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• 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

• VC2M7N02

  represent natural numbers in expanded notation using powers of 10, and as products of powers of prime numbers using exponent notation

  Elaborations

  • relating the sequences 10, 100, 1000, 10 000 … and 10¹, 10², 10³, 10⁴ …
  • applying and explaining the connections between place value and expanded notations, for example, 7000=7×103 and 3750=3×103+7×102+5×101
  • applying knowledge of factors to strategies for expressing natural numbers as products of powers of prime factors, such as repeated division by prime factors or creating factor trees, for example, 48=6×8=2×3×2×2×2=31×24=3×24
  • developing familiarity with the sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 and powers of 2; the sequence 1, 3, 9, 27, 81, 243, 729 and powers of 3; and the sequence 1, 5, 25, 125, 625 and powers of 5
  • solving problems involving lowest common multiples and greatest common divisors (highest common factors) for pairs of natural numbers by comparing their prime factorisation

  VC2M7N02 | Mathematics | Mathematics Version 2.0 | Level 7 | 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

• VC2M9A02

  simplify algebraic expressions, apply the distributive law to expand algebraic expressions including binomial products, and factorise monic quadratic expressions

  Elaborations

  • expanding combinations of binomials such as x+7x+8, x+7x-8,  x-7x+8,  (x-7)(x-8) to identify expansion and factorisation patterns related to x+ax+b=x2+a+bx+ab, where a and b are integers
  • using manipulatives such as algebra tiles or area models to expand or factorise algebraic expressions with readily identifiable binomial factors; for example, x+1x+3=x2+4x+3, (x-5)2=x2-10x+25 or (x-3)2+4=x2-6x+9+4=x2-6x+13
  • recognising the relationship between expansion and factorisation, and using digital tools to systematically explore the factorisation of x2+mx+n where m and n are integers

  VC2M9A02 | Mathematics | Mathematics Version 2.0 | Level 9 | Algebra

• VC2M4A02

  recall and demonstrate proficiency with multiplication facts up to 10 × 10 and related division facts, and explain the patterns in these; extend and apply facts to develop efficient mental and written strategies for computation with larger numbers without a calculator

  Elaborations

  • using arrays on grid paper or created with blocks or counters to develop, represent and explain patterns in multiplication facts up to 10 × 10; and using the arrays to explain the related division facts
  • using materials or diagrams to develop and record multiplication strategies such as doubling, halving, commutativity and adding one more or subtracting from a group to reach a known fact; for example, creating multiples of 3 on grid paper and doubling to find multiples of 6, and recording and explaining the connections to the × 3 and × 6 multiplication facts: 3, 6, 9, … doubled is 6, 12, 18, …
  • using known multiplication facts for 2, 3, 5 and 10 to establish multiplication facts for 4, 6, 7, 8 and 9 in different ways; for example, using multiples of 10 to establish the multiples of 9 as ‘to multiply a number by 9 you multiply by 10 then take the number away’: 9 × 4 = 10 × 4 − 4, so 9 × 4 is 40 − 4 = 36; or using multiples of 3 as ‘to multiply a number by 9 you multiply by 3, and then multiply the result by 3 again’
  • using arrays and known multiplication facts for twos and fives to develop the multiplication facts for sevens, applying the distributive property of multiplication; for example, when finding 6 × 7, knowing that 7 is made up of 2 and 5, and using an array to show that 6 × 7 is the same as 6 × 2 + 6 × 5 = 12 + 30, which is 42

  VC2M4A02 | Mathematics | Mathematics Version 2.0 | Level 4 | Algebra

• VC2M7N09

  recognise, represent and solve problems involving ratios

  Elaborations

  • using diagrams, physical or virtual materials to represent ratios, recognising that ratios express the quantitative relationship between 2 or more groups; for example, using counters or coloured beads to show the ratios 1:4 and 1:1:2
  • using fractions to solve ratio problems involving comparison of quantities and considering part-part and part-whole relations; for example, dividing a set into the ratio 1:2 by determining the number of parts as 3
  • sharing quantities in a given ratio; for example, sharing an amount of money in a given ratio, such as sharing $20 in the ratio 2:3
  • applying ratios to realistic and meaningful contexts – for example, mixing 500 millilitres of a liquid with a concentration of 1:4 means 15 concentrate and 45 water so, 0.2 of 500 millilitres is concentrate and 0.8 of 500 millilitres is water – and interpreting results in context

  VC2M7N09 | Mathematics | Mathematics Version 2.0 | Level 7 | Number

• VC2M4N03

  find equivalent representations of fractions using related denominators and make connections between fractions and decimal notation

  Elaborations

  • extending fraction families within collections of materials, for example, by seeing 34 as 3 in each 4, showing this within related fractions like 68 or seeing that 25 means 2 in each 5 so it can be shown within 410
  • creating models of equivalent fractions by subdividing capacity measures into smaller fractions; for example, half a cup of flour could be shown as two-quarters or four-eighths of a cup of flour
  • folding paper to show equivalence between different fractions; for example, folding A4 paper in half and half again, repeating to form eighths and demonstrating that 48=24=12 ; or folding paper strips into fifths and tenths, and recording as both fractions and decimals
  • identifying and using the connection between fractions of metres and decimals; for example, finding 14 of a metre and connecting this to 0.25 metres or 25 centimetres, or finding 110 of a metre and connecting this with 0.10 metres or 10 centimetres
  • using array diagrams to show the relationship between fractions and division and multiplication of natural numbers, for example, 3 × 4 = 12, 12 ÷ 4 = 3, 14 of 12 is 3, 13 of 12 is 4

  VC2M4N03 | Mathematics | Mathematics Version 2.0 | Level 4 | 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

• 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

• 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

• VC2M9A01

  apply the exponent laws to numerical expressions with integer exponents and the zero exponent, and extend to variables

  Elaborations

  • representing decimals in exponential form; for example, 0.475 can be represented as 0.475=410+7100+51000=4×10-1+7×10-2+5×10-3 and 0.00023 as 23 × 10^-5
  • simplifying and evaluating numerical expressions, involving both positive and negative integer exponents, explaining why; for example, 5-3=153=(15)3=1125 and connecting terms of the sequence 125, 25, 5, 1, 15, 125, 1125… to terms of the sequence 53, 52,51, 50, 5-1, 5-2, 5-3...
  • relating the computation of numerical expressions involving exponents to the exponent laws and the definition of an exponent; for example, 23÷25=2-2=122=14 and (3×5)2=32×52=9×25=225
  • recognising exponents in algebraic expressions and applying the relevant exponent laws and conventions; for example, for any non-zero natural number a, a0=1, x1=x,  r2=r×r,  h3=h×h×h,  y4=y×y×y×y, and 1w×1w=1w2=w-2
  • relating simplification of expressions from first principles and counting to the use of exponent laws; for example,
    (a2)3=a×a×a×a×a×a=a×a×a×a×a×a=a6;
    b2×b3=b×b×b×b×b=b×b×b×b×b=b5; y4y2=y×y×y×yy×y=y21=y2 and (5a)2=5×a×5×a=5 ×5×a×a=25×a2=25a2 
  • applying the exponent laws to simplifying expressions involving products, quotients, and powers of constants and variables; for example, (2xy)3xy4=8x3y3xy4=8x2y-1
  • relating the prefixes for SI units from pico- (trillionth) to tera- (trillion) to the corresponding powers of 10; for example, one pico-gram = 10^-12 grams and one terabyte = 10¹² bytes

  VC2M9A01 | Mathematics | Mathematics Version 2.0 | Level 9 | Algebra