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• VC2M10AN02

  perform operations on numbers involving fractional exponents and surds

  Elaborations

  • exploring the notion that $a^{\frac{m}{n}}= {\left(a^{\frac{1}{n}}\right)}^m= {\left(\sqrt[n]{a}\right)}^m= \sqrt[n]{a^m}={\left(a^m\right)}^{\frac{1}{n}}$ , where $m$ is a non-zero integer and $n$ is a natural number, and evaluating corresponding expressions; for example, $8^{\frac{2}{3}}={\left(\sqrt[3]{8}\right)}^2= 2^2=4$ and $8^{\frac{2}{3}}= \sqrt[3]{8^2}=\sqrt[3]{64}=4$

  VC2M10AN02 | Mathematics | Mathematics Version 2.0 | Level 10A | 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 $\frac{3}{4}$ as 3 in each 4, showing this within related fractions like $\frac{6}{8}$ or seeing that $\frac{2}{5}$ means 2 in each 5 so it can be shown within $\frac{4}{10}$
  • 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 $\frac{4}{8}=\frac{2}{4}=\frac{1}{2}$ ; 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 $\frac{1}{4}$ of a metre and connecting this to 0.25 metres or 25 centimetres, or finding $\frac{1}{10}$ 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, $\frac{1}{4}$ of 12 is 3, $\frac{1}{3}$ of 12 is 4

  VC2M4N03 | Mathematics | Mathematics Version 2.0 | Level 4 | Number

• VC2M10ASP06

  design, test and refine solutions to spatial problems using algorithms and digital tools; communicate and justify solutions 

  Elaborations

  • designing and making scale models of three-dimensional objects using digital tools; for example, making components of a puzzle using a three-dimensional printer, and planning and designing the puzzle using principles of tessellations
  • applying a computational thinking approach to solving problems involving networks, for example, connectedness, coverage and weighted measures, such as taking different routes and choosing the most efficient route to take when travelling by car using virtual map software 
  • defining and decomposing spatial problems, creating and applying algorithms to generate solutions, and evaluating and communicating solutions in terms of the problem; for example, designing a floor plan for a department store that limits congestion at key areas such as check-outs, changing rooms and near popular sale items
  • designing, creating and testing algorithms using pseudocode or flow charts for producing self-similar patterns; and validating algorithms using a range of test cases to compare their output
  • exploring geospatial technologies used by Aboriginal and/or Torres Strait Islander communities, to consider spatial problems including position and transformation

  VC2M10ASP06 | Mathematics | Mathematics Version 2.0 | Level 10A | Space

• VC2M2M02

  identify common uses and represent halves, quarters and eighths in relation to shapes, objects and events

  Elaborations

  • demonstrating how food items can be cut in halves, quarters or eighths; for example, cutting pizzas, slices, cakes or sandwiches into equal parts by halving, then halving again to form quarters and eighths, ensuring that the parts are equal
  • investigating cup and spoon measures used in cooking and discussing what half or quarter of a cup or tablespoon measure means, and using sand or water to compare these to the full cup and tablespoon measures
  • demonstrating and using halves and quarters in folding activities; for example, folding paper in half and quarters and ensuring that the pieces are the same size
  • recognising that halves and quarters can be used to describe lengths, positions and distances; for example, describing the halfway point in a race or instructing someone to stand halfway between 2 chairs
  • discussing that halves and quarters are used to describe durations of time, including durations of time in sporting events, and what this means; for example, discussing how the sirens used during an Australian Rules football game represent quarter time, half-time and three-quarter time during the game, or recognising and using half an hour or quarter of an hour to describe a duration of time

  VC2M2M02 | Mathematics | Mathematics Version 2.0 | Level 2 | Measurement

• VC2M9M03

  solve spatial problems, applying angle properties, scale, similarity, ratio, Pythagoras’ theorem and trigonometry in right-angled triangles

  Elaborations

  • investigating the applications of Pythagoras’ theorem in authentic problems, including applying Pythagoras’ theorem and trigonometry to problems in surveying and design
  • applying the formula for calculation of distances between points on the Cartesian plane from their coordinates, emphasising the connection to vertical and horizontal displacements between the points 
  • understanding the relationship between the corresponding sides of similar right-angled triangles and establishing the relationship between areas of similar figures and the ratio of corresponding sides, the scale factor
  • using images of proportional relationships to estimate actual measurements (for example, taking a photograph of a person standing in front of a tree and using the image and scale to estimate the height of the tree), discussing the findings and ways to improve the estimates
  • investigating theorems and conjectures involving triangles, for example, the triangle inequality and generalising links between the Pythagorean rule for right-angled triangles, and related inequalities for acute and obtuse triangles, determining the minimal sets of information for a triangle from which other measures can all be determined
  • using knowledge of similar triangles, Pythagoras’ theorem, rates and algebra to design and construct a Biltmore stick, used to measure the diameter and height of a tree, and calculating the density and dry mass to predict how much paper could be manufactured from the tree

