasing goods
TEACHING POINTS  The Goods and Services Tax (GST) in Australia is a valueadded tax on the supply of goods and services. It was introduced by the Australian Government and took effect from 1 July 2000. Prior to the GST, Australia operated a wholesale sales tax implemented in the 1930s, when its economy was dominated by the production and sale of goods. In Australia, the GST is levied at a flat rate of 10% on most goods and services, apart from GSTexempt items (which include basic necessities such as milk and bread). 
LANGUAGE  GST stands for ‘Goods and Services Tax’. The difference between the GSTinclusive price, the preGST price, and the amount of the GST itself should be made explicit. When solving financial problems, students should be encouraged to write a few key words on the lefthand side of the equals sign to identify what is being found in each step of their working, and to conclude with a statement in words. Students’ understanding may be increased if they write calculations in words first, before substituting the appropriate values, e.g. percentage discount=discount/retail price×100% Students may need assistance with the subtleties of language used in relation to financial transactions, e.g. the difference between ‘$100 has been discounted by $10’ and ‘$100 has been discounted to $10’. 
PURPOSE RELEVANCE  Financial mathematics’ is used in important areas relating to an individual’s daily financial transactions, money management, and financial decision making. Such areas include earning and spending money (eg calculating ‘best buys’, discounts, GST, personal taxation, profit and loss, investing money, credit and borrowing, hire purchase, simple and compound interest, loan repayments, and depreciation. 
Expectations of Attainment
Select and apply efficient mental and written strategies, and appropriate digital technologies, to solve problems involving multiplication and division with whole numbers (ACMNA123)  select and use efficient mental and written strategies, and digital technologies, to multiply whole numbers of up to four digits by one and twodigit numbers 
select and use efficient mental and written strategies, and digital technologies, to divide whole numbers of up to four digits by a onedigit divisor, including where there is a remainder  
– estimate solutions to problems and check to justify solutions {Problem Solving, Reasoning, Critical and creative thinking}  
use mental strategies to multiply and divide numbers by 10, 100, 1000 and their multiples  
solve word problems involving multiplication and division, e.g. ‘A recipe requires 3 cups of flour for 10 people. How many cups of flour are required for 40 people?’ {Critical and creative thinking}  
– use appropriate language to compare quantities, e.g. ‘twice as much as’, ‘half as much as’ {Communicating, Critical and creative thinking}  
– use a table or similar organiser to record methods used to solve problems {Communicating, Problem Solving, Information and communication technology capability}  
recognise symbols used to record speed in kilometres per hour, e.g. 80 km/h {Literacy}  
solve simple problems involving speed, e.g. ‘How long would it take to travel 600 km if the average speed for the trip is 75 km/h?’ {Critical and creative thinking} 
Explore the use of brackets and the order of operations to write number sentences (ACMNA134)  use the term ‘operations’ to describe collectively the processes of addition, subtraction, multiplication and division 
investigate and establish the order of operations using reallife contexts, e.g. ‘I buy six goldfish costing $10 each and two water plants costing $4 each. What is the total cost?’; this can be represented by the number sentence 6 × 10 + 2 × 4 but, to obtain the total cost, multiplication must be performed before addition {Literacy, Critical and creative thinking, Work and enterprise}  
– write number sentences to represent reallife situations {Communicating, Problem Solving, Literacy}  
recognise that the grouping symbols ( ) and [ ] are used in number sentences to indicate operations that must be performed first {Literacy}  
recognise that if more than one pair of grouping symbols are used, the operation within the innermost grouping symbols is performed first  
perform calculations involving grouping symbols without the use of digital technologies, (2+3)×(16−9)=5×7 3+[20÷(9−5)]=3+[20÷4]=3+5  
apply the order of operations to perform calculations involving mixed operations and grouping symbols, without the use of digital technologies, e.g. 32÷2×4=16×4 32÷(2×4)=32÷8 (32+2)×4=34×4 32+2×4=32+8 {Work and enterprise}  
– investigate whether different digital technologies apply the order of operations {Reasoning, Information and communication technology capability, Critical and creative thinking}  
recognise when grouping symbols are not necessary, eg 32 + (2 × 4) has the same answer as 32 + 2 × 4 
RATIOS & RATES
OUTCOME
A student:
MA47NA:
operates with ratios and rates, and explores their graphical representation
TEACHING POINTS  In Stage 3 Fractions and Decimals, students study fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100. A unit fraction is any proper fraction in which the numerator is 1, eg 12, 13, 14, 15, … 
The process of writing a fraction in its ‘simplest form’ involves reducing the fraction to its lowest equivalent form. In Stage 4, this is referred to as ‘simplifying’ a fraction.  
When subtracting mixed numerals, working with the wholenumber parts separately from the fractional parts can lead to difficulties, particularly where the subtraction of the fractional parts results in a negative value, e.g. in the calculation of 2 1/3−1 5/6, 1/3−5/6 results in a negative value. 
LANGUAGE  Students should be able to communicate using the following language: whole, equal parts, half, quarter, eighth, third, sixth, twelfth, fifth, tenth, hundredth, thousandth, fraction, numerator, denominator, mixed numeral, whole number, number line, proper fraction, improper fraction, is equal to, equivalent, ascending order, descending order, simplest form, decimal, decimal point, digit, round to, decimal places, dollars, cents, best buy, percent, percentage, discount, sale price. 
The decimal 1.12 is read as ‘one point one two’ and not ‘one point twelve’.  
The word ‘cent’ is derived from the Latin word centum, meaning ‘one hundred’. ‘Percent’ means ‘out of one hundred’ or ‘hundredths’.  
A ‘terminating’ decimal has a finite number of decimal places, eg 3.25 (2 decimal places), 18.421 (3 decimal places). 
Expectations of Attainment
Recognise and solve problems involving simple ratios (ACMNA173)  use ratios to compare quantities measured in the same units 
write ratios using the : symbol, e.g. 4:7 {Literacy}  
– express one part of a ratio as a fraction of the whole, e.g. in the ratio 4:7, the first part is 4/11 of the whole {Communicating}  
simplify ratios, e.g. 4:6=2:3, 12:2=1:4, 0.3:1=3:10  
apply the unitary method to ratio problems  
divide a quantity in a given ratio 
Solve a range of problems involving ratios and rates, with and without the use of digital technologies (ACMNA188)  interpret and calculate ratios that involve more than two numbers 
solve a variety of reallife problems involving ratios, e.g. scales on maps, mixes for fuels or concrete  
use rates to compare quantities measured in different units  
– distinguish between ratios, where the comparison is of quantities measured in the same units, and rates, where the comparison is of quantities measured in different units  
convert given information into a simplified rate, e.g. 150 kilometres travelled in 2 hours = 75 km/h  
– solve a variety of reallife problems involving rates, including problems involving rate of travel {speed, Critical and creative thinking} 
Investigate, interpret and analyse graphs from authentic data (ACMNA180)  interpret distance/time graphs (travel graphs) made up of straightline segments 
– write or tell a story that matches a given distance/time graph {Communicating, Literacy}  
– match a distance/time graph to a description of a particular journey and explain the reasons for the choice {Communicating, Reasoning, Literacy}  
– compare distance/time graphs of the same situation, decide which one is the most appropriate, and explain why {Communicating, Reasoning, Literacy, Critical and creative thinking}  
recognise concepts such as change of speed and direction in distance/time graphs  
– describe the meaning of straightline segments with different gradients in the graph of a particular journey {Communicating}  
– calculate speeds for straightline segments of given distance/time graphs {Problem Solving}  
recognise the significance of horizontal line segments in distance/time graphs  
determine which variable should be placed on the horizontal axis in distance/time graphs  
draw distance/time graphs made up of straightline segments  
sketch informal graphs to model familiar events, e.g. noise level during a lesson  
– record the distance of a moving object from a fixed point at equal time intervals and draw a graph to represent the situation, e.g. move along a measuring tape for 30 seconds using a variety of activities that involve a constant rate, such as walking forwards or backwards slowly, and walking or stopping for 10second increments {Problem Solving}  
use the relative positions of two points on a line graph, rather than a detailed scale, to interpret information 
ALGEBRAIC TECHNIQUES (Pt 2)
OUTCOME
A student:
MA48NA:
generalises number properties to operate with algebraic expressions
TEACHING POINTS  When evaluating expressions, there should be an explicit direction to replace the pronumeral with the number to ensure that full understanding of notation occurs. 
LANGUAGE  The meaning of the imperatives ‘expand’, ‘remove the grouping symbols’ and ‘factorise’ and the expressions ‘the expansion of’ and ‘the factorisation of’ should be made explicit to students. 
Expectations of Attainment
Create algebraic expressions and evaluate them by substituting a given value for each variable (ACMNA176)  substitute into algebraic expressions and evaluate the result calculate and compare the values of x^2 for values of x with the same magnitude but opposite sign (Reasoning) 
generate a number pattern from an algebraic expression, eg 
Extend and apply the distributive law to the expansion of algebraic expressions (ACMNA190)  expand algebraic expressions by removing grouping symbols, eg
−5(x+2)=−5x−10 a(a+b)=a2+ab

