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• VC2M7M03

  describe the relationship between π and the circumference, radius and diameter of a circle

  Elaborations

  • recognising the features of circles and their relationships to one another; for example, labelling the parts of a circle including centre, radius, diameter and circumference and using one of radius, diameter or circumference to determine the measure of the other 2; and understanding that the diameter of a circle is twice the radius, or that the radius is the circumference divided by 2π
  • comparing the circumference of circles in relation to their radius and diameter with materials and measuring, to establish measurement formulas; for example, using a compass to draw several circles, then using string to approximate the circumference, comparing the length of string to the diameter of the circle
  • investigating π as the constant in the proportional relationship between the circumference of a circle and its diameter, and historical approximations from different civilisations, including Egypt, Babylon, Greece, India and China
  • investigating the applications and significance of circles in everyday life of Aboriginal and Torres Strait Islander Peoples, such as in basketry, symbols and architecture, recognising the relationships between the centre, radius, diameter and circumference

  VC2M7M03 | Mathematics | Mathematics Version 2.0 | Level 7 | Measurement

• VC2M6N07

  solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies with and without digital tools

  Elaborations

  • explaining how 13 of a quantity can be achieved by dividing by 3, and how knowledge of 13 of a quantity can be used to find 23 or 43 of the same quantity, using situations involving money, length, duration, mass or capacity
  • investigating percentage discounts of 10%, 25% and 50% in an online toy sale, using their equivalent decimal representations of 0.10, 0.25 and 0.50 to calculate the amount of discount on sale items, with and without digital tools
  • linking percentages to their decimal equivalent of tenths and hundredths and using these to determine percentage discounts; for example, finding a 30% discount by using its equivalence to 0.3, dividing by 10 and multiplying the result by 3 to give 30%
  • explaining the equivalence between percentages and fractions, for example, 3313 % and 13 , keeping to percentages that are equivalent to fractions with small denominators such as 6623 % and 12.5%
  • representing a situation with a mathematical expression, for example, numbers and symbols such as 14 × 24, that involves finding a familiar fraction or percentage of a quantity; and using mental strategies or a calculator and explaining the result in terms of the situation in question

  VC2M6N07 | Mathematics | Mathematics Version 2.0 | Level 6 | Number

• Mathematics Version 2.0: Level 8 description

  In Level 8, learning in Mathematics builds on each student’s prior learning and experiences. Students engage in a range of approaches to the learning and doing of mathematics that develop their understanding of and fluency with concepts, procedures and processes by making connections...

  Level description | Mathematics | Mathematics Version 2.0

• VC2M7A01

  recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown

  Elaborations

  • linking variables to attributes and measures being modelled when using formulas, such as the area of a rectangle is equal to the length × width as
    A=l×w; or using p=6g+b to describe a total of points expressed as goals (worth 6 points) and behinds (worth one point)
  • interpreting and using formulas obtained from other sources, for example, maximum heart rates and target heart rates for moderate exercise
  • substituting numerical values for variables when using formulas and calculating the value of an unknown in practical situations; for example, calculating weekly wage W given base wage b and overtime hours h at 1.5 times rate r, W=b+1.5 ×h ×r, using values for mass m and volume v to determine density d of a substance where d=mv
  • using everyday formulas and their application to contexts on Country/Place, investigating the relationships between variables

  VC2M7A01 | Mathematics | Mathematics Version 2.0 | Level 7 | Algebra

• VC2M7N03

  find equivalent representations of rational numbers and represent positive and negative rational numbers and mixed numbers on a number line

  Elaborations

  • investigating equivalence of fractions using common multiples and a fraction wall, diagrams or a number line to show that a fraction such as 23 is equivalent to 46 and 69 and therefore 23 < 56
  • expressing a fraction in simplest form using common divisors
  • applying and explaining the equivalence between fraction, decimal and percentage representations of rational numbers, for example, 16%, 0.16, 16100 and 425, using manipulatives, number lines or diagrams
  • representing positive and negative fractions and mixed numerals on various intervals of the real number line, including intervals that are not symmetrical about zero

  VC2M7N03 | Mathematics | Mathematics Version 2.0 | Level 7 | Number

• VC2M6N03

  apply knowledge of equivalence to compare, order and represent common fractions, including halves, thirds and quarters, on the same number line and justify their order

  Elaborations

  • applying factors and multiples to fraction denominators (such as halves with quarters, eighths and twelfths, and thirds with sixths, ninths and twelfths) to determine equivalent representations of fractions in order to make comparisons
  • representing fractions on the same number line, paying attention to relative position, and using this to explain relationships between denominators
  • explaining equivalence and order between fractions using number lines, drawings and models
  • comparing and ordering fractions by placing cards on a string line across the room and referring to benchmark fractions to justify their position; for example, 58 is greater than 12 can be written as 58 > 12 , because half of 8 is 4; 16 is less than 14 , because 6 > 4 and can be written as 16 < 14

