# Australian Curriculum Standards (ACARA) — Grade 12

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Based on the ACARA curriculum.

Plan, assess, and analyze learning aligned to these standards using Kiddom.

#### ACMEM.1.1.1.a.

Solve practical problems requiring basic number operations (ACMEM001)

#### ACMEM.1.1.1.b.

Apply arithmetic operations according to their correct order (ACMEM002)

#### ACMEM.1.1.1.c.

Ascertain the reasonableness of answers to arithmetic calculations (ACMEM003)

#### ACMEM.1.1.1.e.

Use a calculator for multi-step calculations (ACMEM005)

#### ACMEM.1.1.1.f.

Check results of calculations for accuracy (ACMEM006)

#### ACMEM.1.1.1.g.

Recognise the significance of place value after the decimal point (ACMEM007)

#### ACMEM.1.1.1.h.

Evaluate decimal fractions to the required number of decimal places (ACMEM008)

#### ACMEM.1.1.1.i.

Round up or round down numbers to the required number of decimal places (ACMEM009)

#### ACMEM.1.1.2.a.

Calculate a percentage of a given amount (ACMEM011)

#### ACMEM.1.1.2.b.

Determine one amount expressed as a percentage of another (ACMEM012)

#### ACMEM.1.1.2.c.

Apply percentage increases and decreases in situations; for example, mark-ups, discounts and GST. (ACMEM013)

#### ACMEM.1.1.3.a.

Identify common usage of rates; for example, km/h as a rate to describe speed, beats/minute as a rate to describe pulse (ACMEM014)

#### ACMEM.1.1.3.b.

Convert units of rates occurring in practical situations to solve problems (ACMEM015)

#### ACMEM.1.1.3.c.

Use rates to make comparisons; for example, using unit prices to compare best buys, comparing heart rates after exercise. (ACMEM016)

#### ACMEM.1.2.1.a.

Use metric units of length, their abbreviations, conversions between them, and appropriate levels of accuracy and choice of units (ACMEM017)

#### ACMEM.1.2.1.b.

Estimate lengths (ACMEM018)

#### ACMEM.1.2.1.c.

Convert between metric units of length and other length units (ACMEM019)

#### ACMEM.1.2.1.d.

Calculate perimeters of familiar shapes, including triangles, squares, rectangles, and composites of these. (ACMEM020)

#### ACMEM.1.2.2.b.

Estimate the areas of different shapes

#### ACMEM.1.2.2.d.

Calculate areas of rectangles and triangles. (ACMEM024)

#### ACMEM.1.2.3.a.

Use metric units of mass, their abbreviations, conversions between them, and appropriate choices of units (ACMEM025)

#### ACMEM.1.2.3.b.

Estimate the mass of different objects. (ACMEM026)

#### ACMEM.1.2.4.a.

Use metric units of volume, their abbreviations, conversions between them, and appropriate choices of units (ACMEM027)

#### ACMEM.1.2.4.b.

Understand the relationship between volume and capacity (ACMEM028)

#### ACMEM.1.2.4.d.

Calculate the volume of objects, such as cubes and rectangular and triangular prisms. (ACMEM030)

#### ACMEM.1.3.1.a.

Substitute numerical values into algebraic expressions; for example, substitute different values of x to evaluate the expressions 3x/5, 5(2x-4). (ACMEM035)

#### ACMEM.1.3.2.a.

substitute given values for the other pronumerals in a mathematical formula to find the value of the subject of the formula. (ACMEM036)

#### ACMEM.1.4.1.a.

Interpret information presented in graphs, such as conversion graphs, line graphs, step graphs, column graphs and picture graphs (ACMEM037)

#### ACMEM.1.4.1.b.

Interpret information presented in two-way tables (ACMEM038)

#### ACMEM.1.4.1.c.

Discuss and interpret graphs found in the media and in factual texts. (ACMEM039)

#### ACMEM.1.4.2.a.

Determine which type of graph is best used to display a dataset (ACMEM040)

#### ACMEM.1.4.2.b.

Use spreadsheets to tabulate and graph data (ACMEM041)

#### ACMEM.2.1.1.a.

Identify examples of categorical data (ACMEM043)

#### ACMEM.2.1.1.b.

Identify examples of numerical data. (ACMEM044)

#### ACMEM.2.1.2.a.

Display categorical data in tables and column graphs (ACMEM045)

#### ACMEM.2.1.2.b.

Display numerical data as frequency distributions, dot plots, stem and leaf plots, and histograms (ACMEM046)

#### ACMEM.2.1.2.c.

Recognise and identify outliers (ACMEM047)

#### ACMEM.2.1.2.d.

Compare the suitability of different methods of data presentation in real-world contexts. (ACMEM048)

#### ACMEM.2.1.3.a.

Identify the mode (ACMEM049)

#### ACMEM.2.1.3.b.

Calculate measures of central tendency, the arithmetic mean and the median (ACMEM050)

#### ACMEM.2.1.3.c.

Investigate the suitability of measures of central tendency in various real-world contexts (ACMEM051)

#### ACMEM.2.1.3.d.

Investigate the effect of outliers on the mean and the median (ACMEM052)

#### ACMEM.2.1.3.e.

Calculate and interpret quartiles, deciles and percentiles (ACMEM053)

#### ACMEM.2.1.3.f.

Use informal ways of describing spread, such as spread out/dispersed, tightly packed, clusters, gaps, more/less dense regions, outliers (ACMEM054)

#### ACMEM.2.1.3.g.

Calculate and interpret statistical measures of spread, such as the range, interquartile range and standard deviation (ACMEM055)

#### ACMEM.2.1.3.h.

Investigate real-world examples from the media illustrating inappropriate uses, or misuses, of measures of central tendency and spread. (ACMEM056)

#### ACMEM.2.1.4.a.

Compare back-to-back stem plots for different data-sets (ACMEM057)

#### ACMEM.2.1.4.c.

Construct box plots using a five number summary (ACMEM059)