Factorise algebraic expressions by identifying numerical factors (ACMNA191)  factorise a single algebraic term, eg 6ab=3×2×a×b 
factorise algebraic expressions by finding a common numerical factor, eg
−4t−12=−4(t+3) check expansions and factorisations by performing the reverse process (Reasoning) 
Factorise algebraic expressions by identifying algebraic factors  factorise algebraic expressions by finding a common algebraic factor, eg x2−5x = x(x−5) 5ab+10a=5a(b+2) 
INDICES
OUTCOME
A student:
MA49NA:
operates with positiveinteger and zero indices of numerical bases
TEACHING POINTS  Students have not used indices prior to Stage 4 and so the meaning and use of index notation will need to be made explicit. However, students should have some experience from Stage 3 in multiplying more than two numbers together at the same time. In Stage 3, students used the notion of factorising a number as a mental strategy for multiplication. Teachers may like to make an explicit link to this in the introduction of the prime factorisation of a number in Stage 4, e.g. in Stage 3, 18×5 The square root sign signifies a positive number (or zero). So, √9=3 (only). However, the two numbers whose square is 9 are √9 and −√9, i.e. 3 and –3. 
LANGUAGE  Students need to be able to express the concept of divisibility in different ways, such as ’12 is divisible by 2′, ‘2 divides (evenly) into 12’, ‘2 goes into 12 (evenly)’. A ‘product of prime factors’ can also be referred to as a ‘product of primes’. Students are introduced to indices in Stage 4. The different expressions used when referring to indices should be modelled by teachers. Teachers should use fuller expressions before shortening them, e.g. 2^4 should be expressed as ‘2 raised to the power of 4’, before ‘2 to the power of 4’ and finally ‘2 to the 4’. Students are expected to use the words ‘squared’ and ‘cubed’ when saying expressions containing indices of 2 and 3, respectively, e.g. 4^2 is ‘four squared’, 4^3 is ‘four cubed’. Words such as ‘product’, ‘prime’, ‘power’, ‘base’ and ‘index’ have different meanings outside of mathematics. Words such as ‘base’, ‘square’ and ‘cube’ also have different meanings within mathematics, e.g. ‘the base of the triangle’ versus ‘the base of 3^2 
PURPOSE RELEVANCE  Indices are important in mathematics and in everyday situations. Among their most significant uses is that they allow us to write large and small numbers more simply, and to perform calculations with large and small numbers more easily. For example, without the use of indices, 21000 would be written as 2×2×2×2…, until ‘2’ appeared exactly 1000 times. 
Expectations of Attainment
Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149)  describe numbers written in ‘index form’ using terms such as ‘base’, ‘power’, ‘index’, ‘exponent’ {Literacy} 
use index notation to express powers of numbers (positive indices only), e.g. 8=23 {Literacy}  
evaluate numbers expressed as powers of integers, e.g. 2^3=8, (−2)^3=−8  
– investigate and generalise the effect of raising a negative number to an odd or even power on the sign of the result {Communicating, Critical and creative thinking}  
apply the order of operations to evaluate expressions involving indices, with and without using a calculator, e.g. 3^2+4^2, 4^3+2×5^2 {Information and communication technology capability}  
determine and apply tests of divisibility for 2, 3, 4, 5, 6 and 10 {Critical and creative thinking}  
– verify the various tests of divisibility using a calculator {Problem Solving, Information and communication technology capability}  
– apply tests of divisibility mentally as an aid to calculation {Problem Solving, Critical and creative thinking}  
express a number as a product of its prime factors, using index notation where appropriate  
– recognise that if a given number is divisible by a composite number, then it is also divisible by the factors of that number, e.g. since 660 is divisible by 6, then 660 is also divisible by factors of 6, which are 2 and 3 {Reasoning, Critical and creative thinking}  
– find the highest common factor of large numbers by first expressing the numbers as products of prime factors {Communicating, Problem Solving, Critical and creative thinking} 
Investigate and use square roots of perfect square numbers (ACMNA150)  use the notations for square root and cube root {Literacy} 
recognise the link between squares and square roots and between cubes and cube roots, e.g. 2^3=8 and ^3√8=2 {Critical and creative thinking}  
determine through numerical examples that:  
express a number as a product of its prime factors to determine whether its square root and/or cube root is an integer  
find square roots and cube roots of any nonsquare whole number using a calculator, after first estimating {Information and communication technology capability}  
– determine the two integers between which the square root of a nonsquare whole number lies {Reasoning, Critical and creative thinking}  
apply the order of operations to evaluate expressions involving square roots and cube roots, with and without using a calculator, e.g. Figure 2 {Information and communication technology capability}  
explain the difference between pairs of numerical expressions that appear similar, e.g. ‘Is √36+√64 equivalent to √36+64 ?’ 
Use index notation with numbers to establish the index laws with positiveinteger indices and the zero index (ACMNA182)  develop index laws with positiveinteger indices and numerical bases by expressing each term in expanded form, e.g. Figure 3 {Literacy} 
– verify the index laws using a calculator, e.g. use a calculator to compare the values of (3^4)^2 and 3^8 {Reasoning}  
– explain the incorrect use of index laws, e.g. explain why 3^2×3^4≠9^6 {Communicating, Reasoning, Literacy, Critical and creative thinking}  
establish the meaning of the zero index, e.g. by patterns {Critical and creative thinking}  
– verify the zero index law using a calculator {Reasoning, Information and communication technology capability}  
use index laws to simplify expressions with numerical bases, e.g. 5^2×5^4×5=5^7 
EQUATIONS
OUTCOME
A student:
MA410NA:
operates with positiveinteger and zero indices of numerical bases
LANGUAGE  Describing the steps in the solution of equations provides students with the opportunity to practise using mathematical imperatives in context, e.g. ‘add 5 to both sides’, ‘increase both sides by 5’, ‘subtract 3 from both sides’, ‘take 3 from both sides’, ‘decrease both sides by 3’, ‘reduce both sides by 3’, ‘multiply both sides by 2’, ‘divide both sides by 2’. 
PURPOSE RELEVANCE  An equation is a statement that two quantities or expressions are equal, usually through the use of numbers and/or symbols. Equations are used throughout mathematics and in our daily lives in obtaining solutions to problems of all levels of complexity. People are solving equations (usually mentally) when, for example, they are working out the right quantity of something to buy, or the right amount of an ingredient to use when adapting a recipe. 
Expectations of Attainment
Solve simple linear equations (ACMNA179)  distinguish between algebraic expressions where pronumerals are used as variables, and equations where pronumerals are used as unknowns {Critical and creative thinking} 
solve simple linear equations using concrete materials, such as the balance model or cups and counters, stressing the notion of performing the same operation on both sides of an equation  
solve linear equations that may have noninteger solutions, using algebraic techniques that involve up to two steps in the solution process, e.g.  
– compare and contrast strategies to solve a variety of linear equations {Communicating, Reasoning, Critical and creative thinking}  
– generate equations with a given solution, e.g. find equations that have the solution x=5 {Problem Solving} 
Solve linear equations using algebraic techniques and verify solutions by substitution (ACMNA194)  solve linear equations that may have noninteger solutions, using algebraic techniques that involve up to three steps in the solution process, e.g. Figure 7 
check solutions to equations by substituting {Critical and creative thinking} 
Solve simple quadratic equations  0 then there are two values of x that solve a simple quadratic equation of the form x^2=c”}”>determine that if c>0 then there are two values of x that solve a simple quadratic equation of the form x^2=c 
– explain why quadratic equations could be expected to have two solutions {Communicating, Reasoning, Critical and creative thinking}  
– recognise that x^2=c does not have a solution if c is a negative number {Communicating, Reasoning, Critical and creative thinking}  
solve simple quadratic equations of the form x^2=c, leaving answers in ‘exact form’ and as decimal approximations 
TEACHING POINTS  When describing number patterns algebraically, it is important that students develop an understanding of the use of pronumerals as algebraic symbols for numbers of objects rather than for the objects themselves. In Linear Relationships, the study of patterns focuses on those that are linear. However, teachers may include a few simple nonlinear patterns so that students realise that not all patterns are linear. The Cartesian plane (commonly referred to as the ‘number plane’) is named after the French philosopher and mathematician René Descartes (1596–1650), who was one of the first to develop analytical geometry on the number plane. On the Cartesian plane, the coordinates of a point refer to an ordered pair (x,y) describing the horizontal position x first, followed by the vertical position y. Students are introduced to the four quadrants of the Cartesian plane in Stage 3. However, they may not be familiar with the terms ‘Cartesian plane’, ‘xaxis’ and ‘yaxis’, as in Stage 3 these are generally referred to as the ‘number plane’, ‘horizontal axis’ and ‘vertical axis’, respectively. 
LANGUAGE  In Stage 3, students were introduced to patterns involving one operation and used the terms ‘position in the pattern’ and ‘value of the term’ when describing a rule for a pattern from a table of values, eg ‘The value of the term is three times the position in the pattern’. Students will need to become familiar with and be able to use new terms, including ‘coefficient’, ‘constant term’ and ‘intercept’. Students should be aware that ‘gradient’ may be referred to as ‘slope’ in some contexts. 
PURPOSE RELEVANCE  Linear relationships are very common in mathematics and science. The graph of two quantities that have a linear relationship is a straight line. A linear relationship may be a direct relationship or an inverse relationship. In a direct relationship, as one quantity increases, the other quantity also increases, or as one quantity decreases, the other quantity also decreases. In an inverse relationship, as one quantity increases, the other quantity decreases. Examples of linear relationships familiar in everyday life include the distance travelled and time taken, the conversion of one currency to another, the cost of printing involving an initial setup cost and a dollar rate per item, the cost of taxi fares involving a hiring charge and a dollar rate per kilometre, and the cost of catering involving a base amount for a set number of people plus a rate for each extra attendee. Coordinate geometry facilitates the exploration and interpretation of linear relationships. 
Given coordinates, plot points on the Cartesian plane, and find coordinates for a given point (ACMNA178)  plot and label points on the Cartesian plane, given coordinates, including those with coordinates that are not whole numbers 
identify and record the coordinates of given points on the Cartesian plane, including those with coordinates that are not whole numbers 
Describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates (ACMMG181)  use the notation ${P}^{\prime}$ “>P′ to name the ‘image’ resulting from a transformation of a point $P$ “>P on the Cartesian plane 
plot and determine the coordinates for ${P}^{\prime}$ “>P′ resulting from translating $P$ “>P one or more times  
plot and determine the coordinates for ${P}^{\prime}$
“>P′
resulting from reflecting $P$
“>P
in either the $x$
“>x
or $y$
“>
$y$axis


