# YEAR 7 MATHS FOCUS

## NUMBER AND ALGEBRA

## COMPUTATION WITH INTEGERS

OUTCOME

A student:

MA4-4NA:

compares, orders and calculates with integers, applying a range of strategies to aid computation

TEACHING POINTS | To divide two- and three-digit numbers by a two-digit 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 en-dash ( – ) 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 non-calculator method to divide two- and three-digit numbers by a two-digit 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 real-life 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:

MA4-5NA:

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 non-terminating decimals, but not all non-terminating 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. One-cent and two-cent 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 left-hand 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 three-quarters 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 one-fifth 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 real-life 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:

MA4-6NA:

solves financial problems involving purchasing goods

TEACHING POINTS | The Goods and Services Tax (GST) in Australia is a value-added 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 GST-exempt items (which include basic necessities such as milk and bread). |

LANGUAGE | GST stands for ‘Goods and Services Tax’. The difference between the GST-inclusive price, the pre-GST 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 left-hand 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 two-digit numbers |

select and use efficient mental and written strategies, and digital technologies, to divide whole numbers of up to four digits by a one-digit 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 real-life 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 real-life 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] | |

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:

MA4-7NA:

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 whole-number 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 real-life 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 real-life 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 straight-line 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 straight-line segments with different gradients in the graph of a particular journey {Communicating} | |

– calculate speeds for straight-line 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 straight-line 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 10-second increments {Problem Solving} | |

use the relative positions of two points on a line graph, rather than a detailed scale, to interpret information |

## ALGEBRAIC TECHNIQUES

OUTCOME

A student:

MA4-8NA:

generalises number properties to operate with algebraic expressions

TEACHING POINTS | It is important to develop an understanding of the use of pronumerals (letters) as algebraic symbols to represent one or more numerical values. The recommended approach is to spend time on the conventions for the use of algebraic symbols for first-degree expressions and to situate the translation of generalisations from words to symbols as an application of students’ knowledge of the symbol system, rather than as an introduction to the symbol system. The recommended steps for moving into symbolic algebra are: the variable notion, associating letters with a variety of different numerical values; symbolism for a pronumeral plus a constant; symbolism for a pronumeral times a constant; symbolism for sums, differences, products and quotients. So, if a=6, a+a=6+6, but 2a=2×6 and not 26. To gain an understanding of algebra, students must be introduced to the concepts of pronumerals, expressions, unknowns, equations, patterns, relationships and graphs in a wide variety of contexts. For each successive context, these ideas need to be redeveloped. Students need gradual exposure to abstract ideas as they begin to relate algebraic terms to real situations. It is suggested that the introduction of representation through the use of algebraic symbols precede Linear Relationships in Stage 4, since this substrand presumes that students are able to manipulate algebraic symbols and will use them to generalise patterns. |

LANGUAGE | For the introduction of algebra in Stage 4, the term ‘pronumeral’ rather than ‘variable’ is preferred when referring to unknown numbers. In an algebraic expression such as 2x+5,x can take any value (ie x is variable and a pronumeral). However, in an equation such as 2x+5=11,x represents one particular value (ie x is not a variable but is a pronumeral). In equations such as x+y=11, x and y can take any values that sum to 11 (ie x and y are variables and pronumerals). ‘Equivalent’ is the adjective for ‘equal’, although ‘equal’ can also be used as an adjective, ie ‘equivalent expressions’ or ‘equal expressions’. Some students may confuse the order in which terms or numbers are to be used when a question is expressed in words. This is particularly apparent for word problems that involve subtraction or division to obtain the required result, e.g. ‘5x less than x’ and ‘take 5x from x’ both require the order of the terms to be reversed to x−5x in the solution. Students need to be familiar with the terms ‘sum’, ‘difference’, ‘product’ and ‘quotient’ to describe the results of adding, subtracting, multiplying and dividing, respectively. |