#### ACMEM.2.1.4.d.

Compare the characteristics of the shape of histograms using symmetry, skewness and bimodality. (ACMEM060)

#### ACMEM.2.2.1.a.

Review calculating a percentage of a given amount (ACMEM061)

#### ACMEM.2.2.1.b.

Review one amount expressed as a percentage of another. (ACMEM062)

#### ACMEM.2.2.2.b.

Calculate simple interest for different rates and periods. (ACMEM064)

#### ACMEM.2.3.1.a.

Demonstrate an understanding of the elementary ideas and notation of ratio (ACMEM065)

#### ACMEM.2.3.1.b.

Understand the relationship between fractions and ratio (ACMEM066)

#### ACMEM.2.3.1.c.

Express a ratio in simplest form (ACMEM067)

#### ACMEM.2.3.1.d.

Find the ratio of two quantities (ACMEM068)

#### ACMEM.2.3.1.e.

Divide a quantity in a given ratio (ACMEM069)

#### ACMEM.2.3.1.f.

Use ratio to describe simple scales. (ACMEM070)

#### ACMEM.2.3.2.a.

Review identifying common usage of rates such as km/h (ACMEM071)

#### ACMEM.2.3.2.b.

Convert between units for rates; for example, km/h to m/s, mL/min to L/h (ACMEM072)

#### ACMEM.2.3.2.c.

Complete calculations with rates, including solving problems involving direct proportion in terms of rate. (ACMEM073)

#### ACMEM.2.3.2.d.

Use rates to make comparisons (ACMEM074)

#### ACMEM.2.3.2.e.

Use rates to determine costs; for example, calculating the cost of a tradesman using rates per hour, call-out fees. (ACMEM075)

#### ACMEM.2.4.1.a.

Use units of time, conversions between units, fractional, digital and decimal representations (ACMEM076)

#### ACMEM.2.4.1.b.

Represent time using 12-hour and 24-hour clocks (ACMEM077)

#### ACMEM.2.4.1.d.

Interpret timetables, such as bus, train and ferry timetables (ACMEM079)

#### ACMEM.2.4.1.e.

Use several timetables and electronic technologies to plan the most time-efficient routes (ACMEM080)

#### ACMEM.2.4.1.f.

Interpret complex timetables, such as tide charts, sunrise charts and moon phases (ACMEM081)

#### ACMEM.2.4.1.g.

Compare the time taken to travel a specific distance with various modes of transport (ACMEM082)

#### ACMEM.2.4.2.a.

Use scales to find distances, such as on maps; for example, road maps, street maps, bushwalking maps, online maps and cadastral maps (ACMEM083)

#### ACMEM.2.4.3.b.

Calculate speed, distance or time using the formula speed = distance/time (ACMEM086)

#### ACMEM.2.4.3.c.

Calculate the time or costs for a journey from distances estimated from maps (ACMEM087)

#### ACMEM.2.4.3.d.

Interpret distance-versus-time graphs (ACMEM088)

#### ACMEM.2.4.3.e.

Calculate and interpret average speed; for example, a 4-hour trip covering 250 km. (ACMEM089)

#### ACMEM.3.1.1.a.

Review metric units of length, their abbreviations, conversions between them, estimation of lengths, and appropriate choices of units (ACMEM090)

#### ACMEM.3.1.1.b.

Calculate perimeters of familiar shapes, including triangles, squares, rectangles, polygons, circles, arc lengths, and composites of these. (ACMEM091)

#### ACMEM.3.1.2.b.

Use formulas to calculate areas of regular shapes, including triangles, squares, rectangles, parallelograms, trapeziums, circles and sectors (ACMEM093)

#### ACMEM.3.1.2.c.

Find the area of irregular figures by decomposition into regular shapes (ACMEM094)

#### ACMEM.3.1.2.d.

Find the surface area of familiar solids, including cubes, rectangular and triangular prisms, spheres and cylinders (ACMEM095)

#### ACMEM.3.1.2.e.

Find the surface area of pyramids, such as rectangular- and triangular-based pyramids (ACMEM096)

#### ACMEM.3.1.3.a.

review metric units of mass (and weight), their abbreviations, conversions between them, and appropriate choices of units (ACMEM098)

#### ACMEM.3.1.3.b.

recognise the need for milligrams (ACMEM099)

#### ACMEM.3.1.3.c.

convert between grams and milligrams. (ACMEM100)

#### ACMEM.3.1.4.a.

Review metric units of volume, their abbreviations, conversions between them, and appropriate choices of units (ACMEM101)

#### ACMEM.3.1.4.b.

Recognise relations between volume and capacity, recognising that 1cm^3 = 1mL and 1m^3 = 1kL (ACMEM102)

#### ACMEM.3.1.4.c.

Use formulas to find the volume and capacity of regular objects such as cubes, rectangular and triangular prisms and cylinders (ACMEM103)

#### ACMEM.3.1.4.d.

Use formulas to find the volume of pyramids and spheres. (ACMEM104)

#### ACMEM.3.2.1.a.

Recognise the properties of common two-dimensional geometric shapes and three-dimensional solids (ACMEM105)

#### ACMEM.3.2.1.c.

Use symbols and conventions for the representation of geometric information; for example, point, line, ray, angle, diagonal, edge, curve, face and vertex. (ACMEM107)

#### ACMEM.3.2.2.a.

Interpret commonly used symbols and abbreviations in scale drawings (ACMEM108)

#### ACMEM.3.2.2.b.

Find actual measurements from scale drawings, such as lengths, perimeters and areas (ACMEM109)

#### ACMEM.3.2.2.c.

Estimate and compare quantities, materials and costs using actual measurements from scale drawings; for example, using measurements for packaging, clothes, painting, bricklaying and landscaping. (ACMEM110)

#### ACMEM.3.2.3.a.

Understand and apply drawing conventions of scale drawings, such as scales in ratio, clear indications of dimensions, and clear labelling (ACMEM111)