plot and determine the coordinates for ${P}^{\prime}$
“>P
${\prime}^{}$ resulting from rotating $P$
“>P
by a multiple of 90° about the origin

Plot linear relationships on the Cartesian plane, with and without the use of digital technologies (ACMNA193)  use objects to build a geometric pattern, record the results in a table of values, describe the pattern in words and algebraic symbols, and represent the relationship on a number grid, eg
check pattern descriptions by substituting further values (Reasoning)

recognise a given number pattern (including decreasing patterns), complete a table of values, describe the pattern in words and algebraic symbols, and represent the relationship on a number grid


use a rule generated from a pattern to calculate the corresponding value for a larger number  
form a table of values for a linear relationship by substituting a set of appropriate values for either of the pronumerals and graph the number pairs on the Cartesian plane, eg given y = 3x + 1, form a table of values using x = 0, 1 and 2 and then graph the number pairs on the Cartesian plane with an appropriate scale


extend the line joining a set of points on the Cartesian plane to show that there is an infinite number of ordered pairs that satisfy a given linear relationship


derive a rule for a set of points that has been graphed on the Cartesian plane  
graph more than one line on the same set of axes using digital technologies and compare the graphs to determine similarities and differences, eg parallel, pass through the same point
$\begin{array}{l}y=3x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=3x+2,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=3x2\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\\ y=x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=2x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=3x\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\\ y=x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=x\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}& \end{array}$ “>y=3x, y=3x+2, y=3x−2y=x, y=2x, y=3x $\begin{array}{l}y=3x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=3x+2,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=3x2\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\\ y=x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=2x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=3x\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\\ y=x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=x\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}& \end{array}$ “>y=−x, y=x(Reasoning)


use digital technologies to graph linear and simple nonlinear relationships, such as $y={x}^{2}$
“>y=x^{2}

Solve linear equations using graphical techniques  recognise that each point on the graph of a linear relationship represents a solution to a particular linear equation 
use graphs of linear relationships to solve a corresponding linear equation, with and without the use of digital technologies, eg use the graph of $\phantom{\rule{thinmathspace}{0ex}}y=2x+3\phantom{\rule{thinmathspace}{0ex}}$ “>y = 2x + 3 to find the solution of the equation $\phantom{\rule{thinmathspace}{0ex}}2x+3=11\phantom{\rule{thinmathspace}{0ex}}$ “>2x + 3 = 11  
graph two intersecting lines on the same set of axes and read off the point of intersection

MEASUREMENT AND GEOMETRY
LENGTH
OUTCOME
A student:
MA412MG:
 calculates the perimeters of plane shapes and the circumferences of circles
TEACHING POINTS  Students should develop a sense of the levels of accuracy that are appropriate to a particular situation, eg the length of a bridge may be measured in metres to estimate a quantity of paint needed, but would need to be measured much more accurately for engineering work. 
The number \Pi is known to be irrational (not a fraction) and also transcendental (not the solution of any polynomial equation with integer coefficients). In Stage 4, students only need to know that the digits in its decimal expansion do not repeat (all this means is that it is not a fraction) and in fact have no known pattern. 
RELEVANCE  This substrand focuses on the ‘perimeter’ (or length of the boundary) of shapes (including the ‘circumference’ of a circle). The ability to determine the perimeters of twodimensional shapes is of fundamental importance in many everyday situations, such as framing a picture, furnishing a room, fencing a garden or a yard, and measuring land for farming or building construction. 
LANGUAGE  Pi (π) is the Greek letter equivalent to ‘p’ and is the first letter of the Greek word perimetron, meaning ‘perimeter’. The symbol for pi was first used to represent the ratio of the circumference to the diameter of a circle in the early eighteenth century. The names for some parts of the circle (centre, radius, diameter, circumference, sector, semicircle and quadrant) are introduced in Stage 3. The terms ‘arc’, ‘tangent’, ‘chord’ and ‘segment’ are introduced in Stage 4. Some students may find the use of the terms ‘length/long’, ‘breadth/broad’, ‘width/wide’ and ‘height/high’ difficult. Teachers should model the use of these terms in sentences, both verbally and in written form, when describing diagrams. Students should be encouraged to speak about, listen to, read about and write about the dimensions of given shapes using various combinations of these words, eg ‘The length of this rectangle is 7 metres and the width is 4 metres’, ‘The rectangle is 7 metres long and 4 metres wide’. Students may also benefit from drawing and labelling a shape, given a description of its features in words, eg ‘The base of an isosceles triangle is 6 metres long and its perimeter is 20 metres. Draw the triangle and mark on it the lengths of the three sides’. In Stage 3, students were introduced to the term ‘dimensions’ to describe the length and width of a rectangle. However, some students may need to be reminded of this. 
Expectations of Attainment
Find perimeters of parallelograms, trapeziums, rhombuses and kites (ACMMG196)  find the perimeters of a range of plane shapes, including parallelograms, trapeziums, rhombuses, kites and simple composite figures compare perimeters of rectangles with the same area (Problem Solving) 
solve problems involving the perimeters of plane shapes, eg find the dimensions of a rectangle, given its perimeter and the length of one side 
Investigate the concept of irrational numbers, including π (ACMNA186)  demonstrate by practical means that the ratio of the circumference to the diameter of a circle is constant, eg measure and compare the diameters and circumferences of various cylinders or use dynamic geometry software to measure circumferences and diameters 
define the number π as the ratio of the circumference to the diameter of any circleLiteracy

Investigate the relationship between features of circles, such as the circumference, radius and diameter; use formulas to solve problems involving circumference (ACMMG197)  identify and name parts of a circle and related lines, including arc, tangent, chord, sector and segment 
develop and use the formulas to find the circumferences of circles in terms of the diameter d or radius r:
 
find the perimeters of quadrants and semicircles  
find the perimeters of simple composite figures consisting of two shapes, including quadrants and semicircles  
find arc lengths and the perimeters of sectors  
solve a variety of practical problems involving circles and parts of circles, giving an exact answer in terms of π and an approximate answer using a calculator’s approximation for π 
AREA
OUTCOME
A student:
MA413MG:
 uses formulas to calculate the areas of quadrilaterals and circles, and converts between units of area
TEACHING POINTS  The area formulas for the triangle, the special quadrilaterals and the circle should be developed by practical means and/or by the use of dynamic geometry software, such as prepared applets. 
The area formulas for the triangle and the parallelogram should be related to the area of a rectangle. Applets may be particularly useful in demonstrating that the respective formulas hold for all triangles and parallelograms, including those for which the perpendicular height needs to be drawn outside the shape.  
The area formula for the rhombus or kite depends upon the fact that the diagonals are perpendicular, and so is linked with the geometry of special quadrilaterals. The formula applies to any quadrilateral in which the diagonals are perpendicular. Students should also be aware that because the rhombus is a special type of parallelogram, the area can be found using the formula A=bh.  
The area formula for the trapezium can be developed using various dissections and techniques. Students need to be able to apply the area formula for the trapezium appropriately to trapeziums in any form or orientation.  
The area formula for the circle may be established by using one or both of the following dissections: cut a circle into a large number of sectors and arrange the sectors alternately pointup and pointdown to form a rectangle with height r and base length πr; inscribe a number of congruent triangles in a circle, all with corresponding vertex at the centre, and show that the area of the inscribed polygon is half the length of the perimeter times the perpendicular height of the triangles.  
Students should be made aware that the perpendicular height of a triangle is the shortest distance from the base to the opposite angle. They may also need to be explicitly taught that the shortest distance between the parallel sides of a quadrilateral is the perpendicular distance between these sides.  
Finding the areas of rectangles and squares with integer side lengths is an important link between geometry and multiplying, dividing, factoring and squaring. Expressing a number as the product of two of its factors is equivalent to forming a rectangle with those factors as the side lengths, and (where possible) expressing a number as the square of one of its factors is equivalent to forming a square with that factor as the side length.  
Graphing the relationship between the length of a rectangle with a constant perimeter and possible areas of the rectangle links to nonlinear graphs. 
LANGUAGE  Teachers should reinforce with students the use of the term ‘perpendicular height’, rather than simply ‘height’, when referring to this attribute of a triangle. Students should also benefit from drawing and labelling a triangle when given a description of its features in words. Students may improve their understanding and retention of the area formulas by expressing them in different ways, eg ‘The area of a trapezium is half the perpendicular height multiplied by the sum of the lengths of the parallel sides’, ‘The area of a trapezium is half the product of the perpendicular height and the sum of the lengths of the parallel sides’. The use of the term ‘respectively’ in measurement word problems should be modelled and the importance of the order of the words explained, eg in the sentence ‘The perpendicular height and base of a triangle are 5 metres and 8 metres, respectively’, the first attribute (perpendicular height) mentioned refers to the first measurement (5 metres), and so on. The abbreviation m^{2} is read as ‘square metre(s)’ and not ‘metre(s) squared’ or ‘metre(s) square’. Similarly, the abbreviation cm^{2} is read as ‘square centimetre(s)’ and not ‘centimetre(s) squared’ or ‘centimetre(s) square’. When units are not provided in an area question, students should record the area in ‘square units’. 
EXPECTATIONS OF ATTAINMENT
Choose appropriate units of measurement for area and convert from one unit to another  choose an appropriate unit to measure the areas of different shapes and surfaces, eg floor space, fieldsCritical and creative thinking