PURPOSE RELEVANCE | Algebra is used to some extent throughout our daily lives. 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. Algebra requires, and its use results in, learning how to apply logical reasoning and problem-solving skills. It is used more extensively in other areas of mathematics, the sciences, business, accounting, etc. The widespread use of algebra is readily seen in the writing of formulas in spreadsheets. |

## Expectations of Attainment

Introduce the concept of variables as a way of representing numbers using letters (ACMNA175) | develop the concept that pronumerals (letters) can be used to represent numerical values |

– recognise that pronumerals can represent one or more numerical values (when more than one numerical value, pronumerals may then be referred to as ‘variables’) {Communicating, Reasoning, Literacy} | |

model the following using concrete materials or otherwise: – expressions that involve a pronumeral, and a pronumeral added to a constant, e.g. a, a+1 – expressions that involve a pronumeral multiplied by a constant, e.g. 2a, 3a – sums and products, e.g. 2a+1, 2(a+1) – equivalent expressions, e.g. x+x+y+y+3=2x+2y+3=2(x+y)+3 – simplifying expressions, e.g. (a+2)+(2a+3)=(a+2a)+(2+3)=3a+5 | |

recognise and use equivalent algebraic expressions, e.g. y+y+y+y=4y w×w=w^2 a×b=ab a÷b=a/b {Literacy} | |

use algebraic symbols to represent mathematical operations written in words and vice versa, e.g. the product of x and y is xy, x+y is the sum of x and y {Literacy} |

Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (ACMNA177) | recognise like terms and add and subtract them to simplify algebraic expressions, e.g. 2n+4m+n=4m+3n |

– verify whether a simplified expression is correct by substituting numbers for pronumerals {Communicating, Reasoning} | |

– connect algebra with the commutative and associative properties of arithmetic to determine that a+b=b+a and (a+b)+c=a+(b+c) {Communicating, Critical and creative thinking} | |

recognise the role of grouping symbols and the different meanings of expressions, such as 2a+1 and 2(a+1) {Critical and creative thinking} | |

simplify algebraic expressions that involve multiplication and division, e.g. 12a÷3, 4x×3, 2ab×3a, (8a)/2, (2a)/8, (12a)/9 | |

– recognise the equivalence of algebraic expressions involving multiplication, e.g. 3bc=3cb {Communicating, Critical and creative thinking} | |

– connect algebra with the commutative and associative properties of arithmetic to determine that a×b=b×a and (a×b)×c=a×(b×c) {Communicating, Critical and creative thinking} | |

– recognise whether particular algebraic expressions involving division are equivalent or not, e.g. a÷bc is equivalent to a/bc and a÷(b×c), but is not equivalent to a÷b×c or a/b×c {Communicating, Critical and creative thinking} | |

translate from everyday language to algebraic language and vice versa {Literacy} | |

– use algebraic symbols to represent simple situations described in words, e.g. write an expression for the number of cents in x dollars {Communicating, Literacy} | |

– interpret statements involving algebraic symbols in other contexts, e.g. cell references when creating and formatting spreadsheets {Communicating, Literacy, Information and communication technology capability} |

Simplify algebraic expressions involving the four operations (ACMNA192) | simplify a range of algebraic expressions, including those involving mixed operations |

apply the order of operations to simplify algebraic expressions {Problem Solving} |

## INDICES

OUTCOME

A student:

MA4-9NA:

- operates with positive-integer 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 non-square whole number using a calculator, after first estimating {Information and communication technology capability} | |

– determine the two integers between which the square root of a non-square 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 positive-integer indices and the zero index (ACMNA182) | develop index laws with positive-integer 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:

MA4-10NA:

- operates with positive-integer and zero indices of numerical bases

## INDICES

OUTCOME

A student:

MA4-9NA:

