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• VC2M9SP01

  recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity

  Elaborations

  • understanding the terms ‘base’, ‘altitude’, ‘hypotenuse’, and ‘adjacent’ and ‘opposite’ sides to an angle, in a right-angled triangle, and identifying these for a given right-angled triangle
  • investigating patterns to reason about nested similar triangles that are aligned on a coordinate plane, connecting ideas of parallel sides and identifying the constancy of ratios of corresponding sides for a given angle
  • establishing an understanding that the sine of an angle can be considered as the length of the altitude of a right-angled triangle with a hypotenuse of length one unit and similarly the cosine as the length of the base of the same triangle, and relating this to enlargement and similar triangles
  • relating the tangent of an angle to the altitude and base of nested similar right-angled triangles, and connecting the tangent of the angle at which the graph of a straight line meets the positive direction of the horizontal coordinate axis to the gradient of the straight line

  VC2M9SP01 | Mathematics | Mathematics Version 2.0 | Level 9 | Space

• VC2M10ST04

  analyse claims, inferences and conclusions of statistical reports in the media and other places, by linking claims to displays, statistics and representative data, including ethical considerations and identification of potential sources of bias

  Elaborations

  • identifying potentially misleading data representations in the media such as graphs with broken axes and scales that do not start at zero or are nonlinear; and recognising when data is not relating to the claim, not representative of the population or is deliberately being used to mislead or support a claim or biased point of view
  • investigating the source and size of the sample from which the data was collected and deciding whether the sample is appropriately representative of the population 
  • investigating population proportions and rates, and discussing potential ethical considerations, for example, when presenting statistical data involving infection rates and the number of cases per head of population
  • discussing the ethical considerations associated with collecting data to predict the number of people likely to be infected with a strain of flu or experience side effects with a certain medication, considering validity claims and sample sizes
  • using the concept of Indigenous data sovereignty to critique and evaluate the Australian Government’s ‘Closing the Gap’ report 

  VC2M10ST04 | Mathematics | Mathematics Version 2.0 | Level 10 | Statistics

• VC2M9ST04

  choose appropriate forms of display or visualisation for a given type of data; justify selections and interpret displays for a given context

  Elaborations

  • comparing data displays using mean, median and range to describe and interpret numerical data sets in terms of centre and spread using histograms, dot plots or stem-and-leaf plots
  • choosing the type of representations based on the data type: categorical (nominal or ordinal) or numerical (discrete or continuous)
  • using different visualisations of data, including non-standard representations such as infographics, and discussing their purpose and intended audience, and evaluating how well they communicate responses to statistical questions of interest
  • comparing and interpreting stacked bar charts, area charts and line graphs, discussing how they represent larger categories that can be subdivided into smaller categories and how information that can be obtained from these displays can be used for comparison

  VC2M9ST04 | Mathematics | Mathematics Version 2.0 | Level 9 | Statistics

• Mathematics | Mathematics: Learning in Mathematics

  The proficiencies of Understanding, Fluency, Problem Solving and Reasoning are fundamental to learning mathematics and working mathematically and are applied across all three strands Number and Algebra, Measurement and Geometry, and Statistics and Probability.

  Understanding refers to students...

  Overview material | Mathematics

• VC2M9N01

  recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers with and without using digital tools

  Elaborations

  • investigating the real number system by representing the relationships between irrational numbers, rational numbers, integers and natural numbers and discussing the difference between exact representations and approximate decimal representations of irrational numbers
  • using a real number line to indicate the solution interval for inequalities of the form ax+b <c ,
    for example, 2x+7<0 ; or of the form ax+b>c , for example, 1.2x-5.4>10.8
  • using positive and negative rational numbers to solve problems, for example, for financial planning such as budgeting
  • solving problems involving the substitution of real numbers into formulas, understanding that solutions can be represented in exact form or as a decimal approximation, such as calculating the area of a circle using the formula A=πr2 and specifying the answer in terms of π as an exact real number; for example, the circumference of a circle with diameter 5 units is 5π units, and the exact area is π(52)2=254π square units, which rounds to 19.63 square units, correct to 2 decimal places
  • investigating the position of rational and irrational numbers on the real number line, using geometric constructions to locate rational numbers and square roots on a number line; for example, 2 is located at the intersection of an arc and the number line, where the radius of the arc is the length of the diagonal of a one-unit square