convert between metric units of area using 1 cm^{2} = 100 mm^{2}, 1 m^{2} = 1 000 000 mm^{2}, 1 ha = 10 000 m^{2}, 1 km^{2} = 1 000 000 m^{2} = 100 ha 
Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving  develop and use the formulas to find the areas of rectangles and squares: Area of rectangle=lb where l is the length and b is the breadth (or width) of the rectangle

find the areas of simple composite figures that may be dissected into rectangles, squares, parallelograms and triangles 
Find areas of trapeziums, rhombuses and kites
 develop, with or without the use of digital technologies, and use the formula to find the areas of kites and rhombuses: 
develop and use the formula to find the areas of trapeziums:
 
select and use the appropriate formula to find the area of any of the special quadrilaterals  
solve a variety of practical problems relating to the areas of triangles and quadrilaterals

Investigate the relationship between features of circles, such as the area and the radius; use formulas to solve problems involving area
 develop, with or without the use of digital technologies, and use the formula to find the areas of circles: Area of circle=πr^{2} where r is the length of the radius

find the areas of quadrants, semicircles and sectors  
solve a variety of practical problems involving circles and parts of circles, giving an exact answer in terms of π and an approximate answer using a calculator’s approximation for π 
VOLUME
OUTCOME
A student:
MA414MG:
uses formulas to calculate the volumes of prisms and cylinders, and converts between units of volume
TEACHING POINTS  When developing the volume formula for a prism, students require an understanding of the idea of a uniform crosssection and should visualise, for example, stacking unit cubes, layer by layer, into a rectangular prism, or stacking planks into a pile. In the formula for the volume of a prism, V=Ah, A refers to the ‘area of the base’, which can also be referred to as the ‘area of the uniform crosssection’. 
‘Oblique’ prisms, cylinders, pyramids and cones are those that are not ‘right’ prisms, cylinders, pyramids and cones, respectively. The focus here is on right prisms and cylinders, although the formulas for volume also apply to oblique prisms and cylinders provided that the perpendicular height is used. In a right prism, the base and top are perpendicular to the other faces. In a right pyramid or cone, the base has a centre of rotation, and the interval joining that centre to the apex is perpendicular to the base (and therefore is its axis of rotation).  
The volumes of rectangular prisms and cubes are linked with multiplication, division, powers and factorisation. Expressing a number as the product of three of its factors is equivalent to forming a rectangular prism with those factors as the side lengths, and (where possible) expressing a number as the cube of one of its factors is equivalent to forming a cube with that factor as the side length.  
The abbreviation for megalitres is ML. Students will need to be careful not to confuse this with the abbreviation mL used for millilitres. 
RELEVANCE  The ability to determine the volumes of threedimensional objects and the capacities of containers, and to solve related problems, is of fundamental importance in many everyday activities, such as calculating the number of cubic metres of concrete, soil, sand, gravel, mulch or other materials needed for building or gardening projects; the amount of soil that needs to be removed for the installation of a swimming pool; and the appropriate size in litres of water tanks and swimming pools. 
Knowledge and understanding with regard to determining the volumes of simple threedimensional objects (including containers) such as cubes, other rectangular prisms, triangular prisms, cylinders, pyramids, cones and spheres can be readily applied to determining the volumes and capacities of composite objects (including containers). 
LANGUAGE  The word ‘base’ may cause confusion for some students. The ‘base’ in relation to twodimensional shapes is linear, whereas in relation to threedimensional objects, ‘base’ refers to a surface. In everyday language, the word ‘base’ is used to refer to that part of an object on, or closest to, the ground. In the mathematics of threedimensional objects, the term ‘base’ is used to describe the face by which a prism or pyramid is named, even though it may not be the face on, or closest to, the ground. In Stage 3, students were introduced to the naming of a prism or pyramid according to the shape of its base. In Stage 4, students should be encouraged to make the connection that the name of a particular prism refers not only to the shape of its base, but also to the shape of its uniform crosssection. Students should be aware that a cube is a special prism that has six congruent faces. 
The abbreviation m^{3} is read as ‘cubic metre(s)’ and not ‘metre(s) cubed’ or ‘metre(s) cube’. When units are not provided in a volume question, students should record the volume in ‘cubic units’. 
EXPECTATIONS OF ATTAINMENT
Draw different views of prisms and solids formed from combinations of prisms (ACMMG161)  draw (in two dimensions) prisms, and solids formed from combinations of prisms, from different views, including top, side, front and back views 
identify and draw the crosssections of different prismsLiteracy Critical and creative thinking
 
visualise, construct and draw various prisms from a given crosssectional diagram  
determine if a particular solid has a uniform crosssectionCritical and creative thinking

