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• VC2M7N08

  compare, order and solve problems involving addition and subtraction of integers

  Elaborations

  • using less-than and greater-than notation in expressions when comparing and ordering integers; for example, negative 5 is less than positive 2 and can be represented as -5<+2; -3>-6
  • discussing language such as ‘addition’, ‘subtraction’, ‘magnitude’, ‘difference’ and ‘sign’ and synonyms of these terms
  • ordering, adding and subtracting integers using a number line

  VC2M7N08 | Mathematics | Mathematics Version 2.0 | Level 7 | Number

• VC2M7A03

  solve one-variable linear equations of increasing complexity with natural number solutions; verify equation solutions by substitution

  Elaborations

  • recognising that solving an equation is a process of determining a value that makes the equation true; and using substitution to determine whether a given number is a solution to an equation or not
  • solving equations using concrete materials, the balance model and backtracking, explaining the process
  • solving linear equations such as 3x+7=19 algebraically, and verifying the solution by substitution

  VC2M7A03 | Mathematics | Mathematics Version 2.0 | Level 7 | Algebra

• VC2M5N04

  recognise that 100% represents the complete whole and use percentages to describe, represent and compare relative size; connect familiar percentages to their decimal and fraction equivalents

  Elaborations

  • recognising applications of percentages used in everyday contexts, for example, the bar model used for charging devices indicating the percentage of power remaining, and advertising in retail contexts relating to discounts or sales
  • creating a model by subdividing a whole (for example, using 10 × 10 grids to represent various percentage amounts) and recognising complementary percentages (such as 30% and 70%) combine to make 100%
  • creating a model by subdividing a collection of materials, such as blocks or money, to connect decimals and percentage equivalents of tenths and the commonly used fractions 12, 14 and 34; for example, connecting that one-tenth or 0.1 represents 10% and one-half or 0.5 represents 50%, and recognising that 60% of a whole is 10% more of the whole than 50%
  • using physical and virtual materials to represent the relationship between decimal notation and percentages, for example, 0.3 is 3 out of every 10, which is 30 out of every 100, which is 30%

  VC2M5N04 | Mathematics | Mathematics Version 2.0 | Level 5 | Number

• VC2M8P01

  recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts

  Elaborations

  • understanding that knowing the probability of an event allows the probability of its complement to be found, including for those events that are not equally likely, such as getting a specific novelty toy in a supermarket promotion
  • using the relationship that for a single event A, Pr(A) + Pr(not A) = 1; for example, if the probability that it rains on a particular day is 80%, the probability that it does not rain on that day is 20%, or the probability of not getting a 6 on a single roll of a fair dice is 1-16=56
  • using the sum of probabilities to solve problems, such as the probability of starting a game by throwing a 5 or 6 on a dice is 13 and probability of not throwing a 5 or 6 is 23

  VC2M8P01 | Mathematics | Mathematics Version 2.0 | Level 8 | Probability

• 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

• 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

• VC2M9A06

  use mathematical modelling to solve applied problems involving change, including financial contexts involving simple interest; formulate problems, choosing to use either linear or quadratic functions or other simple variations; interpret solutions in terms of the context; evaluate the model and...

  Elaborations

  • modelling practical contexts using linear functions such as cooking times that include resting or cooling times, or water leakage from water tanks, using tables and graphs or digital tools and algebraically
  • modelling measurement situations and determining the perimeter and areas of rectangles where the length, l, of the rectangle is a linear function of its width, w; for example, l=w,  l=w+5,  l=3w,  l=2w+7
  • modelling practical contexts (for example, area, paths of projectiles, parabolic mirrors, satellite dishes) using simple quadratic functions, tables and graphs (hand drawn or using digital tools) and, algebraically, interpreting features of the graphs such as the turning point and intercepts in context; for example, area, paths of projectiles, parabolic mirrors, satellite dishes
  • modelling and solving problems involving financial contexts using linear functions, for example, combinations of purchases of different items when they have a set amount of money to spend, profit/loss situations and trade quotes involving call-out fees
  • modelling situations involving change, for example, change in daily temperature during the ski season, fluctuation of speed above and below the speed limit, and acceleration and deceleration of a car coming to and moving off from a set of traffic lights
  • modelling the hunting techniques of Aboriginal and Torres Strait Islander Peoples using quadratic functions and exploring the effect of increasing the number of hunters to catch more prey 