  VC2M6N03 | Mathematics | Mathematics Version 2.0 | Level 6 | Number

• VC2M8A02

  graph linear relations on the Cartesian plane using digital tools where appropriate; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution

  Elaborations

  • recognising that in a table of values, if the first difference between consecutive values of the dependent variable is constant, then it is a linear relation
  • graphing linear functions and relations of the form x=a , y=a , x ≤a , x>a , y ≤a , y>a on the Cartesian plane for known values of a
  • completing a table of values, plotting the resulting points on the Cartesian plane and determining whether the relationship is linear
  • graphing the linear relationship ax+b=c for given values of a, b and c and identifying from the graph where ax+b <c or where ax+b >c
  • solving linear equations of the form ax+b=c and one-variable inequalities of the form ax+b <c or ax+b >c where a > 0 using inverse operations and digital tools, and checking solutions by substitution
  • solving linear equations such as 3x+7=6x-9 , representing these graphically, and verifying solutions by substitution

  VC2M8A02 | Mathematics | Mathematics Version 2.0 | Level 8 | Algebra

• VC2M8N03

  convert between fractions and terminating or recurring decimals, using digital tools as appropriate

  Elaborations

  • identifying when a fraction has a terminating decimal expansion from the prime factorisation of its denominator; for example, 724=0.2916_ does not have a terminating decimal expansion, while 725=0.28 does
  • identifying terminating, recurring and non-terminating decimals and choosing their appropriate representations such as 13 is represented as 0.3_

  VC2M8N03 | Mathematics | Mathematics Version 2.0 | Level 8 | 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

• VC2M7A02

  apply the associative, commutative and distributive laws to aid mental and written computation, and formulate algebraic expressions using constants, variables, operations and brackets

  Elaborations

  • simplifying calculations
  • forming simple estimates for calculations involving multiple and/or combined operations
  • generalising arithmetic expressions to algebraic expressions involving constants, variables, operations and brackets; for example, 7+7+7=3×7, x+x+x=3×x noting that 3x includes implied multiplication and recognising the difference between 3x+4 and 3x+4
  • formulating algebraic expressions that represent mathematical relationships; for example, translating from words to symbols in ‘think of a number’ type activities
  • recognising and applying the concept of variable as something that can change in value, investigating the relationships between variables, and the application to processes on Country/Place, including how cultural expressions of Aboriginal and Torres Strait Islander Peoples, such as storytelling, communicate mathematical relationships that can be represented as mathematical expressions

  VC2M7A02 | Mathematics | Mathematics Version 2.0 | Level 7 | Algebra

• VC2M9A07

  experiment with the effects of the variation of parameters on graphs of related functions, using digital tools, making connections between graphical and algebraic representations, and generalising emerging patterns 

  Elaborations

  • investigating transformations of the graph of y=x to the graph of y=ax+b by systematic variation of a and b and interpreting the effects of these transformations using digital tools; for example,
    y=x→y=2x (vertical enlargement as a>1) →y=2x-1 (vertical translation) and
     y=x→y=12x (vertical compression as 0<a<1) →y=-12x (reflection in the horizontal axis) →y=-12x+3 (vertical translation)
  • investigating transformations of the parabola y=x2 to the graph of y=ax-h2+b in the Cartesian plane using digital tools to determine the relationship between graphical and algebraic representations of quadratic functions, including the completed square form; for example,
    y=x2→y=13x2 (vertical compression as 0<a<1) →y=13(x-5)2 (horizontal translation) →y=13(x-5)2+7 (vertical translation) or
    y=x2→y=2x2 (vertical enlargement as a>1) →y=-2x2 (reflection in the horizontal axis) →y=-2(x+6)2 (horizontal translation) →y=-2x+62+10 (vertical translation)
  • experimenting with digital tools by applying transformations to the graphs of functions, such as reciprocal y=1x, square root y=x, cubic y=x3 and exponential functions y=2x, y=(12)x, identifying patterns

  VC2M9A07 | Mathematics | Mathematics Version 2.0 | Level 9 | Algebra

• 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

• 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

• VC2M9A05

  identify and graph quadratic functions, solve quadratic equations graphically and numerically, and use null factor law to solve monic quadratic equations with integer roots algebraically, using graphing software and digital tools as appropriate

  Elaborations

  • recognising that in a table of values, if the second difference between consecutive values of the dependent variable is constant, then it is a quadratic
  • graphing quadratic functions using digital tools and comparing what is the same and what is different between these different functions and their respective graphs; interpreting features of the graphs such as symmetry, turning point, maximum and minimum values; and determining when values of the quadratic function lie within a given range
  • solving quadratic equations algebraically and comparing these to graphical solutions
  • using graphs to determine the solutions of quadratic equations; recognising that the roots of a quadratic function correspond to the x-intercepts of its graph and that if the graph has no x-intercepts, then the corresponding equation has no real solutions
  • relating horizontal axis intercepts of the graph of a quadratic function to the factorised form of its rule using the null factor law; for example, the graph of the function y=x2-5x+6 can be represented as y=x-2x-3 with x-axis intercepts where either x-2=0 or x-3=0
  • recognising that the equation x2=a, where a>0, has 2 solutions, x=a and x=-a (for example, if x2=39 then x=39=6.245 correct to 3 decimal places, or x=-39=-6.245 correct to 3 decimal places) and representing these graphically
  • graphing percentages of illumination of moon phases in relation to Aboriginal and Torres Strait Islander Peoples’ understandings that describe the different phases of the moon