Choose appropriate units of measurement for volume and convert from one unit to another (ACMMG195)  recognise that 1000 litres is equal to one kilolitre and use the abbreviation for kilolitres (kL) 
ecognise that 1000 kilolitres is equal to one megalitre and use the abbreviation for megalitres (ML)  
choose an appropriate unit to measure the volumes or capacities of different objects, eg swimming pools, household containers, damsLiteracy Sustainability
 
convert between metric units of volume and capacity, using 1 cm3 = 1000 mm^{3}, 1 L = 1000 mL = 1000 cm^{3}, 1 m^{3} = 1000 L = 1 kL, 1000 kL = 1 ML 
Develop the formulas for the volumes of rectangular and triangular prisms and of prisms in general; use formulas to solve problems involving volume (ACMMG198)  develop the formula for the volume of prisms by considering the number and volume of ‘layers’ of identical shape:

find the volumes of prisms, given their perpendicular heights and the areas of their uniform crosssections  
find the volumes of prisms with uniform crosssections that are rectangular or triangular  
solve a variety of practical problems involving the volumes and capacities of right prisms 
Calculate the volumes of cylinders and solve related problems (ACMMG217)  develop and use the formula to find the volumes of cylinders:

solve a variety of practical problems involving the volumes and capacities of right prisms and cylinders, eg find the capacity of a cylindrical drink can or a water tank 
TIME
OUTCOME
A student:
MA415MG:
performs calculations of time that involve mixed units, and interprets time zones
TEACHING POINTS  Calculations involving time can be made on a scientific calculator either by using fractions and decimals or by using the ‘degrees, minutes, seconds’ button. Students should be familiar with both approaches. 
Typically, 24hour time is recorded without the use of the colon (:), e.g. 3:45 pm is written as 1545 or 1545 h and read as ‘fifteen fortyfive hours’. 
RELEVANCE  The relevance of this substrand to everyday situations has been seen in earlier stages, as it has involved sequencing events; describing, comparing and ordering the durations of events; reading the time on analog and digital clocks (including 24hour time); converting between hours, minutes and seconds; using am and pm notation in reallife situations; and constructing timelines. 
In Stage 4, students learn the very important everydaylife skills of adding and subtracting time in mixed units (both mentally and by using a calculator) and solve related problems, as well as problems involving international time zones. The ability to compare times in, and calculate time differences between, major cities and areas of the world is of fundamental importance in international travel and also in everyday and work situations, such as communicating with people in other countries, watching overseas sporting events live on television, and conducting international business. 
LANGUAGE  The words ‘minute’ (meaning ‘small’) and ‘minute’ (a time measure), although pronounced differently, are really the same word. A minute (time) is a minute (small) part of one hour. A minute (angle) is a minute (small) part of a right angle. 
EXPECTATIONS OF ATTAINMENT
Solve problems involving duration, including using 12hour and 24hour time within a single time zone (ACMMG199)  add and subtract time mentally using bridging strategies, eg from 2:45 to 3:00 is 15 minutes and from 3:00 to 5:00 is 2 hours, so the time from 2:45 until 5:00 is 15 minutes + 2 hours = 2 hours 15 minutes, interpret and use timetables from reallife situations, including those involving 24hour time {Literacy, Personal and social capability} 
add and subtract time with a calculator, including by using the ‘degrees, minutes, seconds’ button  
round answers to time calculations to the nearest minute or hour  
interpret calculator displays for time calculations, eg 2.25 on a calculator display for a time calculation means 214 hours or 2 hours 15 minutes  
solve a variety of problems involving duration, including where times are expressed in 12hour and 24hour notation, that require the use of mixed units (years, months, days, hours and/or minutes) 
Solve problems involving international time zones  compare times in, and calculate time differences between, major cities of the world, eg ‘Given that London is 10 hours behind Sydney, what time is it in London when it is 6:00 pm in Sydney?’Critical and creative thinking interpret and use information related to international time zones from maps (Problem Solving)Critical and creative thinking solve problems involving international time as it relates to everyday life, eg determine whether a particular soccer game can be watched live on television during normal waking hours (Problem Solving) 
PROPERTIES OF GEOMETRIC FIGURES (PT 2)
OUTCOME
A student:
MA417MG:
 classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles
TEACHING POINTS  For some students, formal setting out of proofs of congruent triangles could be introduced. 
Dynamic geometry software and prepared applets are useful tools for investigating properties of congruent figures through transformations.  
Congruent figures are embedded in a variety of designs, eg tapa cloth, Aboriginal designs, Indonesian ikat designs, Islamic designs, designs used in ancient Egypt and Persia, window lattice, woven mats and baskets.  
Computer drawing programs enable students to prepare tessellation designs and to compare these with other designs, such as those of the Dutch artist M C Escher (1898–1972). 
LANGUAGE  The meaning of the term ‘included angle’ should be taught explicitly. Similarly, the use of the adjective ‘matching’ when referring to the sides and angles of congruent shapes should be made explicit. The term ‘corresponding’ is often used in relation to congruent and similar figures to refer to angles or sides in the same position, but it also has a specific meaning when used to describe a pair of angles in relation to lines cut by a transversal. This syllabus has used ‘matching’ to describe angles and sides in the same position; however, the use of the word ‘corresponding’ is not incorrect. The term ‘superimpose’ is used to describe the placement of one figure upon another in such a way that the parts of one coincide with the parts of the other. 
EXPECTATIONS OF ATTAINMENT
Define congruence of plane shapes using transformations (ACMMG200)  identify congruent figures by superimposing them through a combination of rotations, reflections and translations

match sides and angles of two congruent polygons  
name vertices in matching order when using the symbol ≡ in statements regarding congruence  
determine the condition for two circles to be congruent (equal radii) 
Develop the conditions for congruence of triangles (ACMMG201)  investigate the minimum conditions needed, and establish the four tests, for two triangles to be congruent:

use the congruency tests to identify a pair of congruent triangles from a selection of three or more triangles or from triangles embedded in a diagram 
Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (ACMMG202)  apply the properties of congruent triangles to find an unknown side and/or angle in a diagram, giving a reason 
use transformations of congruent triangles to verify some of the properties of special quadrilaterals, including properties of the diagonals, eg the diagonals of a parallelogram bisect each other 
RIGHT ANGLED TRIANGLES (PYTHAGORAS)
OUTCOME
A student:
MA416MG:
 applies Pythagoras’ theorem to calculate side lengths in rightangled triangles, and solves related problems
TEACHING POINTS  In Stage 3, students are continuing to develop their skills of visual imagery, including the ability to perceive and hold an appropriate mental image of an object or arrangement, and to predict the orientation or shape of an object that has been moved or altered. Also see Year 5 
LANGUAGE  Students should be able to communicate using the following language: object, shape, threedimensional object (3D object), prism, cube, pyramid, base, uniform crosssection, face, edge, vertex (vertices), top view, front view, side view, net. 
EXPECTATIONS OF ATTAINMENT
Investigate pythagorastheorem and its application to solving simple problems involving rightangled triangles (ACMMG222)  identify the hypotenuse as the longest side in any rightangled triangle and also as the side opposite the right angle 
establish the relationship between the lengths of the sides of a rightangled triangle in practical ways, including with the use of digital technologiesInformation and communication technology capability Critical and creative thinking
 
use Pythagoras’ theorem to find the length of an unknown side in a rightangled triangle
 
write answers to a specified or sensible level of accuracy, using an ‘approximately equals’ sign, ie ≑ or ≈  
solve a variety of practical problems involving Pythagoras’ theorem, approximating the answer as a decimal
 