#### ACMEM.3.2.3.b.

Construct scale drawings by hand and by using software packages. (ACMEM112)

#### ACMEM.3.2.4.c.

Interpret diagrams of three-dimensional objects. (ACMEM115)

#### ACMEM.3.2.5.a.

Apply Pythagoras theorem to solve problems (ACMEM116)

#### ACMEM.3.2.5.b.

Apply the tangent ratio to find unknown angles and sides in right-angled triangles (ACMEM117)

#### ACMEM.3.2.5.c.

Work with the concepts of angle of elevation and angle of depression (ACMEM118)

#### ACMEM.3.2.5.d.

Apply the cosine and sine ratios to find unknown angles and sides in right-angled triangles (ACMEM119)

#### ACMEM.3.2.5.e.

Solve problems involving bearings. (ACMEM120)

#### ACMEM.3.3.1.a.

Demonstrate familiarity with Cartesian coordinates in two dimensions by plotting points on the Cartesian plane (ACMEM121)

#### ACMEM.3.3.1.b.

Generate tables of values for linear functions, including for negative values of (ACMEM122)

#### ACMEM.3.3.1.c.

Graph linear functions for all values of with pencil and paper and with graphing software. (ACMEM123)

#### ACMEM.3.3.2.a.

Interpret and use graphs in practical situations, including travel graphs and conversion graphs (ACMEM124)

#### ACMEM.3.3.2.b.

Draw graphs from given data to represent practical situations (ACMEM125)

#### ACMEM.3.3.2.c.

Interpret the point of intersection and other important features of given graphs of two linear functions drawn from practical contexts; for example, the break-even point. (ACMEM126)

#### ACMEM.3.4.1.a.

Investigate the procedure for conducting a census (ACMEM127)

#### ACMEM.3.4.2.a.

Understand the purpose of sampling to provide an estimate of population values when a census is not used (ACMEM129)

#### ACMEM.3.4.2.b.

Investigate the different kinds of samples; for example, systematic samples, self-selected samples, simple random samples (ACMEM130)

#### ACMEM.3.4.2.c.

Investigate the advantages and disadvantages of these kinds of samples; for example, comparing simple random samples with self-selected samples. (ACMEM131)

#### ACMEM.3.4.3.a.

Identify the target population to be surveyed (ACMEM132)

#### ACMEM.3.4.3.b.

Investigate questionnaire design principles; for example, simple language, unambiguous questions, consideration of number of choices, issues of privacy and ethics, and freedom from bias. (ACMEM133)

#### ACMEM.3.4.4.a.

Describe the faults in the collection of data process (ACMEM134)

#### ACMEM.3.4.4.b.

Describe sources of error in surveys; for example, sampling error and measurement error (ACMEM135)

#### ACMEM.3.4.4.c.

Investigate the possible misrepresentation of the results of a survey due to misunderstanding the procedure, or misunderstanding the reliability of generalising the survey findings to the entire population (ACMEM136)

#### ACMEM.3.4.4.d.

Investigate errors and misrepresentation in surveys, including examples of media misrepresentations of surveys. (ACMEM137)

#### ACMEM.3.4.5.a.

Describe the patterns and features of bivariate data (ACMEM138)

#### ACMEM.3.4.5.b.

Describe the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak). (ACMEM139)

#### ACMEM.3.4.6.a.

Identify the dependent and independent variable (ACMEM140)

#### ACMEM.3.4.6.b.

Find the line of best fit by eye (ACMEM141)

#### ACMEM.3.4.6.c.

Use technology to find the line of best fit (ACMEM142)

#### ACMEM.3.4.6.d.

Interpret relationships in terms of the variables (ACMEM143)

#### ACMEM.3.4.6.e.

Use technology to find the correlation coefficient (an indicator of the strength of linear association) (ACMEM144)

#### ACMEM.3.4.6.f.

Use the line of best fit to make predictions, both by interpolation and extrapolation (ACMEM145)

#### ACMEM.3.4.6.g.

Recognise the dangers of extrapolation (ACMEM146)

#### ACMEM.3.4.6.h.

Distinguish between causality and correlation through examples. (ACMEM147)

#### ACMEM.4.1.1.a.

Interpret commonly used probability statements, including possible, probable, likely, certain (ACMEM148)

#### ACMEM.4.1.1.b.

Describe ways of expressing probabilities formally using fractions, decimals, ratios, and percentages. (ACMEM149)

#### ACMEM.4.1.2.a.

Perform simulations of experiments using technology (ACMEM150)

#### ACMEM.4.1.2.b.

Recognise that the repetition of chance events is likely to produce different results (ACMEM151)

#### ACMEM.4.1.2.c.

Identify relative frequency as probability (ACMEM152)

#### ACMEM.4.1.2.d.

Identify factors that could complicate the simulation of real-world events. (ACMEM153)

#### ACMEM.4.1.3.c.

Use arrays or tree diagrams to determine the outcomes and the probabilities for experiments. (ACMEM156)

#### ACMEM.4.1.4.a.

Determine the probabilities associated with simple games (ACMEM157)

#### ACMEM.4.1.4.b.

Determine the probabilities of occurrence of simple traffic-light problems. (ACMEM158)

#### ACMEM.4.2.1.a.

Locate positions on Earths surface given latitude and longitude using GPS, a globe, an atlas, and digital technologies (ACMEM159)

#### ACMEM.4.2.1.b.

Find distances between two places on Earth on the same longitude (ACMEM160)

#### ACMEM.4.2.1.c.

Find distances between two places on Earth using appropriate technology. (ACMEM161)

#### ACMEM.4.2.2.a.

Understand the link between longitude and time (ACMEM162)

#### ACMEM.4.3.1.a.

Review the principles of simple interest (ACMEM168)

#### ACMEM.4.3.1.b.

Understand the concept of compound interest as a recurrence relation (ACMEM169)