- operates with positive-integer and zero indices of numerical bases

TEACHING POINTS | The solution of simple equations can be introduced using a variety of models. Such models include using a two-pan balance with objects such as centicubes and a wrapped ‘unknown’, or using some objects hidden in a container as an ‘unknown’ to produce a number sentence. The solution of simple quadratic equations in Stage 4 enables students to determine side lengths in right-angled triangles through the application of Pythagoras’ theorem. |

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 non-integer 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 non-integer 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 | 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 |

## MEASUREMENT AND GEOMETRY

## LENGTH

OUTCOME

A student:

MA3-9MG:

selects and uses the appropriate unit and device to measure lengths and distances, calculates perimeters, and converts between units of length

TEACHING POINTS | When students are able to measure efficiently and effectively using formal units, they should be encouraged to apply their knowledge and skills in a variety of contexts. Following this, they should be encouraged to generalise their method for calculating the perimeters of squares, rectangles and triangles. |

When recording measurements, a space should be left between the number and the abbreviated unit, e.g. 3 cm, not 3cm. |

LANGUAGE | Students should be able to communicate using the following language: length, distance, kilometre, metre, centimetre, millimetre, perimeter, dimensions, width. |

## Expectations of Attainment

Connect decimal representations to the metric system (ACMMG135) | recognise the equivalence of whole-number and decimal representations of measurements of length, e.g. 165 cm is the same as 1.65 m |

interpret decimal notation for lengths and distances, e.g. 13.5 cm is 13 centimetres and 5 millimetres | |

record lengths and distances using decimal notation to three decimal places, e.g. 2.753 km |

Convert between common metric units of length (ACMMG136) | convert between metres and kilometres |

convert between millimetres, centimetres and metres to compare lengths and distances | |

– explain and use the relationship between the size of a unit and the number of units needed to assist in determining whether multiplication or division is required when converting between units, eg ‘More metres than kilometres will be needed to measure the same distance, and so to convert from kilometres to metres, I need to multiply’ {Communicating, Reasoning, Critical and creative thinking} |

Solve problems involving the comparison of lengths using appropriate units (ACMMG137) | investigate and compare perimeters of rectangles with the same area {Critical and creative thinking} |

– determine the number of different rectangles that can be formed using whole-number dimensions for a given area {Problem Solving, Reasoning, Critical and creative thinking} | |

solve a variety of problems involving length and perimeter, including problems involving different units of length, e.g. ‘Find the total length of three items measuring 5 mm, 20 cm and 1.2 m’ {Critical and creative thinking} |

## AREA

OUTCOME

A student:

MA3-10MG:

selects and uses the appropriate unit to calculate areas, including areas of squares, rectangles and triangles

TEACHING POINTS | Students should have a clear understanding of the distinction between perimeter and area. |

It is important in Stage 3 that students establish a real reference for the square kilometre and the hectare, eg locating an area of one square kilometre or an area of one hectare on a local map. | |

When students are able to measure efficiently and effectively using formal units, they should be encouraged to apply their knowledge and skills in a variety of contexts. | |

Students could be encouraged to find more efficient ways of counting when determining area, such as finding how many squares in one row and multiplying this by the number of rows. They should then begin to generalise their methods to calculate the areas of rectangles (including squares) and triangles. | |

When generalising their methods to calculate areas, students in Stage 3 should use words. Algebraic formulas for areas are not introduced until Stage 4. |

LANGUAGE | Students should be able to communicate using the following language: area, square centimetre, square metre, dimensions, length, width, base (of triangle), perpendicular height. |

## EXPECTATIONS OF ATTAINMENT

Solve problems involving the comparison of areas using appropriate units (ACMMG137) | investigate the area of a triangle by comparing the area of a given triangle to the area of the rectangle of the same length and perpendicular height, eg use a copy of the given triangle with the given triangle to form a rectangle |