identify a Pythagorean triad as a set of three numbers such that the sum of the squares of the first two equals the square of the third  
use the converse of Pythagoras’ theorem to establish whether a triangle has a right angle 
Investigate the concept of irrational numbers (ACMNA186)  use technology to explore decimal approximations of surds

solve a variety of practical problems involving Pythagoras’ theorem, giving exact answers (ie as surds where appropriate), eg √5 
ANGLE RELATIONSHIPS
OUTCOME
A student:
MA418MG:
identifies and uses angle relationships, including those related to transversals on sets of parallel lines
TEACHING POINTS  Students could explore the results relating to angles associated with parallel lines cut by a transversal by starting with corresponding angles and moving one vertex and all four angles to the other vertex by a translation. The other two results then follow, using vertically opposite angles and angles on a straight line. Alternatively, the equality of the alternate angles can be seen by rotation about the midpoint of the transversal. Students should give reasons when finding the sizes of unknown angles. For some students, formal setting out could be introduced. For example, ∠ABQ=70∘ (corresponding angles, AC∥PR). In his calculation of the circumference of the Earth, the Greek mathematician, geographer and astronomer Eratosthenes (c276–c194 BC) used parallel line results. 
LANGUAGE  Students in Stage 4 should write geometrical reasons without the use of abbreviations to assist them in learning new terminology, and in understanding and retaining geometrical concepts, eg ‘When a transversal cuts parallel lines, the cointerior angles formed are supplementary’. Some students may find the use of the terms ‘complementary’ and ‘supplementary’ (adjectives) and ‘complement’ and ‘supplement’ (nouns) difficult. Teachers should model the use of these terms in sentences, both verbally and in written form, eg, ’50° and 40° are complementary angles’, ‘The complement of 50° is 40°’. Students should be aware that complementary and supplementary angles may or may not be adjacent. 
PURPOSE RELEVANCE OF SUBSTRAND  The development of knowledge and understanding of angle relationships, including the associated terminology, notation and conventions, is of fundamental importance in developing an appropriate level of knowledge, skills and understanding in geometry. Angle relationships and their application play an integral role in students learning to analyse geometry problems and developing geometric and deductive reasoning skills, as well as problemsolving skills. Angle relationships are key to the geometry that is important in the work of architects, engineers, designers, builders, physicists, land surveyors, etc, as well as the geometry that is common and important in everyday situations, such as in nature, sports, buildings, astronomy, art, etc. 
Expectations of Attainment
Use the language, notation and conventions of geometry  define, label and name points, lines and intervals using capital letters 
label the vertex and arms of an angle with capital letters  
label and name angles using ∠P or ∠QPR notation  
use the common conventions to indicate right angles and equal angles on diagrams 
Recognise the geometrical properties of angles at a point  use the terms ‘complementary’ and ‘supplementary’ for angles adding to 90° and 180°, respectively, and the associated terms ‘complement’ and ‘supplement’ 
use the term ‘adjacent angles’ to describe a pair of angles with a common arm and a common vertex, and lie on opposite sides of the common arm  
identify and name right angles, straight angles, angles of complete revolution and vertically opposite angles embedded in diagramsLiteracy Critical and creative thinking

Identify corresponding, alternate and cointerior angles when two straight lines are crossed by a transversal (ACMMG163)  identify and name perpendicular lines using the symbol for ‘is perpendicular to’ (⊥), eg AB⊥CD 
use the common conventions to indicate parallel lines on diagrams  
identify and name pairs of parallel lines using the symbol for ‘is parallel to’ (∥), eg PQ∥RS  
define and identify ‘transversals’, including transversals of parallel lines  
identify, name and measure alternate angle pairs, corresponding angle pairs and cointerior angle pairs for two lines cut by a transversalCritical and creative thinking
 
recognise the equal and supplementary angles formed when a pair of parallel lines is cut by a transversal 
Investigate conditions for two lines to be parallel (ACMMG164)  use angle properties to identify parallel lines Critical and creative thinking

Solve simple numerical problems using reasoning (ACMMG164)  find the sizes of unknown angles embedded in diagrams using angle relationships, including angles at a point and angles associated with parallel lines, giving reasons

STATISTICS AND PROBABILITY
DATA COLLECTION AND REPRESENTATION
OUTCOME
A student:
MA419SP:
collects, represents and interprets single sets of data, using appropriate statistical displays
TEACHING POINTS  Students in Stage 4 can be expected to have some prior knowledge of both dot plots and line graphs, as these types of graph are introduced in Stage 3. They construct, describe and interpret column graphs in Stage 2 and Stage 3; however, histograms, divided bar graphs and sector graphs (pie charts) are not encountered until Stage 4. Statistical data is part of everyday life. Data may be displayed in tables or graphs, and may appear in all types of media. Graphs provide a visual overview of the substrand under investigation. Students should be aware that while many graphs are accurate and informative, some can be misleading. They need to experience interpreting a wide variety of graphical representations, including column graphs, dot plots, stemandleaf plots, divided bar graphs, sector graphs and line graphs. Students should be able to select an appropriate graph to represent the collected data. 
RELEVANCE  In investigations, it is important to develop knowledge and understanding of the ways in which relevant and sufficient data can be collected, as well as the associated implications and limitations. It is also important to develop knowledge and understanding of what constitute appropriate sources of data, both primary and secondary. Data and statistics are used in many aspects of our everyday and working lives. Data is collected to provide information on many topics of interest and to assist in making decisions regarding important issues (eg projects aimed at improving or developing products or services). Users at all levels need to have skills in the organisation and display of the collected data for its interpretation and analysis. This can be achieved in a wide variety of ways, including through the use of frequency distribution tables and simple data displays/graphs, such as frequency histograms and polygons, dot plots, stemandleaf plots, divided bar graphs, sector graphs and line graphs. 
LANGUAGE  In everyday language, the term ‘pie chart’ is often used in reference to sector graphs. 
Expectations of Attainment
Investigate techniques for collecting data, including census, sampling and observation (ACMSP284)  define ‘variable’ in the context of statistics as something measurable or observable that is expected to change over time or between individual observations 
recognise variables as categorical or numerical (either discrete or continuous)Literacy
 
recognise and explain the difference between a ‘population’ and a ‘sample’ selected from a population when collecting data  
investigate and determine the differences between collecting data by observation, census and samplingLiteracy Critical and creative thinking

Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes (ACMSP206)  collect data using a random process, eg numbers from a page in a phone book, or from a random number generator 
identify issues that may make it difficult to obtain representative data from either primary or secondary sourcesCritical and creative thinking Ethical understanding
 
investigate and question the selection of data used to support a particular viewpoint, eg the selective use of data in product advertising 
Identify and investigate issues involving numerical data collected from primary and secondary sources (ACMSP169)  identify the difference between data collected from primary and secondary sources, eg data collected in the classroom compared with data drawn from a media source 
explore issues involved in constructing and conducting surveys, such as sample size, bias, type of data required, and ethicsLiteracy Ethical understanding Civics and citizenship Difference and diversity
 
construct appropriate survey questions and a related recording sheet in order to collect both numerical and categorical data about a matter of interestLiteracy Critical and creative thinking Personal and social capability Difference and diversity
 
collect and interpret information from secondary sources, presented as tables and/or graphs, about a matter of interest, eg sporting data, information about the relationship between wealth or education and the health of populations of different countriesLiteracy Intercultural understanding Ethical understanding Civics and citizenship Difference and diversity
 
use spreadsheets or statistical software packages to tabulate and graph dataInformation and communication technology capability