  VC2M9M03 | Mathematics | Mathematics Version 2.0 | Level 9 | Measurement

• VC2M8P02

  determine all possible outcome combinations for 2 events, using two-way tables, tree diagrams and Venn diagrams, and use these to determine probabilities of specific events in practical situations

  Elaborations

  • describing events using language of ‘at least’, exclusive ‘or’ (A or B but not both), inclusive ‘or’ (A or B or both) and ‘and’
  • using the relation Pr(A and B) + Pr(A and not B) + Pr(not A and B) + Pr(not A and not B) = 1 to calculate probabilities, including the special case of mutually exclusive events where Pr(A and B) = 0
  • using Venn diagrams or two-way tables to demonstrate the difference between events that are mutually exclusive, such as whether a coin toss will land on a head or a tail, and those that are not mutually exclusive, such as people who have blonde hair and people who have blue eyes
  • exploring Aboriginal and/or Torres Strait Islander children’s instructive games, for example, Battendi from the Ngarrindjeri Peoples of Lake Murray and Lake Albert in southern Australia, applying possible combinations and relationships and calculating probabilities using two-way tables and Venn diagrams

  VC2M8P02 | Mathematics | Mathematics Version 2.0 | Level 8 | Probability

• VC2M3SP01

  make, compare and classify objects, identifying key features and explaining why these features make them suited to their uses

  Elaborations

  • classifying a collection of geometric objects, including cylinders, spheres, prisms and pyramids, according to key features such as the shape and number of faces and/or surfaces, edges and vertices
  • making and comparing objects built out of cubic blocks and discussing key features; for example, comparing the amount of space that objects occupy by counting how many blocks it takes to build different rectangular prisms that have the same height but different bases
  • making geometric objects in solid form out of connecting cubes and in skeleton form with straws, and constructing objects using dynamic geometry software, recognising, comparing and discussing the features of the objects using the different representations
  • using familiar shapes and objects to build or construct models and compare the suitability of different shapes and objects for aspects of the model; for example, building rectangular towers out of connecting cubes and recognising that the taller the tower, the less stable it becomes unless the base is increased; or building bridges out of straws bent into different shapes and comparing the strength of different designs
  • identifying, classifying and comparing common objects found on Country/Place as cubes, rectangular prisms, cylinders, cones and spheres
  • investigating and explaining how Aboriginal and/or Torres Strait Islander Peoples’ dwellings are oriented in the environment to accommodate climatic conditions

  VC2M3SP01 | Mathematics | Mathematics Version 2.0 | Level 3 | Space

• VC2M1ST02

  represent collected data for a categorical variable using one-to-one displays and digital tools where appropriate; compare the data using frequencies and discuss the findings

  Elaborations

  • creating a pictograph with objects or drawings; discussing the possible categories for the pictograph, arranging the objects or drawings into the categories; and then reflecting on the chosen categories and deciding whether they were helpful
  • translating data from a list or pictorial display into a tally chart to make counting easier; describing what the tally chart is showing, by referring to the categories; using skip counting by fives to compare the numbers within each category; and explaining how the tally chart answers the question
  • recognising that when there is no data for a particular category you may choose whether or not to include it in your data display, and you may use a zero or blank for that category depending on the purpose of the data collection or presentation; for example, creating a birthday calendar for the class and discussing that there are no students born in May and therefore there are no names listed
  • representing data with objects and drawings where one object or drawing represents one data value, and describing the displays and explaining patterns that have been created using counting strategies to determine the frequency of responses
  • exploring Aboriginal and/or Torres Strait Islander children’s instructive games, for example, Kolap from Mer Island in the Torres Strait region, recording the outcomes, and representing and discussing the results

  VC2M1ST02 | Mathematics | Mathematics Version 2.0 | Level 1 | Statistics

• 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,$7+7+7=3\times 7$, $x+x+x=3\times x$ noting that $3x$ includes implied multiplication and recognising the difference between $3x+4$ and $3\left(x+4\right)$
  • 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

