Standard 2 content will be added during 2022.

Standard 2 Syllabus

Algebra

MS-A1 Formulae and EquationsYear 11

• review substitution of numerical values into linear and non-linear algebraic expressions and equations
  • review evaluating the subject of a formula, given the value of other pronumerals in the formula
  • change the subject of a formula
  • solve problems involving formulae, including calculating distance, speed and time (with change of units of measurement as required) or calculating stopping distances of vehicles using a suitable formula
• develop and solve linear equations, including those derived from substituting values into a formula, or those developed from a word description
• calculate and interpret blood alcohol content (BAC) based on drink consumption and body weight
  • use formulae, both in word form and algebraic form, to calculate an estimate for blood alcohol content $(\mathit{BAC})$, including $BAC_{Male} = \frac{10N-7.5H}{6.8M}$ and $BAC_{Female} = \frac{10N-7.5H}{5.5M}$ where $N$ is the number of standard drinks consumed, $H$ is the number of hours of drinking, and $M$ is the person's weight in kilograms
  • determine the number of hours required for a person to stop consuming alcohol in order to reach zero BAC, eg using the formula $\text{time}=\frac{BAC}{0.015}$
  • describe limitations of methods estimating BAC
• calculate required medication dosages for children and adults from packets, given age or weight, using Fried's, Young's or Clark's formula as appropriate
  • Fried's formula: $\text{Dosage for children 1 - 2 years} = \frac{\text{age (in months) }\times\text{ adult dosage}}{150}$
  • Young's formula: $\text{Dosage for children 1 - 12 years} = \frac{\text{age of child (in years) }\times\text{ adult dosage}}{\text{age of child (in years) + 12}}$
  • Clark's formula: $\text{Dosage} = \frac{\text{weight in kg }\times\text{ adult dosage}}{70}$

MS-A2 Linear RelationshipsYear 11

• model, analyse and solve problems involving linear relationships, including constructing a straight-line graph and interpreting features of a straight-line graph, including the gradient and intercepts
  • recognise that a direct variation relationship produces a straight-line graph
  • determine a direct variation relationship from a written description, a straight-line graph passing through the origin, or a linear function in the form $y=mx$
  • review the linear function $y=mx+c$ and understand the geometrical significance of $m$ and $c$
  • recognise the gradient of a direct variation graph as the constant of variations
  • construct straight-line graphs both with and without the aid of technology
• construct and analyse a linear model, graphically or algebraically, to solve practical direct variation problems, including the cost of filling a car with fuel or a currency conversion graph
  • identify and evaluate the limitations of a linear model in a practical context

MS-A4 Types of RelationshipsYear 12

A4.1: Simultaneous linear equations

• solve a pair of simultaneous linear equations graphically, by finding the point of intersection between two straight-line graphs, with and without technology
• develop a pair of simultaneous linear equations to model a practical situation
• solve practical problems that involve determining and interpreting the point of intersection of two straight-line graphs, including the break-even point of a simple business problem where cost and revenue are represented by linear equations

A4.2: Non-linear relationships

• use an exponential model to solve problems
  • graph and recognise an exponential function in the form $y=a^x$ and $y=a^{-x}\ (a>0)$ with and without technology
  • interpret the meaning of the intercepts of an exponential graph in a variety of contexts
  • construct and analyse an exponential model of the form $y=ka^x$ and $y=ka^{-x}\ (a>0)$ where $k$ is a constant, to solve a practical growth or decay problem
• construct and analyse a quadratic model to solve practical problems involving quadratic functions or expressions of the form $y=ax^2+bx+c$, for example braking distance against speed
  • recognise the shape of a parabola and that it always has a turning point and an axis of symmetry
  • graph a quadratic function with and without technology
  • interpret the turning point and intercepts of a parabola in a practical context
  • consider the range of values for $x$ and $y$ for which the quadratic model makes sense in a practical context
• recognise that reciprocal functions of the form $y=\frac{k}{x}$, where $k$ is a constant, represent inverse variation, identify the rectangular hyperbolic shape of these graphs and their important feature
  • use a reciprocal model to solve practical inverse variation problems algebraically and graphically, eg the amount of pizza received when sharing a pizza between increasing numbers of people