Construct and compare a range of data displays, including stemandleaf plots and dot plots (ACMSP170)  use a tally to organise data into a frequency distribution table 
construct and interpret frequency histograms and polygons
 
construct dot plots
 
construct ordered stemandleaf plots, including stemandleaf plots with twodigit stems
 
construct divided bar graphs, sector graphs and line graphs, with and without the use of digital technologiesLiteracy
 
interpret a variety of graphs, including dot plots, stemandleaf plots, divided bar graphs, sector graphs and line graphsLiteracy

SINGLE VARIABLE DATA ANALYSIS
OUTCOME
A student:
MA420SP:
analyses single sets of data using measures of location, and range
TEACHING POINTS  Many opportunities occur in this substrand for students to strengthen their skills in: collecting, analysing and organising information; communicating ideas and information; planning and organising activities; working with others and in teams; using mathematical ideas and techniques; using technology, including spreadsheets. 
RELEVANCE  Singlevariable (or ‘univariate’) data analysis involves the statistical examination of a particular ‘variable’ (ie a value or characteristic that changes for different individuals, etc) and is of fundamental importance in the statistics used widely in everyday situations and in fields including education, business, economics and government. Most singlevariable data analysis methods are used for descriptive purposes. In organising and displaying the data collected, frequencies, tables and a variety of data displays/graphs are used. These data displays/graphs, and numerical summary measures, are used to analyse and describe a data set in relation to a single variable, such as the scores on a test, and to compare a data set to other data sets. Singlevariable data analysis is commonly used in the first stages of investigations, research, etc to describe and compare data sets, before being supplemented by more advanced ‘bivariate’ or ‘multivariate’ data analysis. 
LANGUAGE  The term ‘average’, when used in everyday language, generally refers to the mean and describes a ‘typical value’ within a set of data. Students need to be provided with opportunities to discuss what information can be drawn from the data presented. They need to think about the meaning of the information and to put it into their own words. Language to be developed would include superlatives, comparatives, expressions such as ‘prefer … over’, etc. 
Expectations of Attainment
Calculate mean, median, mode and range for sets of data and interpret these statistics in the context of data (ACMSP171) 

determine the median, mode and range for sets of data
 
identify and describe the mean, median and mode as ‘measures of location’ or ‘measures of centre’, and the range as a ‘measure of spread’  
describe, in practical terms, the meaning of the mean, median, mode and/or range in the context of the data, eg when referring to the mode of shoesize data: ‘The most popular shoe size is size 7’ 
Investigate the effect of individual data values, including outliers, on the mean and median (ACMSP207)  identify any clusters, gaps and outliers in sets of data 
investigate the effect of outliers on the mean, median, mode and range by considering a small set of data and calculating each measure, with and without the inclusion of an outlierCritical and creative thinking
 
analyse collected data to identify any obvious errors and justify the inclusion of any individual data values that differ markedly from the rest of the data collected 
Describe and interpret data displays using mean, median and range (ACMSP172)  calculate measures of location (mean, median and mode) and the range for data represented in a variety of statistical displays, including frequencydistribution tables, frequency histograms, stemandleaf plots and dot plots 
draw conclusions based on the analysis of data displays using the mean, median and/or mode, and range 
Explore the variation of means and proportions of random samples drawn from the same population(ACMSP293)  investigate ways in which different random samples may be drawn from the same population, eg random samples from a census may be chosen by gender, postcode, state, etc 
calculate and compare summary statistics (mean, median, mode and range) of at least three different random samples drawn from the same population use a spreadsheet to calculate and compare summary statistics of different random samples drawn from the same population (Communicating, Problem Solving)

Construct and compare a range of data displays, including stemandleaf plots and dot plots (ACMSP170)  use a tally to organise data into a frequency distribution table 
construct and interpret frequency histograms and polygons
 
construct dot plots
 
construct ordered stemandleaf plots, including stemandleaf plots with twodigit stems
 
construct divided bar graphs, sector graphs and line graphs, with and without the use of digital technologiesLiteracy
 
interpret a variety of graphs, including dot plots, stemandleaf plots, divided bar graphs, sector graphs and line graphsLiteracy

PROBABILITY (Pt 2)
OUTCOME
A student:
MA421SP:
represents probabilities of simple and compound events
TEACHING POINTS  John Venn (1834−1923) was a British mathematician best known for his diagrammatic way of representing sets, and their unions and intersections. 
Students are expected to be able to interpret Venn diagrams involving three attributes; however, they are not expected to construct Venn diagrams involving three attributes.  
A compound event is an event that can be expressed as a combination of simple events, eg drawing a card that is black or a King from a standard set of playing cards, throwing at least 5 on a standard sixsided die. 
LANGUAGE  A compound event is an event that can be expressed as a combination of simple events, eg drawing a card that is black or a King from a standard set of playing cards, throwing at least 5 on a standard sixsided die. 
EXPECTATIONS OF ATTAINMENT
Describe events using language of ‘at least’, exclusive ‘or’ (A or B but not both), inclusive ‘or’ (A or B or both) and ‘and’ (ACMSP205)  recognise the difference between mutually exclusive and nonmutually exclusive events, eg when a die is rolled, ‘rolling an odd number’ and ‘rolling an even number’ are mutually exclusive events; however, ‘rolling an even number’ and ‘rolling a 2’ are nonmutually exclusive events 
describe compound events using the following terms:
 
classify compound events using inclusive ‘or’ and exclusive ‘or’, eg ‘choosing a male or a female’ is exclusive as one cannot be both, whereas ‘choosing a male or someone lefthanded’ could imply exclusivity or inclusivity

Represent events in twoway tables and Venn diagrams and solve related problems (ACMSP292)  interpret Venn diagrams involving two or three attributes
The image shows a Venn diagram with 2 separate ovals inside a rectangle.
There are 25 students who play both basketball and football; 46 students who play basketball or football, but not both; 19 students who play neither sport; and 71 students who play basketball or football or both (Communicating, Problem Solving, Reasoning) 
construct Venn diagrams to represent all possible combinations of two attributes from given or collected data
 
interpret given twoway tables representing nonmutually exclusive attributesLiteracy Critical and creative thinking
There are 63 male righthanded students, ie 63 students are neither female nor lefthanded; there are 114 students who are male or righthanded, or both (Communicating, Problem Solving, Reasoning)  
construct twoway tables to represent the relationships between attributes
 
convert between representations of the relationships between two attributes in Venn diagrams and twoway tables, eg 