  VC2M9N01 | Mathematics | Mathematics Version 2.0 | Level 9 | Number

• VC2M2N06

  use mathematical modelling to solve practical problems involving additive and multiplicative situations, including money transactions; represent situations and choose calculation strategies; interpret and communicate solutions in terms of the context

  Elaborations

  • modelling practical problems by interpreting an everyday additive or multiplicative situation; for example, making a number of purchases at a store and deciding whether to use addition, subtraction, multiplication or division to solve the problem and justifying the choice of operation, such as ‘I used subtraction to solve this problem as I knew the total and one of the parts, so I needed to subtract to find the missing part’
  • modelling and solving simple money problems involving whole dollar amounts with addition, subtraction, multiplication or division, for example, ‘If each member of our class contributes $5, how much money will we have in total?’
  • modelling and solving practical problems such as deciding how many people should be in each team for a game or sports event, how many teams for a given game can be filled from a class, or how to share out some food or distribute money in whole dollar amounts, including deciding what to do if there is a remainder
  • modelling and solving the problem ‘How many days are there left in this year?’ by using a calendar
  • modelling problems involving equal grouping and sharing in Aboriginal and/or Torres Strait Islander children’s instructive games; for example, in Yangamini from the Tiwi Peoples of Bathurst Island, representing relationships with a number sentence and interpreting and communicating solutions in terms of the context

  VC2M2N06 | Mathematics | Mathematics Version 2.0 | Level 2 | Number

• VC2M3N02

  recognise, represent and order natural numbers using naming and writing conventions for numerals beyond 10 000

  Elaborations

  • moving materials from one place to another on a place value model to show renaming of numbers (for example, 1574 can be shown as one thousand, 5 hundreds, 7 tens and 4 ones, or as 15 hundreds, 7 tens and 4 ones)
  • using the repeating pattern of place value names and spaces within sets of 3 digits to name and write larger numbers: ones, tens, hundreds, ones of thousands, tens of thousands, hundreds of thousands, ones of millions, tens of millions; for example, writing four hundred and twenty-five thousand as 425 000
  • predicting and naming the number that is one more than 99, 109, 199, 1009, 1099, 1999, 10 009 … 99 999 and discussing what will change when one, one ten and one hundred is added to each
  • comparing the Hindu-Arabic numeral system to other numeral systems; for example, investigating the Japanese numeral system, 一、十、百、千、万
  • comparing, reading and writing the numbers involved in more than 60 000 years of Aboriginal and Torres Strait Islander Peoples’ presence on the Australian continent through timescales relating to pre-colonisation and post-colonisation

  VC2M3N02 | Mathematics | Mathematics Version 2.0 | Level 3 | Number

• VC2M7N10

  use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts such as ‘best buys’; formulate problems, choosing representations and efficient calculation strategies, designing algorithms and using digital tools as appropriate; interpret...

  Elaborations

  • modelling additive situations involving positive and negative quantities; for example, a lift travelling up and down floors in a high-rise apartment where the ground floor is interpreted as zero, or in geography when determining altitude above and below sea level
  • modelling contexts involving proportion, such as the proportion of students attending the school disco, proportion of the bottle cost to the recycling refund, proportion of the school site that is green space, 55% of Year 7 students attended the end of term function or 23% of the school population voted ‘yes’ to a change of school uniform; and interpreting and communicating answers in terms of the context of the situation
  • modelling financial problems involving profit and loss, credits and debits, gains and losses; for example, holding a fundraising sausage sizzle and determining whether the event made a percentage profit or loss
  • finding the sum of a set of consecutive numbers using a loop structure
  • constructing geometric patterns such as a honeycomb, using dynamic geometry functionality
  • using mathematical modelling to investigate the proportion of land mass/area of Aboriginal Peoples’ traditional grain belt compared with Australia’s current grain belt