– explain the relationship between the area of a triangle and the area of the rectangle of the same length and perpendicular height {Communicating, Reasoning, Critical and creative thinking} | |

establish the relationship between the base length, perpendicular height and area of a triangle {Critical and creative thinking} | |

record, using words, the method for finding the area of any triangle, e.g. ‘Area of triangle = 1/2 × base × perpendicular height’ {Literacy} | |

investigate and compare the areas of rectangles that have the same perimeter, e.g. compare the areas of all possible rectangles with whole-number dimensions and a perimeter of 20 centimetres {Critical and creative thinking} | |

– determine the number of different rectangles that can be formed using whole-number dimensions for a given perimeter {Problem Solving, Reasoning, Critical and creative thinking} | |

solve a variety of problems involving the areas of rectangles (including squares) and triangles {Critical and creative thinking} |

## VOLUME & CAPACITY

OUTCOME

A student:

MA3-11MG:

selects and uses the appropriate unit to estimate, measure and calculate volumes and capacities, and converts between units of capacity

TEACHING POINTS | The attribute of volume is the amount of space occupied by an object or substance and is usually measured in cubic units, eg cubic centimetres (cm^3) and cubic metres (m^3). |

Capacity refers to the amount a container can hold and is measured in units, such as millilitres (mL), litres (L) and kilolitres (kL). Capacity is only used in relation to containers and generally refers to liquid measurement. The capacity of a closed container will be slightly less than its volume – capacity is based on the inside dimensions, while volume is determined by the outside dimensions of the container. It is not necessary to refer to these definitions with students (capacity is not taught as a concept separate from volume until Stage 4). | |

Once students are able to measure efficiently and effectively using formal units, they could use centimetre cubes to construct rectangular prisms, counting the number of cubes to determine volume, and then begin to generalise their method for calculating the volume. | |

The cubic metre can be related to the metre as a unit to measure length and the square metre as a unit to measure area. It is important that students are given opportunities to reflect on their understanding of length and area so that they can use this to calculate volume. |

LANGUAGE | Students should be able to communicate using the following language: capacity, container, litre, millilitre, volume, dimensions, length, width, height, layers, cubic centimetre, cubic metre. |

The abbreviation m^3 is read as ‘cubic metre(s)’ and not ‘metre(s) cubed’. |

## EXPECTATIONS OF ATTAINMENT

Connect volume and capacity and their units of measurement (ACMMG138) | select the appropriate unit to measure volume and capacity |

demonstrate that a cube of side 10 cm will displace 1 litre of water | |

demonstrate, by using a medicine cup, that a cube of side 1 cm will displace 1 mL of water | |

equate 1 cubic centimetre to 1 millilitre and 1000 cubic centimetres to 1 litre | |

find the volumes of irregular solids in cubic centimetres using a displacement strategy |

Connect decimal representations to the metric system (ACMMG135) | recognise the equivalence of whole-number and decimal representations of measurements of capacities, e.g. 375 mL is the same as 0.375 L |

interpret decimal notation for volumes and capacities, e.g. 8.7 L is the same as 8 litres and 700 millilitres | |

record volume and capacity using decimal notation to three decimal places, e.g. 1.275 L |

Convert between common metric units of capacity (ACMMG136) | convert between millilitres and litres |

– explain and use the relationship between the size of a unit and the number of units needed to assist in determining whether multiplication or division is required when converting between units, e.g. ‘Fewer litres than millilitres will be needed to measure the same capacity, and so to convert from millilitres to litres, I need to divide’ {Communicating, Reasoning, Critical and creative thinking} |

## MASS

OUTCOME

A student:

MA3-12MG:

selects and uses the appropriate unit and device to measure the masses of objects, and converts between units of mass

TEACHING POINTS | One litre of water has a mass of one kilogram and a volume of 1000 cubic centimetres. While the relationship between volume and capacity is constant for all substances, the same volumes of substances other than water may have different masses, e.g. 1 litre of oil is lighter than 1 litre of water, which in turn is lighter than 1 litre of honey. This can be demonstrated using digital scales. See also Year Five |