  VC2M9A06 | Mathematics | Mathematics Version 2.0 | Level 9 | Algebra

• VC2M10AA04

  explore the inverse relationship between exponential functions and logarithmic functions and the solution of related equations

  Elaborations

  • using the definition of a logarithm and the exponent laws to establish the logarithm laws
  • evaluating 10x for decimal values of x and relating this to a logarithm base 10 scale; solving exponential equations algebraically using logarithms with base 10, for example, 5000×1.01x=10 000⇒x= log10(2)log10(1.01)≈69.66 ; and connecting to the graph of the corresponding function

  VC2M10AA04 | Mathematics | Mathematics Version 2.0 | Level 10A | Algebra

• VC2M9SP02

  apply the enlargement transformation to shapes and objects using dynamic geometry software as appropriate; identify and explain, using language of similarity, ratio and scale, aspects that remain the same and those that change

  Elaborations

  • comparing the ratio of lengths of corresponding sides of similar triangles and angles 
  • using the properties of similarity to solve problems involving enlargement
  • investigating and generalising patterns in length, angle, area and volume when side lengths of shapes and objects are enlarged or dilated by whole and rational numbers; for example, comparing an enlargement of a square and a cube of side length 2 units by a factor of 3 increases the area of the square, (3×2)2=9×22=9 times the original area and the volume of the cube, 23, to (3×2)3=27×23=27 times the volume

  VC2M9SP02 | Mathematics | Mathematics Version 2.0 | Level 9 | Space

• VC2M10A16

  solve equations graphically or using systematic numerical guess-check-and-refine with digital tools, with consideration of whether all solutions have been found

  Elaborations

  • refining intervals on graphs and/or in tables of values to determine with increasing accuracy when the values of 2 functions are approximately equal
  • applying the graphing zoom functionality of digital tools and systematically refining intervals to identify approximate location of points of intersection of the graphs of 2 functions, such as x2=2x

  VC2M10A16 | Mathematics | Mathematics Version 2.0 | Level 10 | Algebra

• VC2M10A08

  solve linear inequalities and graph their solutions on a number line

  Elaborations

  • representing word problems with simple linear inequalities and solving them to answer questions
  • graphing regions corresponding to inequalities in the Cartesian plane (for example, graphing 2x+3y<24) and verifying using a test point such as (0, 0)

  VC2M10A08 | Mathematics | Mathematics Version 2.0 | Level 10 | Algebra

• 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

• VC2M10AA10

  experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns

  Elaborations

  • applying a bisection algorithm to determine the approximate location of the horizontal axis intercepts of the graph of a quadratic function such as fx=2x2-3x-7
  • applying transformations to the graph of x2+y2=1
  • identifying the coordinates of any points of intersection of the graph of a linear function with the graph of a quadratic function or a circle
  • identifying intervals on the real number line over which a given quadratic function is positive or negative
  • using a table of values to determine when an exponential growth or decay function exceeds or falls below a given value, such as monitoring the trend in value of a share price in a context of exponential growth or decay

  VC2M10AA10 | Mathematics | Mathematics Version 2.0 | Level 10A | 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

• VC2M6N08

  approximate numerical solutions to problems involving rational numbers and percentages, using appropriate estimation strategies