Measurement

MS-M1 Applications of MeasurementYear 11

M1.1: Practicalities of Measurement

• review the use of different metric units of measurement including units of area, take measurements, and calculate conversions between common units of measurement, for example kilometres to metres or litres to millilitres
• calculate the absolute error of a reported measurement using $\text{Absolute error}=\frac{1}{2}\times\text{Precision}$ and state the corresponding limits of accuracy
  • find the limits of accuracy as given by:
    $\text{Uppwer bound}=\text{Measurement}+\text{Absolute error}$
    $\text{Lower bound}=\text{Measurement}-\text{Absolute error}$
  • investigate types of errors, eg human error or device limitations
  • calculate the percentage error of a reported measurement using
    $\text{Percentage error}=\frac{\text{Absolute error}}{\text{Measurement}}\times100\%$
• use standard form and standard metric prefixes in the context of measurement, with and without a required number of significant figures
  • standard prefixes include nano-, micro-, milli-, centi-, kilo-, mega-, giga- and tera-

M1.2: Perimeter, area and volume

• review and extend how to solve practical problems requiring the calculation of perimeters and areas of triangles, rectangles, parallelograms, trapezia, circles, sectors of circles and composite shapes
  • review the use of Pythagoras' theorem to solve problems involving right-angled triangles
  • review the use of a scale factor to find unknown lengths in similar figures
• solve problems involving surface area of solids including prisms, cylinders, spheres and composite solids
• solve problems involving volume and capacity of solids including prisms, cylinders, spheres, pyramids and composite solids
  • convert between units of volume and capacity
• calculate perimeters and areas of irregularly shaped blocks of land by dissection into regular shapes including triangles and trapezia
  • derive the Trapezoidal rule for a single application, $A\approx\frac{h}{2}\left(d_f+d_l\right)$
  • use the Trapezoidal rule to solve a variety of practical problems with and without technology, eg the volume of water in a swimming pool
• solve problems involving perimeters, area, surface area, volumes and capacity in a variety of contexts

M1.3: Units of energy and mass

• review the use of metric units of mass in solving problems, including grams, kilograms and tonnes, their abbreviations and how to convert between them
• use metric units of energy to solve problems, including calories, kilocalories, joules and kilojoules, their abbreviations and how to convert between them
• use units of energy and mass to solve problems related to food and nutrition, including calories
• use units of energy to solve problems involving the amount of energy expended in activities, for example kilojoules
• use units of energy to solve problems involving the consumption of electricity, for example kilowatt hours, and investigate common appliances in terms of their energy consumption

MS-M2 Working with TimeYear 11

• indicate positions on the Earth's surface
  • locate points on Earth's surface using latitude, longitude or position coordinates with a globe, an atlas and digital technologies, eg a smartphone or GPS device
  • understand and use the link between longitude and time to find time differences
• calculate times and time differences around the world
  • review using units of time, converting between 12-hour and 24-hour clocks and calculating time intervals
  • understand and use the link between longitude and time to find time differences
  • solve problems involving time zones in Australia and in neighbouring nations, making any necessary allowances for daylight saving
  • solve problems involving Coordinated Universal Time (UTC), and the International Date Line (IDL)
  • find time differences between two places on Earth using recognised international time zones
  • review how to interpret timetables, eg bus, train and ferry timetables, and use them to solve problems
  • solve practical problems, eg travelling east and west, incorporating time zones, or internet and phone usage across time zones, or the timing of events broadcast live from states of countries between different time zones

MS-M6 Non-right-angled TrigonometryYear 12

• review and use the trigonometric ratios to find the length of an unknown side or the size of an unknown angle in a right-angled triangle
• use technology to investigate the sign of $\sin A$ and $\cos A$ for $0\degree\leq A\leq180\degree$
• determine the area of any triangle, given two sides and an included angle, by using the rule $A=\frac{1}{2}ab\sin C$, and solve related practical problems
• solve problems involving non-right-angled triangles using the sine rule, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ (ambiguous case excluded)
  • find the size of an obtuse angle, given that it is obtuse
• solve problems involving non-right-angled triangles using the cosine rule, $c^2=a^2+b^2-2ab\cos C$
• understand various navigational methods
  • understand the difference between compass and true bearings
  • investigate navigational methods used by different cultures, including those of Aboriginal and Torres Strait Islander Peoples
• solve practical probelsm involving Pythagoras' theorem, the trigonometry of right-angled and non- right-angled triangles, angles of elevation and depression and the use of true bearings and compass bearings
  • work with angles correct to the nearest degree and/or minute
• construct and interpret compass radial surveys and solve related problems