LANGUAGE | Students should be able to communicate using the following language: mass, measure, scales, tonne, kilogram, gram. |

As the terms ‘weigh’ and ‘weight’ are common in everyday usage, they can be accepted in student language should they arise. Weight is a force that changes with gravity, while mass remains constant. |

## EXPECTATIONS OF ATTAINMENT

Connect decimal representations to the metric system (ACMMG135) | recognise the equivalence of whole-number and decimal representations of measurements of mass, e.g. 3 kg 250 g is the same as 3.25 kg |

interpret decimal notation for masses, e.g. 2.08 kg is the same as 2 kilograms and 80 grams | |

measure mass using scales and record using decimal notation of up to three decimal places, e.g. 0.875 kg |

Convert between common metric units of mass (ACMMG136) | convert between kilograms and grams and between kilograms and tonnes |

– explain and use the relationship between the size of a unit and the number of units needed to assist in determining whether multiplication or division is required when converting between units, e.g. ‘More grams than kilograms will be needed to measure the same mass, and so to convert from kilograms to grams, I need to multiply’ {Communicating, Reasoning, Critical and creative thinking} | |

solve problems involving different units of mass, e.g. find the total mass of three items weighing 50 g, 750 g and 2.5 kg {Critical and creative thinking} | |

relate the mass of one litre of water to one kilogram |

## TIME

OUTCOME

A student:

MA3-13MG:

uses 24-hour time and am and pm notation in real-life situations, and constructs timelines

TEACHING POINTS | Australia is divided into three time zones. In non-daylight saving periods, time in Queensland, New South Wales, Victoria and Tasmania is Eastern Standard Time (EST), time in South Australia and the Northern Territory is half an hour behind EST, and time in Western Australia is two hours behind EST. |

Typically, 24-hour 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 forty-five hours’. |

LANGUAGE | Students should be able to communicate using the following language: timetable, timeline, scale, 12-hour time, 24-hour time, hour, minute, second, am (notation), pm (notation). |

## EXPECTATIONS OF ATTAINMENT

Interpret and use timetables (ACMMG139) | read, interpret and use timetables from real-life situations, including those involving 24-hour time {Literacy, Personal and social capability} |

use bus, train, ferry and airline timetables, including those accessed on the internet, to prepare simple travel itineraries {Literacy, Information and communication technology capability, Personal and social capability} | |

– interpret timetable information to solve unfamiliar problems using a variety of strategies {Problem Solving, Literacy Critical and creative thinking} |

Draw and interpret timelines using a given scale | determine a suitable scale and draw an accurate timeline using the scale, e.g. represent events using a many-to-one scale of 1 cm = 10 years |

interpret a given timeline using the given scale {Literacy} |

## THREE-DIMENSIONAL SPACE

OUTCOME

A student:

MA3-14MG:

identifies three-dimensional objects, including prisms and pyramids, on the basis of their properties, and visualises, sketches and constructs them given drawings of different views

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, three-dimensional object (3D object), prism, cube, pyramid, base, uniform cross-section, face, edge, vertex (vertices), top view, front view, side view, net. |

## EXPECTATIONS OF ATTAINMENT

Construct simple prisms and pyramids (ACMMG140) | create prisms and pyramids using a variety of materials, e.g. plasticine, paper or cardboard nets, connecting cubes |

– construct as many rectangular prisms as possible using a given number of connecting cubes {Problem Solving, Critical and creative thinking} | |

create skeletal models of prisms and pyramids, e.g. using toothpicks and modelling clay or straws and tape {Critical and creative thinking} | |

– connect the edges of prisms and pyramids with the construction of their skeletal models {Problem Solving} | |

construct three-dimensional models of prisms and pyramids and sketch the front, side and top views | |

– describe to another student how to construct or draw a three-dimensional object {Communicating, Literacy} | |

construct three-dimensional models of prisms and pyramids, given drawings of different views |