#### ACMEM.4.3.1.c.

Consider similar problems involving compounding; for example, population growth (ACMEM170)

#### ACMEM.4.3.1.d.

Use technology to calculate the future value of a compound interest loan or investment and the total interest paid or earned (ACMEM171)

#### ACMEM.4.3.1.e.

Use technology to compare, numerically and graphically, the growth of simple interest and compound interest loans and investments (ACMEM172)

#### ACMEM.4.3.1.f.

Use technology to investigate the effect of the interest rate and the number of compounding periods on the future value of a loan or investment. (ACMEM173)

#### ACMEM.4.3.2.a.

Use technology and a recurrence relation to model a reducing balance loan (ACMEM174)

#### ACMEM.4.3.2.b.

Investigate the effect of the interest rate and repayment amount on the time taken to repay a loan. (ACMEM175)

#### ACMGM.1.1.1.a.

Review rates and percentages (ACMGM001)

#### ACMGM.1.1.1.b.

Calculate weekly or monthly wage from an annual salary, wages from an hourly rate including situations involving overtime and other allowances and earnings based on commission or piecework (ACMGM002)

#### ACMGM.1.1.1.c.

Calculate payments based on government allowances and pensions (ACMGM003)

#### ACMGM.1.1.1.d.

Prepare a personal budget for a given income taking into account fixed and discretionary spending (ACMGM004)

#### ACMGM.1.1.1.e.

Compare prices and values using the unit cost method (ACMGM005)

#### ACMGM.1.1.1.f.

Apply percentage increase or decrease in various contexts; for example, determining the impact of inflation on costs and wages over time, calculating percentage mark-ups and discounts, calculating GST, calculating profit or loss in absolute and percentage terms, and calculating simple and compound interest (ACMGM006)

#### ACMGM.1.1.2.a.

Use a spreadsheet to display examples of the above computations when multiple or repeated computations are required; for example, preparing a wage-sheet displaying the weekly earnings of workers in a fast food store where hours of employment and hourly rates of pay may differ, preparing a budget, or investigating the potential cost of owning and operating a car over a year. (ACMGM009)

#### ACMGM.1.2.1.a.

Substitute numerical values into linear algebraic and simple non-linear algebraic expressions, and evaluate (ACMGM010)

#### ACMGM.1.2.1.b.

Find the value of the subject of the formula, given the values of the other pronumerals in the formula (ACMGM011)

#### ACMGM.1.2.1.c.

Use a spreadsheet or an equivalent technology to construct a table of values from a formula, including two-by-two tables for formulas with two variable quantities; for example, a table displaying the body mass index (BMI) of people of different weights and heights. (ACMGM012)

#### ACMGM.1.2.2.a.

Use matrices for storing and displaying information that can be presented in rows and columns; for example, databases, links in social or road networks (ACMGM013)

#### ACMGM.1.2.2.b.

Recognise different types of matrices (row, column, square, zero, identity) and determine their size (ACMGM014)

#### ACMGM.1.2.2.c.

Perform matrix addition, subtraction, multiplication by a scalar, and matrix multiplication, including determining the power of a matrix using technology with matrix arithmetic capabilities when appropriate (ACMGM015)

#### ACMGM.1.2.2.d.

Use matrices, including matrix products and powers of matrices, to model and solve problems; for example, costing or pricing problems, squaring a matrix to determine the number of ways pairs of people in a communication network can communicate with each other via a third person. (ACMGM016)

#### ACMGM.1.3.1.a.

Review Pythagoras Theorem and use it to solve practical problems in two dimensions and for simple applications in three dimensions. (ACMGM017)

#### ACMGM.1.3.2.a.

Solve practical problems requiring the calculation of perimeters and areas of circles, sectors of circles, triangles, rectangles, parallelograms and composites (ACMGM018)

#### ACMGM.1.3.2.b.

Calculate the volumes of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the volume of water contained in a swimming pool (ACMGM019)

#### ACMGM.1.3.2.c.

Calculate the surface areas of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the surface area of a cylindrical food container. (ACMGM020)

#### ACMGM.1.3.3.a.

Review the conditions for similarity of two-dimensional figures including similar triangles (ACMGM021)

#### ACMGM.1.3.3.b.

Use the scale factor for two similar figures to solve linear scaling problems (ACMGM022)

#### ACMGM.1.3.3.c.

Obtain measurements from scale drawings, such as maps or building plans, to solve problems (ACMGM023)

#### ACMGM.1.3.3.d.

Obtain a scale factor and use it to solve scaling problems involving the calculation of the areas of similar figures (ACMGM024)

#### ACMGM.1.3.3.e.

Obtain a scale factor and use it to solve scaling problems involving the calculation of surface areas and volumes of similar solids. (ACMGM025)

#### ACMGM.2.1.1.a.

Review the statistical investigation process; for example, identifying a problem and posing a statistical question, collecting or obtaining data, analysing the data, interpreting and communicating the results. (ACMGM026)

#### ACMGM.2.1.2.a.

Classify a categorical variable as ordinal, such as income level (high, medium, low), or nominal, such as place of birth (Australia, overseas), and use tables and bar charts to organise and display the data (ACMGM027)

#### ACMGM.2.1.2.c.

With the aid of an appropriate graphical display (chosen from dot plot, stem plot, bar chart or histogram), describe the distribution of a numerical dataset in terms of modality (uni or multimodal), shape (symmetric versus positively or negatively skewed), location and spread and outliers, and interpret this information in the context of the data (ACMGM029)

#### ACMGM.2.1.2.d.

Determine the mean and standard deviation of a dataset and use these statistics as measures of location and spread of a data distribution, being aware of their limitations. (ACMGM030)

#### ACMGM.2.1.3.b.

Compare groups on a single numerical variable using medians, means, IQRs, ranges or standard deviations, as appropriate; interpret the differences observed in the context of the data; and report the findings in a systematic and concise manner (ACMGM032)