MS-M7 Rates and RatiosYear 12

• use rates to solve and describe practical problems
  • use rates to make comparisons, eg using unit prices to compare best buys, working with speed, comparing heart rates after exercise and considering target heart rate ranges during training
  • know that a watt (W) is the International System of Units (SI) derived unit of power and is equal to one joule per second
  • interpret the energy rating of household appliances and compare running costs of different models of the same type of appliance, considering costs of domestic electricity, eg calculate the cost of running a 200-watt television for six hours if the average peak rate for domestic electricity is $0.15/kWh
  • investigate local council requirements for energy-efficient housing
  • calculate the amount of fuel used on a trip, given the fuel consumption rate, and compare fuel consumption statistics for various vehicles
• solve practical problems involving ratio, for example capture-recapture, mixtures for building materials or cost per item
  • work with ratio to express a ratio in simplest form, to find the ratio of two quantities and to divide a quantity in a given ratio
  • use ratio to describe map scales
• obtain measurements from scale drawings, including maps (including cultural mappings or models) or building plans, to solve problems
  • interpret commonly used symbols and abbreviations on building plans and elevation views
  • calculate the perimeter or area of a section of land, using the Trapezoidal rule where appropriate, from a variety of sources, including a site plan, an aerial photograph, radial surveys or maps that include a scale
  • calculate the volume of rainfall over an area, using $V=Ah$, from a variety of sources, including a site plan, an aerial photograph, radial surveys or maps that include a scale

Financial Mathematics

MS-F1 Money MattersYear 11

F1.1: Interest and depreciation

• apply percentage increase or decrease in various contexts, eg calculating the goods and services tax (GST) payable on a range of goods and services, and calculating profit or loss in absolute and percentage terms
• calculate simple interest for different rates and periods
  • use technology or otherwise to compare simple interest graphs for different rates and periods
• calculate the depreciation of an asset using the straight-line method as an application of the simple interest formula
  • use $S=V_0-Dn$, where $S$ is the salvage value of the asset after $n$ periods, $V_0$ is the initial value of the asset, $D$ is the amount of depreciation per period, and $n$ is the number of periods
• use a spreadsheet to calculate and graph compound interest as a recurrence relation involving repeated applications of simple interest

F1.2: Earning and managing money

• calculate monthly, fortnightly, weekly, daily or hourly pay rates from a given salary, wages involving hourly rates and penalty rates, including situations involving overtime and other special allowances, and earnings based on commission (including commission based on a sliding scale), piecework or royalties
  • calculate annual leave loading
  • calculate payments based on government allowances and pensions
• calculate income tax
  • identify allowable tax deductions
  • calculate taxable income after allowable tax deductions are taken from gross pay
  • calculate the Medicare levy (basic levy only)
  • calculate the amount of Pay As You Go (PAYG) tax payable per fortnight or week using current tax scales, and use this to determine if more tax is payable or if a refund is owing after completing a tax return
• calculate net pay following deductions from income
• use technology to perform financial computations, for example calculating percentage change, calculating tax payable and preparing a wage-sheet

F1.3: Budgeting and household expenses

• interpret and use information about a household's electricity, water or gas usage and related charges and costs from household bills
• plan for the purchase of a car
  • investigate on-road costs for new and used vehicles, including sale price (or loan repayments), registration, insurance and stamp duty at current rates
  • consider sustainability when choosing a vehicle to purchase, eg fuel consumption rates
  • calculate and compare the cost of purchasing different vehicles using a spreadsheet
• plan for the running and maintenance of a car
  • describe the different types of insurance available, including compulsory and non-compulsory third-party insurance, and comprehensive insurance
  • investigate other running costs associated with ownership of a vehicle, eg cost of servicing, repairs and tyres
  • calculate and compare the cost of running different vehicles using a spreadsheet
• prepare a personal budget for a given income, taking into account fixed and discretionary spending