  Elaborations

  • using familiar fractions, decimals and percentages to approximate calculations, such as 0.3 of 180 is about 13 of 180, or 52% is about 12
  • choosing appropriate estimation strategies including rounding to the nearest whole number, knowledge of multiples of 2, 5 or 10 and partitioning numbers, in contexts such as measuring or cost per unit
  • recognising the effect of rounding on calculations involving fractions or decimals and saying what numbers the answer will be between
  • recognising the usefulness of estimation to check calculations for contexts such as dividing wood into a number of lengths, calculating cost per unit, reducing a recipe or dividing the cost of dinner for a group into individual amounts
  • verifying solutions by estimating percentages in suitable contexts, such as discounts, using common percentages of 10%, 25%, 30%, 50% and 1%
  • investigating estimation strategies to make decisions about steam cooking in ground ovens by Aboriginal and/or Torres Strait Islander Peoples, including catering for different numbers of people and resources needed for cooking

  VC2M6N08 | Mathematics | Mathematics Version 2.0 | Level 6 | Number

• 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

• 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

• VC2M7N01

  describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems

  Elaborations

  • investigating squares of natural numbers from one to 20, and connecting them to visual representations such as dots arranged in a square pattern
  • using the square and square root notation, and the distributive property and area diagrams, to calculate the squares of two-digit numbers; for example, 432=40+32=402+2×40×3+32=1600+240+9=1849
  • determining between which 2 consecutive natural numbers the square root of a given number lies; for example, 43 is between the square numbers 36 and 49 so 43 is between 36 and 49 and therefore between 6 and 7
  • generating a list of perfect square numbers and describing any emerging patterns, for example, the last digit of perfect square numbers, or the difference between consecutive square numbers, and recognising the constant second difference
  • using the relationship between perfect square numbers and their square roots to determine the perimeter of a square tiled floor using square tiles; for example, an area of floor with 144 square tiles has a perimeter of 48 tile lengths

  VC2M7N01 | Mathematics | Mathematics Version 2.0 | Level 7 | Number

• VC2M8N02

  establish and apply the exponent laws with positive integer exponents and the zero exponent, using exponent notation with numbers

  Elaborations

  • recognising the connection between exponent form and expanded form with the exponent laws of product of powers rule, quotient of powers rule, and power of a power rule, for example, 23×22 can be represented as 2×2×2×2×2=25 , and connecting the result to the addition of exponents
  • applying the exponent laws of the product of powers rule, quotient of powers rule, power of a power rule and zero exponent individually and in combination; for example, using exponents to determine the effect on the volume of a 2 centimetre cube when the cube is enlarged to a 6 centimetre cube, 6323= 23×3323=33, so the volume is increased by a factor of 27
  • using digital tools to systematically explore the application of the exponent laws; observing that the bases need to be the same
  • using examples such as 3434=1, and 34-4=30 to illustrate the necessity that for any non-zero natural number n,  n0=1

  VC2M8N02 | Mathematics | Mathematics Version 2.0 | Level 8 | Number

• VC2M1N04

  add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies

  Elaborations

  • using drawings, physical and virtual materials, and number combinations within 10 to add and subtract collections to 20
  • adding and subtracting numbers within 20, using a variety of representations and strategies, such as counting on, counting back, partitioning and part-part-whole knowledge of numbers to 10; for example, using partitioning and combining 7+5=7+3+2=10+2=12
  • developing and using strategies for one-digit addition and subtraction based on part-part-whole relationships for each of the numbers to 10 and subitising with physical and virtual materials; for example, 8 and 6 is the same as 8 and 2 and 4
  • representing story problems involving addition and subtraction of numbers within 20 using a Think Board; recognising and using + and − symbols and the equal sign (=) to represent the operations of addition and subtraction; and showing and explaining the connections between any materials used using the language of plus and minus, and the numbers within the story problem
  • creating and performing addition and subtraction stories told through Aboriginal and/or Torres Strait Islander dances

  VC2M1N04 | Mathematics | Mathematics Version 2.0 | Level 1 | Number