  VC2M7N10 | Mathematics | Mathematics Version 2.0 | Level 7 | Number

• VC2M3ST03

  conduct guided statistical investigations involving the collection, representation and interpretation of data for categorical and discrete numerical variables with respect to questions of interest

  Elaborations

  • creating a poster, flow chart or infographic that describes the process of statistical investigation, and the components, tools and types of data that can be collected, represented and interpreted for a purpose
  • collaboratively working through a whole-class investigation by choosing a question of interest, using an efficient collection method and recording collected data, and then interpreting the data in terms of the question
  • planning and carrying out investigations that involve collecting data; for example, narrowing the focus of a question such as ‘Which is the most popular breakfast cereal?’ to ‘Which is the most popular breakfast cereal among Year 3 students in our class?’
  • conducting a whole-class statistical investigation into the best day to hold an open day for parents by creating a simple survey, collecting the data by asking the parents, representing and interpreting the results, and deciding as a class which day would be best
  • investigating seasonal calendars of Aboriginal and/or Torres Strait Islander Peoples by collecting data and creating frequency tables and spreadsheets based on environmental indicators, and creating one-to-one data displays about frequency of environmental indicators for the current season

  VC2M3ST03 | Mathematics | Mathematics Version 2.0 | Level 3 | Statistics

• VC2M1ST01

  acquire and record data for categorical variables in various ways including using digital tools, objects, images, drawings, lists, tally marks and symbols

  Elaborations

  • discussing methods of collecting data to answer a question such as ‘What types of rubbish are found in the playground?’, sharing ideas and trying out some of the suggested methods, reviewing the data collected and explaining how they might change the way they collect data next time
  • collecting and recording information on a topic of interest using lists (for example, ‘How many people follow a particular football team’ or ‘What colour eyes each person has’), examining the data to generate some questions that it could answer, and then rearranging the data or collecting different data to answer the question
  • creating a tally to record data while observing events such as the year level of students using the bike shed, deciding on the possible categories before the observations are taken, and then reviewing the data afterwards to notice whether the tally was effective
  • using star charts with stickers or emojis to represent class data; for example, using emojis on a personal feeling chart to represent how they are feeling each day or using emojis to represent activities on the class calendar
  • exploring ways of representing, sharing and communicating data through stories and symbols used by Aboriginal and Torres Strait Islander Peoples

  VC2M1ST01 | Mathematics | Mathematics Version 2.0 | Level 1 | Statistics

• VC2M10AN02

  perform operations on numbers involving fractional exponents and surds

  Elaborations

  • exploring the notion that amn= a1nm= anm= amn=am1n , where m is a non-zero integer and n is a natural number, and evaluating corresponding expressions; for example, 823=832= 22=4  and 823= 823=643=4

  VC2M10AN02 | Mathematics | Mathematics Version 2.0 | Level 10A | Number

• VC2M8A01

  create, expand, factorise, rearrange and simplify linear expressions, applying the associative, commutative, identity, distributive and inverse properties

  Elaborations

  • rearranging and simplifying linear expressions involving variables with integer coefficients and constants; and using manipulatives such as algebra tiles to support calculations, for example, using manipulatives to demonstrate that 2x+4=2x+2 , 3a-b=3a-3b, or 5m+2n+3m-4n= 5m+10n+3m-4n=8m+6n
  • demonstrating the relationship between factorising and expanding linear expressions using manipulatives, such as algebra tiles or area models, and describing with mathematical language
  • using the distributive, associative, commutative, identity and inverse properties to expand and factorise algebraic expressions using strategies such as the area model