MS-F4 Investments and LoansYear 12

F4.1: Investments

• calculate the future value ($FV$) or present value ($PV$) and the interest rate ($r$) of a compound interest investment using the formula $FV=PV(1+r)^n$
  • compare the growth of simple interest and compound interest investments numerically and graphically, linking graphs to linear and exponential modelling using technology
  • investigate the effect of varying the interest rate, the term or the compounding period on the future value of an investment, using technology
  • compare and contrast different investment strategies, performing appropriate calculations when needed
• solve practical problems involving compounding, for example determine the impact of inflation on prices and wages
• work with shares and calculate the appreciated value of items, for example antiques
  • record and graph the price of a share over time
  • calculate the dividend paid on a portfolio of shares, and the dividend yield (excluding franked dividends)

F4.2: Depreciation and loans

• calculate the depreciation of an asset using the declining-balance method using the formula $S=V_0(1-r)^n$, where $S$ is the salvage value of the asset after $n$ periods, $V_0$ is the initial value of the asset, $r$ is the depreciation rate per period, expressed as a decimal, and $n$ is the number of periods, as an application of the compound interest formula
• solve practical problems involving reducing balance loans, for example determining the total loan amount and monthly repayments
• recognise credit cards as an example of a reducing balance loan and solve practical problems relating to credit cards
  • identify the various fees and charges associated with credit card usage
  • compare credit card interest rates with interest rates for other loan types
  • interpret credit card statements, recognising the implications of only making the minimum payment
  • understand what is meant by an interest-free period
  • calculate the compounding interest charged on a retail purchase, transaction or the outstanding balance for a given number of days, using technology or otherwise

MS-F5 AnnuitiesYear 12

• solve compound interest related problems involving financial decisions, for example a home loan, a savings account, a car loan or an annuity
  • identify an annuity as an investment account with regular, equal contributions and interest compounding at the end of each period, or as a single sum investment from which regular, equal withdrawals are made
  • using technology, model an annuity as a recurrence relation, and investigate (numerically or graphically) the effect of varying the amount and frequency of each contribution, the interest rate or the payment amount on the duration and/or future value of the annuity
  • use a table of interest factors to perform annuity calculations, eg calculating the present or future value of an annuity, the contribution amount required to achieve a given future value or the single sum that would produce the same future value as a given annuity

Statistical Analysis

MS-S1 Data AnalysisYear 11

S1.1: Classifying and represending data (grouped and ungrouped)

• describe and use appropriate data collection methods for a population or samples
  • investigate whether a sample obtained from a population may or may not be representative of the population by considering different kinds of sampling methods: systematic sampling, self-selected sampling, capture-recapture, simple random sampling and stratified sampling
  • investigate the advantages and disadvantages of each type of sampling
  • describe the potential faults in the design and practicalities of data collection processes, eg surveys, experiments and observational studies, misunderstandings and misrepresentations, including examples from the media
• classify data relating to a single random variable
  • classify a categorical variable as either ordinal, eg income level (low, medium, high) or nominal, eg place of birth (Australia, overseas)
  • classify a numerical variable as either discrete, eg the number of rooms in a house, or continuous, eg the temperature in degrees Celsius
• review how to organise and display data into appropriate tabular and/or graphical representations
  • display categorical data in tables and, as appropriate, in both bar charts or Pareto charts
  • display numerical data as frequency distribution tables and histograms, cumulative frequency distribution tables and graphs, dot plots and stem and leaf plots (including back-to-back where comparing two datasets)
  • construct and interpret tables and graphs related to real-world contexts, including: motor vehicle safety including driver behaviour, accident statistics, blood alcohol content over time, running costs of a motor vehicle, costs of purchase and insurance, vehicle depreciation, rainfall, hourly temperature, household and personal water usage
• interpret and compare data by considering it in tabular and/or graphical representations
  • choose appropriate tabular and/or graphical representations to enable comparisons
  • compare the suitability of different methods of data presentation in real-world contexts, including their visual appeal, eg a heat map to illustrate climate change data or the median house prices across suburbs

