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• VC2M5SP02

  construct a grid coordinate system that uses coordinates to locate positions within a space; use coordinates and directional language to describe position and movement

  Elaborations

  • understanding how the numbers on the axes on a grid coordinate system are numbers on a number line and are used to pinpoint locations
  • discussing the conventions of indicating a point in a grid coordinate system; for example, writing the horizontal axis number first and the vertical axis number second, and using brackets and commas
  • comparing a grid reference system to a grid coordinate system (first quadrant only) by using both to play strategy games involving location; for example, in playing the game Quadrant Commander, deducing that in a grid coordinate system the lines are numbered (starting from zero), not the spaces
  • placing a coordinate grid over a contour line, drawing and listing the coordinates of each point in the picture, asking a peer to re-create the drawing using only the list of coordinates, and discussing the reasons for the potential similarities and differences between the 2 drawings

  VC2M5SP02 | Mathematics | Mathematics Version 2.0 | Level 5 | Space

• VC2M6N06

  multiply and divide decimals by multiples of powers of 10 without a calculator, applying knowledge of place value and proficiency with multiplication facts, using estimation and rounding to check the reasonableness of answers

  Elaborations

  • applying place value knowledge, including that the value of the digit is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a decimal by powers of 10
  • applying and explaining estimation strategies in multiplicative situations involving a decimal greater than one that is multiplied by a two- or three-digit number, using a multiple of 10 or 100 when the situation requires just an estimation
  • explaining the effect of multiplying or dividing a decimal by 10, 100, 1000 … in terms of place value and not the decimal point shifting

  VC2M6N06 | Mathematics | Mathematics Version 2.0 | Level 6 | Number

• 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

• VC2M10ST03

  construct two-way tables and discuss possible relationship between categorical variables

  Elaborations

  • using two-way tables to investigate and comparing the survey responses to questions involving five-point Likert scale against 2 different categories of respondents, for example, junior students’ responses compared to senior students’ responses to a survey question
  • recording data in two-way tables and using percentages and proportions to identify patterns and associations in the data
  • conducting a litter survey around the school, considering the relationship between different categorical variables such as the day of the week (as canteen specials might lead to different types of litter) or the weather (due to hot days leading to more icy poles and cold drinks being sold)

  VC2M10ST03 | Mathematics | Mathematics Version 2.0 | Level 10 | Statistics

• 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 $x^2=2^x$

  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 $2x+3y<24)\mathrm{ }$and verifying using a test point such as (0, 0)

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

• VC2M10ST04

  analyse claims, inferences and conclusions of statistical reports in the media and other places, by linking claims to displays, statistics and representative data, including ethical considerations and identification of potential sources of bias

  Elaborations

  • identifying potentially misleading data representations in the media such as graphs with broken axes and scales that do not start at zero or are nonlinear; and recognising when data is not relating to the claim, not representative of the population or is deliberately being used to mislead or support a claim or biased point of view
  • investigating the source and size of the sample from which the data was collected and deciding whether the sample is appropriately representative of the population 
  • investigating population proportions and rates, and discussing potential ethical considerations, for example, when presenting statistical data involving infection rates and the number of cases per head of population
  • discussing the ethical considerations associated with collecting data to predict the number of people likely to be infected with a strain of flu or experience side effects with a certain medication, considering validity claims and sample sizes
  • using the concept of Indigenous data sovereignty to critique and evaluate the Australian Government’s ‘Closing the Gap’ report 

  VC2M10ST04 | Mathematics | Mathematics Version 2.0 | Level 10 | Statistics

• 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
    $A=l\times w$; 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 $r$, $W=b+1.5 \times h \times r$, using values for mass $m$ and volume $v$ to determine density $d$ of a substance where $d=\frac{m}{v}$
  • 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

• 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

• 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 $f\left(x\right)=2x^2-3x-7$
  • applying transformations to the graph of $x^2+y^2=1$
  • 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

• VC2M4N01

  recognise and extend the application of place value to tenths and hundredths and use the conventions of decimal notation to name and represent decimals

  Elaborations

  • using a bar to represent the whole, dividing it into 10 equal pieces with each piece representing 0.1 or one-tenth of the whole length and understanding that 2 pieces are 0.2 or two-tenths of the whole
  • using materials to show the multiplicative relationship between the whole, tenths and hundredths; for example, using a bundle of 10 straws to represent the whole, one straw as the tenth, and cutting the tenth into 10 parts to show the hundredths; or using deci-pipes to represent tenths
  • recognising that one is the same as ten-tenths and one-tenth is the same as ten-hundredths and using this relationship to rename decimals; for example, renaming 0.25 as two-tenths and five-hundredths or twenty-five-hundredths
  • making models of measurement attributes to show the relationship between the base unit and parts of the unit, for example, 1.5 metres is one metre and five-tenths of the next metre; 1.75 units is one unit and seventy-five-hundredths of the next unit
  • counting large quantities of mixed notes and coins, writing the total using dollars and cents, and recognising the cents as parts of the next dollar
  • comparing the way money and measures are read and said, and explaining how they are the same and different; for example, explaining that $2.75 is said ‘two dollars seventy-five’ and 2.75 metres is said ‘two point seven five metres’, and recognising that the 7 means seven-tenths and the 5 means five-hundredths in both

  VC2M4N01 | Mathematics | Mathematics Version 2.0 | Level 4 | Number

• 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

• 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 $\frac{1}{4}$ on a fraction wall represents the area of one-quarter of the whole, while on the number line $\frac{1}{4}$ 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 $\frac{2}{10}$ is located at the same position on a parallel number line as $\frac{1}{5}\mathrm{ }$because $\frac{1}{5}\mathrm{ }$is equivalent to $\frac{2}{10}$
  • 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

• 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 $x=a$ , $y=a$ , $x \le a$ , $x>a$ , $y \le a$ , $y>a$ 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

• 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 $y=x^2-5x+6$ can be represented as $y=\left(x-2\right)\left(x-3\right)$ with $x$-axis intercepts where either $\left(x-2\right)=0$ or $\left(x-3\right)=0$
  • recognising that the equation $x^2=a$, where $a>0$, has 2 solutions, $x=\sqrt[]{a}$ and $x=-\sqrt[]{a}$ (for example, if $x^2=39$ then $x=\sqrt[]{39}=6.245$ correct to 3 decimal places, or $x=-\sqrt[]{39}=-6.245$ 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