  VC2M8A01 | Mathematics | Mathematics Version 2.0 | Level 8 | Algebra

• VC2M6A02

  find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations

  Elaborations

  • using brackets and the order of operations to write number sentences and appreciating the need for an agreed set of rules to complete multiple operations within the same number sentence; for example, for 40 ÷ 2 × (4 + 6) = □, you solve what is in the brackets first then complete the number sentence from left to right as there is no hierarchy between division and multiplication
  • constructing equivalent number sentences involving brackets and combinations of the 4 operations, and explaining the need to have shared agreement on the order of operations when solving problems involving more than one operation to have unique solutions
  • finding pairs of unknown values in numerical equations that make the equation hold true; for example, listing possible combinations of natural numbers that make this statement true: 6 + 4 × 8 = 6 × Δ + □

  VC2M6A02 | Mathematics | Mathematics Version 2.0 | Level 6 | Algebra

• VC2M3A03

  recall and demonstrate proficiency with multiplication facts for 3, 4, 5 and 10; extend and apply facts to develop the related division facts

  Elaborations

  • using concrete or virtual materials, groups and repeated addition to recognise patterns and establish the 3, 4, 5 and 10 multiplication facts; for example, using the language of ‘3 groups of 2 equals 6’ to develop into ‘3 twos are 6’ and extend to establish the 3 × 10 multiplication facts and related division facts
  • recognising that when they multiply a number by 5, the resulting number will either end in a 5 or a zero; and using a calculator or spreadsheet to generate a list of the multiples of 5 to develop the multiplication and related division facts for fives
  • practising calculating and deriving multiplication facts for 3, 4, 5 and 10, explaining and recalling the patterns in them and using them to derive related division facts
  • systematically exploring algorithms used for repeated addition, comparing and describing what is happening, and using them to establish the multiplication facts for 3, 4, 5 and 10; for example, following the sequence of steps, the decisions being made and the resulting solution, recognising and generalising any emerging patterns

  VC2M3A03 | Mathematics | Mathematics Version 2.0 | Level 3 | Algebra

• VC2M10AP01

  explore counting principles, and factorial notation as a representation that provides efficient counting in multiplicative contexts, including calculations of probabilities

  Elaborations

  • applying the multiplication principle to problems involving combinations including probabilities relating to sampling with and without replacement, and representing these using tree diagrams
  • understanding that a set with n elements has 2ⁿ different subsets formed by considering each element for inclusion or not in combination, and that these can be systematically listed using a tree diagram or a table; for example, the set {a, b, c} has 2³ = 8 subsets which are { ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
  • using the definition of n! to represent and calculate in contexts that involve choices from a set; for example, how many different combinations of 3 playing cards from a pack? How many if the suits are ignored? How many with and without replacement?
  • performing calculations on numbers expressed in factorial form, such as n!r! to evaluate the number of possible arrangements of n objects in a row, r of which are identical; for example, 5 objects, 3 of which are identical, can be arranged in a row in
    5!3!=5×4×3×2×13×2×1= 20 different ways

  VC2M10AP01 | Mathematics | Mathematics Version 2.0 | Level 10A | Probability

• VC2M5M01

  choose appropriate metric units when measuring the length, mass and capacity of objects; use smaller units or a combination of units to obtain a more accurate measure

  Elaborations

  • ordering metric units from the largest unit to the smallest, for example, kilometre, metre, centimetre, millimetre
  • recognising that some units of measurement are better suited to some tasks than others; for example, kilometres are more appropriate than metres to measure the distance between 2 towns
  • deciding on the unit required to estimate the amount of paint or carpet for a room or a whole building, and justifying the choice of unit in relation to the context and the degree of accuracy required
  • measuring and comparing distances (for example, measuring and comparing jumps or throws using a metre length of string and then measuring the part metre with centimetres and/or millimetres) and explaining which unit of measure is most accurate
  • researching how the base units are derived for the International System of Units (SI), commonly known as the metric system of units, recognising that the metric unit names for the attributes of length and mass are international standards for measurement

  VC2M5M01 | Mathematics | Mathematics Version 2.0 | Level 5 | Measurement

• VC2M3M05

  identify angles as measures of turn and use right angles as a reference to compare angles in everyday situations