S1.2: Summary statistics

• describe the distinguishing features of a population and sample
  • define notations associated with population values (parameters) and sample-based estimates (statistics), including population mean $\mu$, population standard deviation $\sigma$, sample mean $\bar{x}$ and sample standard deviation
• summarise and interpret grouped and ungrouped data through appropriate graphs and summary statistics
  • discuss the mode and determine where possible
  • calculate measures of central tendency, including the arithmetic mean and the median
  • investigate the suitability of measures of central tendency in real-world contexts and use them to compare datasets
  • calculate measures of spread including the range, quantiles (including quartiles, deciles and percentiles), interquartile range (IQR) and standard deviation (calculations for standard deviation are only required by using technology)
• investigate and describe the effect of outliers on summary statistics
  • use different approaches for identifying outliers, including consideration of the distance from the mean or median, or the use of $Q_1-1.5\times IQR$ and $Q_3+1.5\times IQR$ as criteria, recognising and justifying when each approach is appropriate
  • investigate and recognise the effect of outliers on the mean and median
• investigate real-world examples from the media illustrating appropriate and inappropriate uses or misuses of measures of central tendency and spread
• describe, compare and interpret the distributions of graphical displays and/or numerical datasets and report findings in a systematic and concise manner
  • identify modality (unimodal, bimodal or multimodal)
  • identify shape (symmetric or positively or negatively skewed)
  • identify central tendency, spread and outliers, using and justifying appropriate criteria
  • calculate measures of central tendency or measures of spread where appropriate
• construct and compare parallel box-plots
  • complete a five-number summary for different datasets (ACMEM058)
  • compare groups in terms of central tendency (median), spread (IQR and range) and outliers (using appropriate criteria)
  • interpret and communicate the differences observed between parallel box-plots in the context of the data

MS-S2 Relative Frequency and ProbabilityYear 11

• review, understand and use the language associated with theoretical probability and relative frequency
  • construct a sample space for an experiment and use it to determine the number of outcomes
  • review probability as a measure of the 'likely chance of occurrence' of an event
  • review the probability scale: $0\leq P(A)\leq 1$ for each event $A$, with $P(A)=0$ if $A$ is an impossibility and $P(A)=1$ if $A$ is a certainty
• determine the probabilities associated with simple games and experiments
  • use the following definition of probability of an event where outcomes are equally likely: $P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$
  • calculate the probability of the complement of an event using the relationship $P(\text{an event does not occur})=1-P(\text{the event does occur})=P(\overline{\text{the event does occur}})=P(\text{event}^c)$
• use arrays and tree diagrams to determine the outcomes and probabilities for multistage experiments
  • construct and use tree diagrams to establish the outcomes for a simple multistage event
  • use probability tree diagrams to solve problems involving two-stage events
• solve problems involving simulations or trials of experiments in a variety of contexts
  • perform simulations of experiments using technology
  • use relative frequency as an estimate of probability
  • recognise that an increasing number of trials produces relative frequencies that gradually become closer in value to the theoretical probability
  • identify factors that could complicate the simulation of real-world events
• solve problems involving probability and/or relative frequency in a variety of contexts
  • use existing known probabilities, or estimates based on relative frequencies to calculate expected frequency for a given sample or population, eg predicting, by calculation, the number of people of each blood type in a population given a two-way table of percentage breakdowns
  • calculate the expected frequency of an event occurring using $np$ where $n$ represents the number of times an experiment is repeated, and on each of those times the probability that the event occurs is $p$