## TWO-DIMENSIONAL SPACE

OUTCOME

A student:

MA3-15MG:

manipulates, classifies and draws two-dimensional shapes, including equilateral, isosceles and scalene triangles, and describes their properties

TEACHING POINTS | When drawing diagonals, students need to be careful that the endpoints of their diagonals pass through the vertices of the shape. |

LANGUAGE | Students should be able to communicate using the following language: shape, two-dimensional shape (2D shape), circle, centre, radius, diameter, circumference, sector, semicircle, quadrant, triangle, equilateral triangle, isosceles triangle, scalene triangle, right-angled triangle, quadrilateral, parallelogram, rectangle, rhombus, square, trapezium, kite, pentagon, hexagon, octagon, regular shape, irregular shape, diagonal, vertex (vertices), line (axis) of symmetry, translate, reflect, rotate, clockwise, anti-clockwise. |

A diagonal of a two-dimensional shape is an interval joining two non-adjacent vertices of the shape. The diagonals of a convex two-dimensional shape lie inside the figure. |

## EXPECTATIONS OF ATTAINMENT

Investigate the diagonals of two-dimensional shapes | identify and name ‘diagonals’ of convex two-dimensional shapes {Literacy} |

– recognise the endpoints of the diagonals of a shape as the vertices of the shape {Communicating, Literacy} | |

determine and draw all the diagonals of convex two-dimensional shapes | |

compare and describe diagonals of different convex two-dimensional shapes | |

– use measurement to determine which of the special quadrilaterals have diagonals that are equal in length {Problem Solving} | |

– determine whether any of the diagonals of a particular shape are also lines (axes) of symmetry of the shape {Problem Solving} |

Identify and name parts of circles | create a circle by finding points that are all the same distance from a fixed point (the centre) |

identify and name parts of a circle, including the centre, radius, diameter, circumference, sector, semicircle and quadrant {Literacy} |

Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies (ACMMG142) | identify whether a two-dimensional shape has been translated, reflected or rotated, or has undergone a number of transformations, e.g. ‘The parallelogram has been rotated clockwise through 90° once and then reflected once’ |

construct patterns of two-dimensional shapes that involve translations, reflections and rotations using computer software {Information and communication technology capability} | |

predict the next translation, reflection or rotation in a pattern, eg ‘The arrow is being rotated 90° anti-clockwise each time’ | |

– choose the correct pattern from a number of options when given information about a combination of transformations {Reasoning, Critical and creative thinking} |

## ANGLES

OUTCOME

A student:

MA3-16MG:

measures and constructs angles, and applies angle relationships to find unknown angles

TEACHING POINTS | Students should be encouraged to give reasons when finding unknown angles. |

LANGUAGE | Students should be able to communicate using the following language: angle, right angle, straight angle, angles on a straight line, angle of revolution, angles at a point, vertically opposite angles. |

Two angles at a point are called adjacent if they share a common arm and a common vertex, and lie on opposite sides of the common arm. |

## EXPECTATIONS OF ATTAINMENT

Investigate, with and without the use of digital technologies, angles on a straight line, angles at a point, and vertically opposite angles; use the results to find unknown angles(ACMMG141) | identify and name angle types formed by the intersection of straight lines, including right angles, ‘angles on a straight line’, ‘angles at a point’ that form an angle of revolution, and ‘vertically opposite angles’ {Literacy} |

– recognise right angles, angles on a straight line, and angles of revolution embedded in diagrams {Reasoning} | |

– identify the vertex and arms of angles formed by intersecting lines {Communicating} | |

– recognise vertically opposite angles in different orientations and embedded in diagrams {Reasoning} | |

investigate, with and without the use of digital technologies, adjacent angles that form a right angle and establish that they add to 90° | |

investigate, with and without the use of digital technologies, adjacent angles on a straight line and establish that they form a straight angle and add to 180° | |

investigate, with and without the use of digital technologies, angles at a point and establish that they form an angle of revolution and add to 360° | |

use the results established for adjacent angles that form right angles, straight angles and angles of revolution to find the size of unknown angles in diagrams {Critical and creative thinking} | |