#### ACMGM.2.1.3.c.

Implement the statistical investigation process to answer questions that involve comparing the data for a numerical variable across two or more groups; for example, are Year 11 students the fittest in the school? (ACMGM033)

#### ACMGM.2.2.1.a.

Review the use of the trigonometric ratios to find the length of an unknown side or the size of an unknown angle in a right-angled triangle (ACMGM034)

#### ACMGM.2.2.1.d.

Solve practical problems involving the trigonometry of right-angled and non-right-angled triangles, including problems involving angles of elevation and depression and the use of bearings in navigation. (ACMGM037)

#### ACMGM.2.3.1.a.

Identify and solve linear equations (ACMGM038)

#### ACMGM.2.3.1.b.

Develop a linear formula from a word description (ACMGM039)

#### ACMGM.2.3.2.a.

Construct straight-line graphs both with and without the aid of technology (ACMGM040)

#### ACMGM.2.3.2.b.

Determine the slope and intercepts of a straight-line graph from both its equation and its plot (ACMGM041)

#### ACMGM.2.3.2.c.

Interpret, in context, the slope and intercept of a straight-line graph used to model and analyse a practical situation (ACMGM042)

#### ACMGM.2.3.2.d.

Construct and analyse a straight-line graph to model a given linear relationship; for example, modelling the cost of filling a fuel tank of a car against the number of litres of petrol required. (ACMGM043)

#### ACMGM.2.3.3.a.

Solve a pair of simultaneous linear equations, using technology when appropriate (ACMGM044)

#### ACMGM.2.3.3.b.

Solve practical problems that involve finding the point of intersection of two straight-line graphs; for example, determining the break-even point where cost and revenue are represented by linear equations. (ACMGM045)

#### ACMGM.3.1.1.a.

Review the statistical investigation process; for example, identifying a problem and posing a statistical question, collecting or obtaining data, analysing the data, interpreting and communicating the results. (ACMGM048)

#### ACMGM.3.1.2.a.

Construct two-way frequency tables and determine the associated row and column sums and percentages (ACMGM049)

#### ACMGM.3.1.2.b.

Use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association (ACMGM050)

#### ACMGM.3.1.2.c.

Describe an association in terms of differences observed in percentages across categories in a systematic and concise manner, and interpret this in the context of the data. (ACMGM051)

#### ACMGM.3.1.3.a.

Construct a scatterplot to identify patterns in the data suggesting the presence of an association (ACMGM052)

#### ACMGM.3.1.3.b.

Describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak) (ACMGM053)

#### ACMGM.3.1.3.c.

Calculate and interpret the correlation coefficient (r) to quantify the strength of a linear association. (ACMGM054)

#### ACMGM.3.1.4.a.

Identify the response variable and the explanatory variable (ACMGM055)

#### ACMGM.3.1.4.b.

Use a scatterplot to identify the nature of the relationship between variables (ACMGM056)

#### ACMGM.3.1.4.c.

Model a linear relationship by fitting a least-squares line to the data (ACMGM057)

#### ACMGM.3.1.4.d.

Use a residual plot to assess the appropriateness of fitting a linear model to the data (ACMGM058)

#### ACMGM.3.1.4.e.

Interpret the intercept and slope of the fitted line (ACMGM059)

#### ACMGM.3.1.4.g.

Use the equation of a fitted line to make predictions (ACMGM061)

#### ACMGM.3.1.4.h.

Distinguish between interpolation and extrapolation when using the fitted line to make predictions, recognising the potential dangers of extrapolation (ACMGM062)

#### ACMGM.3.1.4.i.

Write up the results of the above analysis in a systematic and concise manner. (ACMGM063)

#### ACMGM.3.1.5.a.

Recognise that an observed association between two variables does not necessarily mean that there is a causal relationship between them (ACMGM064)

#### ACMGM.3.1.5.b.

Identify possible non-causal explanations for an association, including coincidence and confounding due to a common response to another variable, and communicate these explanations in a systematic and concise manner. (ACMGM065)

#### ACMGM.3.1.6.a.

Implement the statistical investigation process to answer questions that involve identifying, analysing and describing associations between two categorical variables or between two numerical variables; for example, is there an association between attitude to capital punishment (agree with, no opinion, disagree with) and sex (male, female)? Is there an association between height and foot length? (ACMGM066)

#### ACMGM.3.2.1.b.

Display the terms of an arithmetic sequence in both tabular and graphical form and demonstrate that arithmetic sequences can be used to model linear growth and decay in discrete situations (ACMGM068)

#### ACMGM.3.2.1.c.

Deduce a rule for the nth term of a particular arithmetic sequence from the pattern of the terms in an arithmetic sequence, and use this rule to make predictions (ACMGM069)

#### ACMGM.3.2.1.d.

Use arithmetic sequences to model and analyse practical situations involving linear growth or decay; for example, analyzing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, or calculating the value of an office photocopier at the end of each year using the straight-line method or the unit cost method of depreciation. (ACMGM070)

#### ACMGM.3.2.2.b.

Display the terms of a geometric sequence in both tabular and graphical form and demonstrate that geometric sequences can be used to model exponential growth and decay in discrete situations (ACMGM072)

#### ACMGM.3.2.2.c.

Deduce a rule for the nth term of a particular geometric sequence from the pattern of the terms in the sequence, and use this rule to make predictions (ACMGM073)

#### ACMGM.3.2.2.d.

Use geometric sequences to model and analyse (numerically, or graphically only) practical problems involving geometric growth and decay; for example, analysing a compound interest loan or investment, the growth of a bacterial population that doubles in size each hour, the decreasing height of the bounce of a ball at each bounce; or calculating the value of office furniture at the end of each year using the declining (reducing) balance method to depreciate. (ACMGM074)

#### ACMGM.3.2.3.a.

Use a general first-order linear recurrence relation to generate the terms of a sequence and to display it in both tabular and graphical form (ACMGM075)