MS-S4 Bivariate Data AnalysisYear 12

• construct a bivariate scatterplot to identify patterns in the data that suggest the presence of an association
• use bivariate scatterplots (constructing them when needed) to describe the patterns, features and associations of bivariate datasets, justifying any conclusions
  • describe bivariate datasets in terms of form (linear/non-linear) and, in the case of linear, the direction (positive/negative) and strength of any association (strong/moderate/weak)
  • identify the dependent and independent variables within bivariate datasets where appropriate
  • describe and interpret a variety of bivariate datasets involving two numerical variables using real-world examples from the media or freely available from government or business datasets
  • calculate and interpret Pearson's correlation coefficient ($r$) using technology to quantify the strength of a linear association of a sample
• model a linear relationship by fitting an appropriate line of best fit to a scatterplot and using it to describe and quantify associations
  • fit a line of best fit both by eye and by using technology to the data
  • fit a least-squares regression line to the data using technology
  • interpret the intercept and gradient of the fitted line
• use the appropriate line of best fit, both found by eye and by applying the equation, to make predictions by either interpolation or extrapolation
  • recognise the limitations of interpolation and extrapolation, and interpolate from plotted data to make predictions where appropriate
• solve problems that involve identifying, analysing and describing associations between two numerical variables
• construct, interpret and analyse scatterplots for bivariate numerical data in practical contexts
  • demonstrate an awareness of issues of privacy and bias, ethics, and responsiveness to diverse groups and cultures when collecting and using data
  • investigate using biometric data obtained by measuring the body or by accessing published data from sources including government organisations, and determine if any associations exist between identified variables

MS-S5 The Normal DistributionYear 12

• recognise a random variable that is normally distributed, justifying their reasoning, and draw an appropriate 'bell-shaped' frequency distribution curve to represent it
  • identify that the mean and median are approximately equal for data arising from a random variable that is normally distributed
• calculate the $z$-score (standardised score) corresponding to a particular value in a dataset
  • use the formula $z=\frac{x-\mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation
  • describe the $z$-score as the number of standard deviations a value lies above or below the mean
  • recognise that the set of $z$-scores for data arising from a random variable that is normally distributed has a mean of 0 and a standard deviation of 1
• use calculated $z$-scores to compare scores from different datasets, for example comparing student's subject examination scores
• use collected data to illustrate that, for normally distributed random variables, approximately 68% of data will have $z$-scores between -1 and 1, approximately 95% of data will have $z$-scores between -2 and 2 and approximately 99.7% of data will have $z$-scores between -3 and 3 (known as the empirical rule)
  • apply the empirical rule to a variety of problems
  • indicate by shading where results sit within the normal distribution, eg where the top 10% of data lies
• use $z$-scores to identify probabilities of events less or more extreme than a given event
  • use statistical tables to determine probabilities
  • use technology to determine probabilities
• use $z$-scores to make judgements related to outcomes of a given event or sets of data

Networks

MS-N2 Network ConceptsYear 12

N2.1: Networks

• identify and use network terminology: vertices, edges, paths, the degree of a vertex, directed networks and weighted edges
• solve problems involving network diagrams
  • recognise circumstances in which networks could be used, eg the cost of connecting various locations on a university campus with computer cables
  • given a map, draw a network to represent the map, eg travel times for the stages of a planned journey
  • draw a network diagram to represent information given in a table
  • investigate and solve practical problems, eg planning a garbage bin collection route

N2.2: Shortest paths

• determine the minimum spanning tree of a given network with weighted edges
  • determine the minimum spanning tree by using Kruskal's or Prim's alogrithms or by inspection
  • determine the definition of a tree and a minimum spanning tree for a given network
  • use minimum spanning trees to solve minimal connector problems, eg minimising the length of cable needed to provide power from a single power station to substations in several towns
• find a shortest path from one place to another in a network with no more than 10 vertices
  • identify a shortest path on a network diagram
  • recognise a circumstance in which a shortest path is not necessarily the best path or contained in any minimum spanning tree

MS-N3 Critical Path AnalysisYear 12

• construct a network to represent the duration and interdependencies of activities that must be completed during a particular project, for example a student schedule, or preparing a meal
• given activity charts, prepare network diagrams and use critical path analysis to determine the minimum time for a project to be completed
  • use forward and backward scanning to determine the earliest starting time (EST) and latest starting time (LST) for each activity in the project
  • understand why the EST for an activity could be zero, and in what circumstances it would be greater than zero
  • calculate float times of non-critical activities
  • understand what is meant by critical path
  • use ESTs and LSTs to locate the critical path(s) for the project
• solve small-scale network flow problems, including the use of the 'maximum-flow minimum-cut' theorem, for example determining the maximum volume of oil that can flow through a network of pipes from an oil storage tank (the source) to a terminal (the sink)
  • convert information presented in a table into a network diagram
  • determine the flow capacity of a network and whether the flow is sufficient to meet the demand in various contexts