• 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, $2^3\times 2^2$ can be represented as $\left(2\times 2\times 2\right)\times \left(2\times 2\right)=2^5 ,$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, $\frac{6^3}{2^3}=\mathrm{ }\frac{2^3\times 3^3}{2^3}=3^3$, 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 $\frac{3^4}{3^4}=1,\text{and} 3^{4-4}=3^0$to illustrate the necessity that for any non-zero natural number $n, { n}^0=1$

  VC2M8N02 | Mathematics | Mathematics Version 2.0 | Level 8 | 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 $1-\frac{1}{6}=\frac{5}{6}$
  • 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 $\frac{1}{3}$ and probability of not throwing a 5 or 6 is $\frac{2}{3}$

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

• 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 $\frac{2}{3}$ is equivalent to $\frac{4}{6}$ and $\frac{6}{9}$ and therefore $\frac{2}{3}$ < $\frac{5}{6}$
  • 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, $\frac{16}{100}$ and $\frac{4}{25}$, 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

• 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 $0.475=\frac{4}{10}+\frac{7}{100}+\frac{5}{1000}=4\times {10}^{-1}+7\times {10}^{-2}+5\times {10}^{-3}$ and 0.00023 as 23 × 10^-5
  • simplifying and evaluating numerical expressions, involving both positive and negative integer exponents, explaining why; for example, $5^{-3}=\frac{1}{5^3}={\left(\frac{1}{5}\right)}^3=\frac{1}{125}$ and connecting terms of the sequence $125, 25, 5, 1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}$… to terms of the sequence $5^3, 5^2,5^1, 5^0, 5^{-1}, 5^{-2}, 5^{-3}$...
  • relating the computation of numerical expressions involving exponents to the exponent laws and the definition of an exponent; for example, $2^3÷2^5=2^{-2}=\frac{1}{2^2}=\frac{1}{4}$ and ${(3\times 5)}^2=3^2\times 5^2=9\times 25=225$
  • recognising exponents in algebraic expressions and applying the relevant exponent laws and conventions; for example, for any non-zero natural number $a$, $a^0=1$, $x^1=x, {r^2=r\times r, h}^3=h\times h\times h,$$y^4=y\times y\times y\times y$, and$\frac{1}{w}\times \frac{1}{w}=\frac{1}{w^2}=w^{-2}$
  • relating simplification of expressions from first principles and counting to the use of exponent laws; for example,
    ${\left(a^2\right)}^3=\left(a\times a\right)\times \left(a\times a\right)\times \left(a\times a\right)=a\times a\times a\times a\times a\times a=a^6$;
    $b^2\times b^3=\left(b\times b\right)\times \left(b\times b\times b\right)=b\times b\times b\times b\times b=b^5; \frac{y^4}{y^2}=\frac{y\times y\times y\times y}{y\times y}=\frac{y^2}{1}=y^2$and${\left(5a\right)}^2=\left(5\times a\right)\times \left(5\times a\right)=5 \times 5\times a\times a=25\times a^2=25a^2$
  • applying the exponent laws to simplifying expressions involving products, quotients, and powers of constants and variables; for example, $\frac{{\left(2xy\right)}^3}{{xy}^4}=\frac{8x^3{y}^3}{x{y}^4}=8x^2{y}^{-1}$
  • 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

• 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,
    $y=x\to y=2x$ (vertical enlargement as $a>1$) $\to y=2x-1$ (vertical translation) and
    $y=x\to y=\frac{1}{2}x$(vertical compression as $0<a<1$) $\to y=-\frac{1}{2}x$ (reflection in the horizontal axis) $\to y=-\frac{1}{2}x+3$ (vertical translation)
  • investigating transformations of the parabola $y=x^2$to the graph of $y=a{\left(x-h\right)}^2+b$ 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,
    $y=x^2\to y=\frac{1}{3}{x}^2$ (vertical compression as $0<a<1$) $\to y=\frac{1}{3}{(x-5)}^2$ (horizontal translation) $\to y=\frac{1}{3}{(x-5)}^2+7$ (vertical translation) or
    $y=x^2\to y=2x^2$ (vertical enlargement as $a>1$) $\to y=-2x^2$ (reflection in the horizontal axis) $\to y=-2{(x+6)}^2$ (horizontal translation) $\to y=-2{\left(x+6\right)}^2+10$ (vertical translation)
  • experimenting with digital tools by applying transformations to the graphs of functions, such as reciprocal $y=\frac{1}{x}$, square root $y=\sqrt[]{x}$, cubic $y=x^3$ and exponential functions$y=2^x, y={\left(\frac{1}{2}\right)}^x$, identifying patterns

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