#### ACMGM.3.2.3.b.

Recognise that a sequence generated by a first-order linear recurrence relation can have a long term increasing, decreasing or steady-state solution (ACMGM076)

#### ACMGM.3.2.3.c.

Use first-order linear recurrence relations to model and analyse (numerically or graphically only) practical problems; for example, investigating the growth of a trout population in a lake recorded at the end of each year and where limited recreational fishing is permitted, or the amount owing on a reducing balance loan after each payment is made. (ACMGM077)

#### ACMGM.3.3.2.b.

Apply Eulers formula, v + f e = 2, to solve problems relating to planar graphs. (ACMGM082)

#### ACMGM.4.1.1.a.

Construct time series plots (ACMGM087)

#### ACMGM.4.1.1.b.

Describe time series plots by identifying features such as trend (long term direction), seasonality (systematic, calendar-related movements), and irregular fluctuations (unsystematic, short term fluctuations), and recognise when there are outliers; for example, one-off unanticipated events. (ACMGM088)

#### ACMGM.4.1.2.a.

Smooth time series data by using a simple moving average, including the use of spreadsheets to implement this process (ACMGM089)

#### ACMGM.4.1.2.b.

Calculate seasonal indices by using the average percentage method (ACMGM090)

#### ACMGM.4.1.2.c.

Deseasonalise a time series by using a seasonal index, including the use of spreadsheets to implement this process (ACMGM091)

#### ACMGM.4.1.2.d.

Fit a least-squares line to model long-term trends in time series data. (ACMGM092)

#### ACMGM.4.1.3.a.

Implement the statistical investigation process to answer questions that involve the analysis of time series data. (ACMGM093)

#### ACMGM.4.2.1.a.

Use a recurrence relation to model a compound interest loan or investment, and investigate (numerically or graphically) the effect of the interest rate and the number of compounding periods on the future value of the loan or investment (ACMGM094)

#### ACMGM.4.2.1.b.

Calculate the effective annual rate of interest and use the results to compare investment returns and cost of loans when interest is paid or charged daily, monthly, quarterly or six-monthly (ACMGM095)

#### ACMGM.4.2.1.c.

With the aid of a calculator or computer-based financial software, solve problems involving compound interest loans or investments; for example, determining the future value of a loan, the number of compounding periods for an investment to exceed a given value, the interest rate needed for an investment to exceed a given value. (ACMGM096)

#### ACMGM.4.2.2.a.

Use a recurrence relation to model a reducing balance loan and investigate (numerically or graphically) the effect of the interest rate and repayment amount on the time taken to repay the loan (ACMGM097)

#### ACMGM.4.2.2.b.

With the aid of a financial calculator or computer-based financial software, solve problems involving reducing balance loans; for example, determining the monthly repayments required to pay off a housing loan. (ACMGM098)

#### ACMMM.1.1.1.a.

Determine the coordinates of the midpoint of two points (ACMMM001)

#### ACMMM.1.1.1.b.

Examine examples of direct proportion and linearly related variables (ACMMM002)

#### ACMMM.1.1.1.c.

Recognise features of the graph of y=mx+c, including its linear nature, its intercepts and its slope or gradient (ACMMM003)

#### ACMMM.1.1.2.a.

Examine examples of quadratically related variables (ACMMM006)

#### ACMMM.1.1.2.b.

Recognise features of the graphs of y=x^2, y=a(x-b)^2+c, and y=a(x-b)(x-c), including their parabolic nature, turning points, axes of symmetry and intercepts (ACMMM007)

#### ACMMM.1.1.2.c.

Solve quadratic equations using the quadratic formula and by completing the square (ACMMM008)

#### ACMMM.1.1.2.d.

Find the equation of a quadratic given sufficient information (ACMMM009)

#### ACMMM.1.1.2.e.

Find turning points and zeros of quadratics and understand the role of the discriminant (ACMMM010)

#### ACMMM.1.1.2.f.

Recognise features of the graph of the general quadratic y=ax^2+bx+c. (ACMMM011)

#### ACMMM.1.1.3.a.

Examine examples of inverse proportion (ACMMM012)

#### ACMMM.1.1.3.b.

Recognise features of the graphs of y=1/x and y=a/(x-b), including their hyperbolic shapes, and their asymptotes. (ACMMM013)

#### ACMMM.1.1.4.a.

Recognise features of the graphs of y=x^n for nN, n=-1 and n=1/2, including shape, and behaviour as x _ and x -_ (ACMMM014)

#### ACMMM.1.1.4.b.

Identify the coefficients and the degree of a polynomial (ACMMM015)

#### ACMMM.1.1.4.c.

Expand quadratic and cubic polynomials from factors (ACMMM016)

#### ACMMM.1.1.4.d.

Recognise features of the graphs of y=x^3, y=a(x-b)^3+c and y=k(x-a)(x-b)(x-c), including shape, intercepts and behaviour as x _ and x -_ (ACMMM017)

#### ACMMM.1.1.4.f.

Solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained. (ACMMM019)

#### ACMMM.1.1.5.a.

Recognise features of the graphs of x^2+y^2=r^2 and (x-a)^2 + (y-b)^2 = r^2, including their circular shapes, their centres and their radii (ACMMM020)

#### ACMMM.1.1.5.b.

Recognise features of the graph of y^2 = x including its parabolic shape and its axis of symmetry. (ACMMM021)

#### ACMMM.1.1.6.a.

Understand the concept of a function as a mapping between sets, and as a rule or a formula that defines one variable quantity in terms of another (ACMMM022)

#### ACMMM.1.1.6.b.

Use function notation, domain and range, independent and dependent variables (ACMMM023)

#### ACMMM.1.1.6.c.

Understand the concept of the graph of a function (ACMMM024)

#### ACMMM.1.1.6.d.

Examine translations and the graphs of y = f(x)+a and y = f(x+b) (ACMMM025)