  VC2M9A05 | Mathematics | Mathematics Version 2.0 | Level 9 | Algebra

• VC2M5N05

  solve problems involving addition and subtraction of fractions with the same or related denominators, using different strategies

  Elaborations

  • using different ways to add and subtract fractional amounts by subdividing different models of measurement attributes; for example, adding half an hour and three-quarters of an hour using a clock face, adding a 34 cup of flour and a 14 cup of flour, subtracting 34 of a metre from 2 14 metres
  • representing and solving addition and subtraction problems involving fractions by using jumps on a number line, or bar models, or making diagrams of fractions as parts of shapes
  • using materials, diagrams, number lines or arrays to show and explain that fraction number sentences can be rewritten in equivalent forms without changing the quantity, for example, 12+14 is the same as 24+14 

  VC2M5N05 | Mathematics | Mathematics Version 2.0 | Level 5 | Number

• VC2M5N03

  compare and order common unit fractions with the same and related denominators, including mixed numerals, applying knowledge of factors and multiples; represent these fractions on a number line

  Elaborations

  • using pattern blocks to represent equivalent fractions; selecting one block or a combination of blocks to represent one whole, and making a design with shapes; and recording the fractions to justify the total
  • creating a fraction wall from paper tape to model and compare a range of different fractions with related denominators, and using the model to play fraction wall games
  • connecting a fraction wall model and a number line model of fractions to say how they are the same and how they are different; for example, explaining 14 on a fraction wall represents the area of one-quarter of the whole, while on the number line 14 is identified as a point that is one-quarter of the distance between zero and one
  • using an understanding of factors and multiples as well as equivalence to recognise efficient methods for the location of fractions with related denominators on parallel number lines; for example, explaining on parallel number lines that 210 is located at the same position on a parallel number line as 15 because 15 is equivalent to 210
  • converting between mixed numerals and improper fractions to assist with locating them on a number line

  VC2M5N03 | Mathematics | Mathematics Version 2.0 | Level 5 | Number

• VC2M8A05

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

  Elaborations

  • using graphing software to investigate the effect of systematically varying parameters of linear functions on the corresponding graphs, making and testing conjectures; for example, making a conjecture that if the coefficient of x is negative, then the line will slope down from left to right
  • using graphing software to systematically contrast the graphs of y=2x, -y=2x, y=-2x and -y=-2x with those of
    y<2x, -y<2x, y<-2x and -y<-2x and those of y>2x, -y>2x,  y>-2x and -y>-2x, making and testing conjectures about sign and direction of the inequality
  • using digital tools to investigate integer solutions to equations such as 2x+3y=48

  VC2M8A05 | Mathematics | Mathematics Version 2.0 | Level 8 | Algebra

• VC2M10AA03

  simplify combinations of linear expressions with rational coefficients and the solution of related equations

  Elaborations

  • simplifying sums and differences of linear expressions of the form ax+bc where
    a and b are integers, and c is a non-zero integer, for example, 6x-112-7x4+9-5x3=-5x-3012
  • solving equations involving sums and differences of linear expressions with rational coefficients, for example, -2x3+49= 1157x-1 ⇒-30x+20=693x-99 ⇒x=119723 , and verifying the solution

  VC2M10AA03 | Mathematics | Mathematics Version 2.0 | Level 10A | Algebra

• VC2M7M05

  demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles

  Elaborations

  • using concrete materials to demonstrate that the sum of the interior angles of a triangle is 180°; for example, using paper triangles and tearing to demonstrate that the interior angles when combined form 180°
  • using decomposition and the angle sum of a triangle to generalise the interior angle sum of an n-sided polygon, as 180n-2=180n-360

  VC2M7M05 | Mathematics | Mathematics Version 2.0 | Level 7 | Measurement

• VC2M8N04

  use the 4 operations with integers and with rational numbers, choosing and using efficient mental and written strategies, and digital tools where appropriate, and making estimates for these computations

  Elaborations

  • using patterns to assist in establishing the rules for the multiplication and division of integers
  • applying and explaining efficient strategies such as using the commutative or associative property for regrouping, partitioning, place value, patterning, multiplication or division facts to solve problems involving positive and negative integers, fractions and decimals
  • recognising the effect of sign in the multiplication of integers, for example, (-1)4=1 and (-1)5=-1

  VC2M8N04 | Mathematics | Mathematics Version 2.0 | Level 8 | Number