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• VC2M4A01

  find unknown values in numerical equations involving addition and subtraction, using the properties of numbers and operations

  Elaborations

  • demonstrating the commutative properties of addition using materials, diagrams and number lines; for example, using number lines to demonstrate that 5 + 2 = 2 + 5, and demonstrating that 2 + 2 + 3 = 7 and 2 + 3 + 2 = 7 and 3 + 2 + 2 = 7
  • using balance scales and informal uniform units to create addition or subtraction number sentences showing equivalence, such as 7 + 8 = 6 + 9, and to find unknowns in equivalent number sentences, such as 6 + 8 = □ + 10
  • using relational thinking and knowledge of equivalent number sentences to explain whether equations involving addition or subtraction are true; for example, explaining that 27 − 14 = 17 − 4 is true and using a number line to show the common difference is 13

  VC2M4A01 | Mathematics | Mathematics Version 2.0 | Level 4 | 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

• 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

• VC2M9A04

  find the gradient of a line segment, the midpoint of the line interval and the distance between 2 distinct points on the Cartesian plane

  Elaborations

  • recognising that the gradient of a line is calculated using the gradient of a line segment on that line and is independent of which 2 distinct points on the line are used for this calculation
  • using digital tools and transformations to illustrate that parallel lines in the Cartesian plane have the same gradient and that the relationship between the gradients of pairs of perpendicular lines is that their product is (−1)
  • using Pythagoras’ theorem to establish the distance between 2 points in the Cartesian plane and applying this using horizontal and vertical distances and coordinates
  • investigating graphical and algebraic techniques for finding the midpoint and gradient of the line segment between 2 points
  • using dynamic graphing software and superimposed images (for example, playground equipment, ramps and escalators) to investigate gradients in context and their relationship to rule of a linear function, and interpret gradient as a constant rate of change in linear modelling contexts

  VC2M9A04 | Mathematics | Mathematics Version 2.0 | Level 9 | Algebra

• VC2M3A02

  extend and apply knowledge of addition and subtraction facts to 20 to develop efficient mental strategies for computation with larger numbers without a calculator

  Elaborations

  • partitioning using materials and part-part-whole diagrams to develop subtraction facts related to addition facts, such as 8 + 7 = 15 therefore 15 − 7 = 8 and 15 − 8 = 7
  • using partitioning to develop and record facts systematically (for example, ‘How many ways can 12 monkeys be spread among 2 trees?’, 12 = 12 + 0, 12 = 11 + 1, 12 = 10 + 2, 12 = 9 + 3, …), explaining how they know they have found all possible partitions
  • understanding basic addition and related subtraction facts and using extensions to these facts; for example, 6 + 6 = 12, 16 + 6 = 22, 6 + 7 = 13, 16 + 7 = 23, and 60 + 60 = 120, 600 + 600 = 1200

  VC2M3A02 | Mathematics | Mathematics Version 2.0 | Level 3 | Algebra

• VC2M10A10

  solve problems involving gradients of parallel and perpendicular lines

  Elaborations

  • solving problems using the fact that parallel lines have the same gradient and conversely that if 2 lines have the same gradient then they are parallel
  • solving problems using the fact that the product of the gradients of perpendicular lines is −1 and conversely that if the product of the gradients of 2 lines is −1 then they are perpendicular

  VC2M10A10 | Mathematics | Mathematics Version 2.0 | Level 10 | Algebra

• VC2M10A09

  solve simultaneous linear equations, using algebraic and graphical techniques including using digital tools

  Elaborations

  • investigating situations involving simultaneous linear equations in context (such as multiple quotes for a job, or profit and loss) and solving the equations graphically, giving solutions in everyday language (such as ‘break-even point’ or ‘point to change providers’ for the job)
  • describing the solution of simultaneous equations within the context of the situation
  • using the algebraic methods of substitution and elimination, with and without the use of digital tools, and identifying which technique is the most efficient for a given pair of equations
  • associating the solution of simultaneous equations with the coordinates of the intersection of their corresponding graphs
  • investigating the strategies inherent in Aboriginal and/or Torres Strait Islander children’s instructive games, for example, Weme from the Warlpiri Peoples of central Australia, and their connection to strategies to solve simultaneous linear equations in 2 variables

  VC2M10A09 | Mathematics | Mathematics Version 2.0 | Level 10 | 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

• 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

• VC2M10A02

  simplify algebraic products and quotients using exponent laws

  Elaborations

  • applying knowledge of exponent laws to algebraic terms, and simplifying algebraic expressions using both positive and negative integral indices

  VC2M10A02 | Mathematics | Mathematics Version 2.0 | Level 10 | Algebra

• VC2M2A04

  apply repetition in arithmetic operations, including multiplication as repeated addition and division as repeated subtraction

  Elaborations

  • using technology to construct a sequence of numbers based on constant addition or subtraction from a given starting value
  • sharing a set of objects equally between a small number of groups