#### ACMMM.1.1.6.e.

Examine dilations and the graphs of y=cf(x) and y=f(kx) (ACMMM026)

#### ACMMM.1.1.6.f.

Recognise the distinction between functions and relations, and the vertical line test. (ACMMM027)

#### ACMMM.1.2.1.a.

Review sine, cosine and tangent as ratios of side lengths in right-angled triangles (ACMMM028)

#### ACMMM.1.2.1.b.

Understand the unit circle definition of cos, sin and tan and periodicity using degrees (ACMMM029)

#### ACMMM.1.2.1.c.

Examine the relationship between the angle of inclination of a line and the gradient of that line (ACMMM030)

#### ACMMM.1.2.2.a.

Define and use radian measure and understand its relationship with degree measure (ACMMM032)

#### ACMMM.1.2.2.b.

calculate lengths of arcs and areas of sectors in circles. (ACMMM033)

#### ACMMM.1.2.3.a.

Understand the unit circle definition of cos, sin and tan and periodicity using radians (ACMMM034)

#### ACMMM.1.2.3.b.

Recognise the exact values of cos, sin and tan at integer multiples of /6 and /4 (ACMMM035)

#### ACMMM.1.2.3.c.

Recognise the graphs of y=sin x, y=cos x, and y=tan x on extended domains (ACMMM036)

#### ACMMM.1.2.3.d.

Examine amplitude changes and the graphs of y=a sin x and y=a cos x(ACMMM037)

#### ACMMM.1.2.3.e.

Examine period changes and the graphs of y=sin bx, y=cos bx, and y=tan bx (ACMMM038)

#### ACMMM.1.2.3.f.

Examine phase changes and the graphs of y=sin(x+c), y=cos(x+c) and (ACMMM039)

#### ACMMM.1.2.3.g.

y=tan(x+c) and the relationships sin(x+/2)=cos x and cos(x-/2)=sin x(ACMMM040)

#### ACMMM.1.2.3.h.

Prove and apply the angle sum and difference identities (ACMMM041)

#### ACMMM.1.2.3.i.

Identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems (ACMMM042)

#### ACMMM.1.2.3.j.

Solve equations involving trigonometric functions using technology, and algebraically in simple cases. (ACMMM043)

#### ACMMM.1.3.1.c.

Expand (x+y)^n for small positive integers n (ACMMM046)

#### ACMMM.1.3.2.a.

Review the concepts and language of outcomes, sample spaces and events as sets of outcomes (ACMMM049)

#### ACMMM.1.3.2.b.

Use set language and notation for events, including __ (or A) for the complement of an event A, A?B for the intersection of events A and B, and A?B for the union, and recognise mutually exclusive events (ACMMM050)

#### ACMMM.1.3.2.c.

Use everyday occurrences to illustrate set descriptions and representations of events, and set operations. (ACMMM051)

#### ACMMM.1.3.3.a.

Review probability as a measure of the likelihood of occurrence of an event (ACMMM052)

#### ACMMM.1.3.3.b.

Review the probability scale: 0 = _ P(A) _ 1 for each event A, with P(A)=0 if A is an impossibility and P(A) = 1 if A is a certainty (ACMMM053)

#### ACMMM.1.3.3.c.

Review the rules: P(__) = 1-P(A) and P(AB) = P(A) + P(B) P(AB) (ACMMM054)

#### ACMMM.1.3.3.d.

Use relative frequencies obtained from data as point estimates of probabilities. (ACMMM055)

#### ACMMM.1.3.4.a.

Understand the notion of a conditional probability and recognise and use language that indicates conditionality (ACMMM056)

#### ACMMM.1.3.4.b.

Use the notation P(A|B) and the formula P(AB) = P(A|B)P(B) (ACMMM057)

#### ACMMM.1.3.4.c.

Understand the notion of independence of an event A from an event B, as defined by P(A|B)=P(A) (ACMMM058)

#### ACMMM.1.3.4.d.

Establish and use the formula P(AB) = P(A)P(B) for independent events A and B, and recognise the symmetry of independence (ACMMM059)

#### ACMMM.1.3.4.e.

Use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events. (ACMMM060)

#### ACMMM.2.1.1.a.

Review indices (including fractional indices) and the index laws (ACMMM061)

#### ACMMM.2.1.1.b.

Use radicals and convert to and from fractional indices (ACMMM062)

#### ACMMM.2.1.1.c.

Understand and use scientific notation and significant figures. (ACMMM063)

#### ACMMM.2.1.2.a.

Establish and use the algebraic properties of exponential functions (ACMMM064)

#### ACMMM.2.1.2.b.

Recognise the qualitative features of the graph of y=a^x (a &gt; 0) including asymptotes, and of its translations (y = a^x + b and y = a^(x+c)) (ACMMM065)

#### ACMMM.2.1.2.d.

Solve equations involving exponential functions using technology, and algebraically in simple cases. (ACMMM067)

#### ACMMM.2.2.1.a.

Recognise and use the recursive definition of an arithmetic sequence: t_(n+1) = t_n + d (ACMMM068)

#### ACMMM.2.2.1.b.

Use the formula t_n = t_1 + (n-1) for the general term of an arithmetic sequence and recognise its linear nature (ACMMM069)

#### ACMMM.2.2.1.c.

Use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest (ACMMM070)

#### ACMMM.2.2.1.d.

Establish and use the formula for the sum of the first n terms of an arithmetic sequence. (ACMMM071)

#### ACMMM.2.2.2.a.

Recognise and use the recursive definition of a geometric sequence: t_(n+1) = rt_n (ACMMM072)

#### ACMMM.2.2.2.b.

Use the formula t_n = r^(n-1)t_1 for the general term of a geometric sequence and recognise its exponential nature (ACMMM073)

#### ACMMM.2.2.2.c.

Understand the limiting behaviour as n _ of the terms t_n in a geometric sequence and its dependence on the value of the common ratio r (ACMMM074)