  Elaborations

  • using quarter, half and three-quarter turns and comparing them to a right angle, for example, a quarter turn is the same as a right angle; a half a turn is greater than a right angle and is the same as 2 right angles; a three-quarter turn is greater than a right angle and is the same as 3 right angles
  • recognising that right angles occur at the corners of many everyday objects, for example, books, windows, tabletops and whiteboards
  • identifying angles that are bigger than, smaller than and the same as a right angle in the environment; for example, opening doors partially and fully and comparing the angles created to a right angle
  • exploring Aboriginal and/or Torres Strait Islander children’s instructive games to investigate angles as measures of turn, for example, in the game Waayin from the Datiwuy People in the northern part of the Northern Territory

  VC2M3M05 | Mathematics | Mathematics Version 2.0 | Level 3 | Measurement

• VC2M7M04

  identify corresponding, alternate and co-interior relationships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons

  Elaborations

  • constructing a pair of parallel lines and a pair of perpendicular lines using their properties, a pair of compasses and a ruler and set squares, or using dynamic geometry software
  • using dynamic geometry software to identify relationships between alternate, corresponding and co-interior angles for a pair of parallel lines cut by a transversal
  • using dynamic geometry software to demonstrate how angles and their properties are involved in the design and construction of scissor lifts, folding umbrellas, toolboxes and cherry pickers
  • using geometric reasoning of angle properties to generalise the angle relationships of parallel lines and transversals, and related properties, such as the size of an exterior angle of a triangle is equal to the sum of the sizes of opposite and non-adjacent interior angles, and the sum of the sizes of interior angles in a triangle in the plane is equal to the size of 2 right angles or 180°

  VC2M7M04 | Mathematics | Mathematics Version 2.0 | Level 7 | Measurement

• VC2M3N09

  follow and create algorithms involving a sequence of steps and decisions to investigate numbers; describe any emerging patterns

  Elaborations

  • following or creating an algorithm to generate number patterns formed by doubling and halving using technology to assist, where appropriate, and identifying and describing emerging patterns
  • following or creating an algorithm that determines whether a given number is a multiple of 2, 5 or 10, identifying and discussing emerging patterns
  • creating an algorithm as a set of instructions that a classmate can follow to generate multiples of 3 using the rule ‘To multiply by 3 you double the number and add on one more of the number’ (for example, for 3 threes, you double 3 and add on 3 to get 9; for 3 fours you double 4 and add one more 4 to get 12 …)
  • creating a sorting algorithm that will sort a collection of 5-cent and 10-cent coins and providing the total value of the collection by applying knowledge of multiples of 5 and 10

  VC2M3N09 | Mathematics | Mathematics Version 2.0 | Level 3 | Number

• VC2M7N06

  use the 4 operations with positive rational numbers, including fractions and decimals, to solve problems using efficient mental and written calculation strategies

  Elaborations

  • solving addition and subtraction problems involving fractions and decimals; for example, using rectangular arrays with dimensions equal to the denominators, algebra tiles, digital tools or informal jottings
  • choosing an appropriate numerical representation for a problem so that efficient computations can be made, such as 12.5%, 18, 0.125 or 1251000
  • developing efficient strategies with appropriate use of the commutative and associative properties, place value, patterning, and multiplication facts to solve multiplication and division problems involving fractions and decimals; for example, using the commutative property to calculate 23 of 12 , giving 12 of 23=13
  • solving multiplicative problems involving fractions and decimals using fraction walls, rectangular arrays, algebra tiles, calculators or informal jottings
  • developing efficient strategies with appropriate use of the commutative and associative properties, regrouping or partitioning to solve additive problems involving fractions and decimals
  • carrying out calculations to solve problems using the representation that makes computations efficient, such as 12.5% of 96 is more efficiently calculated as 18 of 96, including contexts such as comparing land use by calculating the total local municipal area set aside for parkland or manufacturing and retail, the amount of protein in daily food intake across several days, or increases or decreases in energy accounts each account cycle

  VC2M7N06 | Mathematics | Mathematics Version 2.0 | Level 7 | Number