– explain how the size of an unknown angle in a diagram was calculated {Communicating, Reasoning, Critical and creative thinking} | |

investigate, with and without the use of digital technologies, vertically opposite angles and establish that they are equal in size | |

use the equality of vertically opposite angles to find the size of unknown angles in diagrams |

## POSITION

OUTCOME

A student:

MA3-17MG:

locates and describes position on maps using a grid-reference system

TEACHING POINTS | In Stage 2, students were introduced to the compass directions north, east, south and west, and north-east, south-east, south-west and north-west. In Stage 3, students are expected to use these compass directions when describing routes between locations on maps. |

By convention when using grid-reference systems, the horizontal component of direction is named first, followed by the vertical component. This connects with plotting points on the Cartesian plane in Stage 3 Patterns and Algebra, where the horizontal coordinate is recorded first, followed by the vertical coordinate. |

LANGUAGE | Students should be able to communicate using the following language: position, location, map, plan, street directory, route, grid, grid reference, legend, key, scale, directions, compass, north, east, south, west, north-east, south-east, south-west, north-west. |

## EXPECTATIONS OF ATTAINMENT

Use a grid-reference system to describe locations (ACMMG113) | find locations on maps, including maps with legends, given their grid references {Literacy} |

describe particular locations on grid-referenced maps, including maps with a legend, eg ‘The post office is at E4’ |

Describe routes using landmarks and directional language (ACMMG113) | find a location on a map that is in a given direction from a town or landmark, eg locate a town that is north-east of Broken Hill {Literacy} |

describe the direction of one location relative to another, eg ‘Darwin is north-west of Sydney’ {Literacy} | |

follow a sequence of two or more directions, including compass directions, to find and identify a particular location on a map {Literacy} | |

use a given map to plan and show a route from one location to another, eg draw a possible route to the local park or use an Aboriginal land map to plan a route(Literacy, Aboriginal and Torres Strait Islander histories and cultures} | |

– use a street directory or online map to find the route to a given location {Problem Solving, Literacy, Information and communication technology capability} | |

describe a route taken on a map using landmarks and directional language, including compass directions, eg ‘Start at the post office, go west to the supermarket and then go south-west to the park’ {Literacy} |

## STATISTICS AND PROBABILITY

## DATA

OUTCOME

A student:

MA3-18SP:

uses appropriate methods to collect data and constructs, interprets and evaluates data displays, including dot plots, line graphs and two-way tables

TEACHING POINTS | Data selected for interpretation can include census data, environmental audits of resources such as water and energy, and sports statistics. Also see Year 5 |

LANGUAGE | Students should be able to communicate using the following language: data, collect, category, display, table, column graph, scale, axes, two-way table, side-by-side column graph, misleading, bias. |

## Expectations of Attainment

Interpret and compare a range of data displays, including side-by-side column graphs for two categorical variables (ACMSP147) | interpret data presented in two-way tables {Literacy, Civics and citizenship} |

create a two-way table to organise data involving two categorical variables | |

interpret side-by-side column graphs for two categorical variables, eg favourite television show of students in Year 1 compared to that of students in Year 6 {Literacy} | |

interpret and compare different displays of the same data set to determine the most appropriate display for the data set | |

– compare the effectiveness of different student-created data displays {Communicating} | |

– discuss the advantages and disadvantages of different representations of the same data {Communicating, Critical and creative thinking, Ethical understanding} | |

– explain which display is the most appropriate for interpretation of a particular data set {Communicating, Reasoning, Literacy, Critical and creative thinking} | |

– compare representations of the same data set in a side-by-side column graph and in a two-way table {Reasoning, Critical and creative thinking} |