  VC2M2A04 | Mathematics | Mathematics Version 2.0 | Level 2 | Algebra

• VC2M6A01

  recognise and use rules that generate visually growing patterns and number patterns involving rational numbers

  Elaborations

  • investigating patterns such as the number of tiles in a geometric pattern, or the number of dots or other shapes in successive repeats of a strip or border pattern, looking for patterns in the way the numbers increase or decrease
  • using a calculator or spreadsheet to experiment with number patterns that result from multiplying or dividing; for example, 1 ÷ 9, 2 ÷ 9, 3 ÷ 9 …, 210 × 11, 211 × 11, 212 × 11 …, 111 × 11, 222 × 11, 333 × 11 …, or 100 ÷ 99, 101 ÷ 99, 102 ÷ 99 …
  • creating an extended number sequence that represents an additive pattern using decimals; for example, representing the additive pattern formed as students pay their $2.50 for an incursion as 2.50, 5.00, 7.50, 10.00, 12.50, 15.00, 17.50 …
  • investigating the number of regions created by successive folds of a sheet of paper (one fold, 2 regions; 2 folds, 4 regions; 3 folds, 8 regions) and describing the pattern using everyday language
  • creating a pattern sequence with materials, writing the associated number sequence and then describing the sequence with a rule so someone else can replicate it with different materials; for example, using matchsticks or toothpicks to create a growing pattern of triangles using 3 for one triangle, 5 for 2 triangles and 7 for 3 triangles and describing the pattern as ‘Multiply the number of triangles by 2 and then add one for the extra toothpick in the first triangle’

  VC2M6A01 | Mathematics | Mathematics Version 2.0 | Level 6 | Algebra

• 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

• 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

• VC2M10A13

  solve simple quadratic equations using a range of strategies, including null factor law

  Elaborations

  • using techniques to solve quadratic equations, including grouping like terms, completing the square, the quadratic formula and null factor law, and choosing 2 integers with the required product and sum
  • using digital tools to explore the conditions for solutions to exist
  • using digital tools to explore both graphically and algebraically the link between the value of the discriminant and the number of real solutions to a quadratic equation

  VC2M10A13 | Mathematics | Mathematics Version 2.0 | Level 10 | Algebra

• 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

• VC2M5A01

  recognise and explain the connection between multiplication and division as inverse operations and use this to develop families of number facts

  Elaborations

  • using materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication
  • using arrays, multiplication tables, and physical and virtual materials to develop families of facts, for example, 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3
  • demonstrating multiplicative partitioning using materials, diagrams or arrays and recording 2 multiplication and 2 division facts for each grouping (for example, 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 4 = 6 and 24 ÷ 6 = 4), explaining how each grouping is different from and connected to other groupings in the materials, diagrams or arrays
  • using materials, diagrams or arrays to recognise and explain the inverse relationship between multiplication and division (for example, solving 240 ÷ 20 = □ by thinking 20 × □ = 240) and using the inverse to make calculations easier (for example, solving 17 × □ = 221 using division, □ = 221 ÷ 17)

  VC2M5A01 | Mathematics | Mathematics Version 2.0 | Level 5 | Algebra

• VC2M10A12

  solve linear equations involving simple algebraic fractions

  Elaborations

  • solving a wide range of linear equations, including those involving one or 2 simple algebraic fractions, and checking solutions by substitution
  • representing word problems, including those involving fractions, as equations and solving them to answer the question

  VC2M10A12 | Mathematics | Mathematics Version 2.0 | Level 10 | Algebra

• VC2M2A03

  recall and demonstrate proficiency with multiplication facts for twos; extend and apply facts to develop the related division facts using doubling and halving

  Elaborations

  • recognising and relating terms such as ‘double’, ‘twice’ and ‘multiply by 2’, and ‘halve’ and ‘divide by 2’ using physical and virtual materials; for example, colouring numbers on a hundreds chart to represent doubles and using it to recognise halves or recognising the doubling pattern and applying the pattern to find related facts such as ‘For 8 twos, think 2 eights’
  • doubling and halving collections to practise and develop fluency with multiplication and division facts for twos, leading to recall of these facts
  • establishing an understanding of doubles and near doubles using physical or virtual manipulatives; for example, using manipulatives to establish that doubling 5 gives you 10, then extending this doubling fact to respond to the question ‘How can you use this fact to double 6 or double 4?’
  • developing fluency with doubling and halving numbers within 20 using physical or virtual materials and playing doubling and halving games; for example, using a physical or virtual dice and choosing whether to double or halve to reach a target number 

  VC2M2A03 | Mathematics | Mathematics Version 2.0 | Level 2 | Algebra

• VC2M10A03

  apply the 4 operations to simple algebraic fractions with numerical or single variable denominators

  Elaborations

  • expressing the sum and difference of algebraic fractions with a common denominator
  • using the exponent laws to simplify products and quotients of algebraic fractions

  VC2M10A03 | Mathematics | Mathematics Version 2.0 | Level 10 | Algebra