Interpret secondary data presented in digital media and elsewhere (ACMSP148) | interpret data representations found in digital media and in factual texts {Literacy, Information and communication technology capability} |

– interpret tables and graphs from the media and online sources, eg data about different sports teams {Reasoning, Information and communication technology capability, Critical and creative thinking} | |

– identify and describe conclusions that can be drawn from a particular representation of data {Communicating, Reasoning, Literacy} | |

critically evaluate data representations found in digital media and related claims {Literacy, Information and communication technology capability. Critical and creative thinking, Personal and social capability, Ethical understanding} | |

– discuss the messages that those who created a particular data representation might have wanted to convey {Communicating, Literacy, Critical and creative thinking, Personal and social capability, Ethical understanding, Civics and citizenship} | |

– identify sources of possible bias in representations of data in the media by discussing various influences on data collection and representation, eg who created or paid for the data collection, whether the representation is part of an advertisement {Communicating, Reasoning, Literacy, Critical and creative thinking, Personal and social capability, Ethical understanding} | |

– identify misleading representations of data in the media, eg broken axes, graphics that are not drawn to scale {Reasoning, Literacy, Information and communication technology capability, Critical and creative thinking, Personal and social capability, Ethical understanding} |

Describe and interpret different data sets in context (ACMSP120) | interpret line graphs using the scales on the axes {Literacy} |

describe and interpret data presented in tables, dot plots, column graphs and line graphs, eg ‘The graph shows that the heights of all children in the class are between 125 cm and 154 cm’ {Literacy} | |

– determine the total number of data values represented in dot plots and column graphs, eg find the number of students in the class from a display representing the heights of all children in the class {Problem Solving, Reasoning} | |

– identify and describe relationships that can be observed in data displays, eg ‘There are four times as many children in Year 5 whose favourite food is noodles compared to children whose favourite food is chicken’ {Communicating, Reasoning, Literacy} | |

– use information presented in data displays to aid decision making, eg decide how many of each soft drink to buy for a school fundraising activity by collecting and graphing data about favourite soft drinks for the year group or school {Reasoning, Critical and creative thinking} |

## CHANCE

OUTCOME

A student:

MA3-19SP:

conducts chance experiments and assigns probabilities as values between 0 and 1 to describe their outcomes

TEACHING POINTS | Random generators include coins, dice, spinners and digital simulators. |

As the number of trials in a chance experiment increases, the observed probabilities should become closer in value to the expected probabilities. | |

Refer also to background information in Chance 1. |

LANGUAGE | Students should be able to communicate using the following language: chance, event, likelihood, equally likely, experiment, outcome, expected outcomes, random, fair, trials, probability, expected probability, observed probability, frequency, expected frequency, observed frequency. |

The term ‘frequency’ is used in this substrand to describe the number of times a particular outcome occurs in a chance experiment. In Stage 4, students will also use ‘frequency’ to describe the number of times a particular data value occurs in a data set. |

## EXPECTATIONS OF ATTAINMENT

Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies (ACMSP145) | assign expected probabilities to outcomes in chance experiments with random generators, including digital simulators, and compare the expected probabilities with the observed probabilities after both small and large numbers of trials {Information and communication technology capability} |

– determine and discuss the differences between the expected probabilities and the observed probabilities after both small and large numbers of trials {Communicating, Reasoning, Literacy} | |

– explain what happens to the observed probabilities as the number of trials increases {Communicating, Reasoning, Literacy} | |

use samples to make predictions about a larger ‘population’ from which the sample comes, eg take a random sample of coloured lollies from a bag, calculate the probability of obtaining each colour of lolly when drawing a lolly from the bag, and use these probabilities and the total number of lollies in the bag to predict the number of each colour of lolly in the bag {Critical and creative thinking} | |

– discuss whether a prediction about a larger population, from which a sample comes, would be the same if a different sample were used {Communicating, Reasoning, Literacy, Critical and creative thinking} |