#### ACMMM.2.2.2.d.

Establish and use the formula S_n = t_1 ((r^n-1)/(r-1)) for the sum of the first n terms of a geometric sequence (ACMMM075)

#### ACMMM.2.2.2.e.

Use geometric sequences in contexts involving geometric growth or decay, such as compound interest. (ACMMM076)

#### ACMMM.2.3.1.d.

Interpret the ratios (f(x+h)-f(x))/h and y/x as the slope or gradient of a chord or of the graph of interpret the ratios and as the slope or gradient of a chord or secant of the graph of y = f(x). (ACMMM080)

#### ACMMM.2.3.2.a.

Examine the behaviour of the difference quotient f(x+h)-f(x)/h as h 0 as an informal introduction to the concept of a limit (ACMMM081)

#### ACMMM.2.3.2.b.

Define the derivative f(x) as lim(h 0) f(x+h)-f(x)/h (ACMMM082)

#### ACMMM.2.3.2.c.

Use the Leibniz notation for the derivative: dy/dx = lim(x 0) y/x and the correspondence dy/dx = f(x) where y=f(x) (ACMMM083)

#### ACMMM.2.3.2.d.

Interpret the derivative as the instantaneous rate of change (ACMMM084)

#### ACMMM.2.3.2.e.

Interpret the derivative as the slope or gradient of a tangent line of the graph of y=f(x). (ACMMM085)

#### ACMMM.2.3.3.a.

Estimate numerically the value of a derivative, for simple power functions (ACMMM086)

#### ACMMM.2.3.4.b.

Recognise and use linearity properties of the derivative (ACMMM090)

#### ACMMM.2.3.4.c.

Calculate derivatives of polynomials and other linear combinations of power functions. (ACMMM091)

#### ACMMM.2.3.5.a.

Find instantaneous rates of change (ACMMM092)

#### ACMMM.2.3.5.c.

Construct and interpret position-time graphs, with velocity as the slope of the tangent (ACMMM094)

#### ACMMM.2.3.5.d.

Sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as x _ and x -_ (ACMMM095)

#### ACMMM.2.3.6.a.

Calculate anti-derivatives of polynomial functions and apply to solving simple problems involving motion in a straight line. (ACMMM097)

#### ACMMM.3.1.1.a.

Estimate the limit of a^h1/h as h 0 using technology, for various values of a &gt; 0 (ACMMM098)

#### ACMMM.3.1.1.b.

Recognise that e is the unique number a for which the above limit is 1 (ACMMM099)

#### ACMMM.3.1.2.a.

Establish the formulas d/dx (sin x) = cos x, and d/dx (cos x) = -sin x by numerical estimations of the limits and informal proofs based on geometric constructions (ACMMM102)

#### ACMMM.3.1.2.b.

Use trigonometric functions and their derivatives to solve practical problems. (ACMMM103)

#### ACMMM.3.1.4.a.

Use the increments formula: y dy/dx x x to estimate the change in the dependent variable y resulting from changes in the independent variable x (ACMMM107)

#### ACMMM.3.1.4.b.

Understand the concept of the second derivative as the rate of change of the first derivative function (ACMMM108)

#### ACMMM.3.1.4.c.

Establish and use the formula (x^n)dx = (1/n+1)x^(n+1) + c for n -1 (ACMMM116)

#### ACMMM.3.1.4.d.

Understand the concepts of concavity and points of inflection and their relationship with the second derivative (ACMMM110)

#### ACMMM.3.1.4.e.

Understand and use the second derivative test for finding local maxima and minima (ACMMM111)

#### ACMMM.3.1.4.f.

Sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection (ACMMM112)

#### ACMMM.3.1.4.g.

Determine indefinite integrals of the form f(ax+b)dx (ACMMM120)

#### ACMMM.3.1.4.h.

Identify families of curves with the same derivative function (ACMMM121)

#### ACMMM.3.1.4.i.

Determine f(x), given f(x) and an initial condition f(a) = b (ACMMM122)

#### ACMMM.3.2.1.a.

Recognise anti-differentiation as the reverse of differentiation (ACMMM114)

#### ACMMM.3.2.2.a.

Examine the area problem, and use sums of the form i f(xi) xi to estimate the area under the curve y = f(x) (ACMMM124)

#### ACMMM.3.2.2.b.

Interpret the definite integral {a to b} f(x)dx as area under the curve y = f(x) if f(x) &gt; 0 (ACMMM125)

#### ACMMM.3.2.2.c.

Recognise the definite integral {a to b} f(x)dx as a limit of sums of the form i f(xi) xi (ACMMM126)

#### ACMMM.3.2.2.d.

Interpret {a to b} f(x)dx as a sum of signed areas (ACMMM127)

#### ACMMM.3.2.2.e.

Recognise and use the additivity and linearity of definite integrals. (ACMMM128)

#### ACMMM.3.2.3.a.

Understand the concept of the signed area function F(x) = {a to x} f(t)dt (ACMMM129)

#### ACMMM.3.2.3.b.

Understand and use the theorem: F(x) = d/dx ({a to x} f(t)dt) = f(x), and illustrate its proof geometrically (ACMMM130)

#### ACMMM.3.2.3.c.

Calculate total change by integrating instantaneous or marginal rate of change (ACMMM133)

#### ACMMM.3.2.3.d.

Identify contexts suitable for modelling by Bernoulli random variables (ACMMM144)

#### ACMMM.3.2.3.e.

Understand the formula {a to b} f(x)dx = F(b)-F(a) and use it to calculate definite integrals. (ACMMM131)

#### ACMMM.3.2.3.f.

Calculate the area between curves in simple cases (ACMMM134)

#### ACMMM.3.2.3.g.

Recognise the mean p and variance p(1-p) of the Bernoulli distribution with parameter p (ACMMM145)

#### ACMMM.3.2.3.h.

Use Bernoulli random variables and associated probabilities to model data and solve practical problems. (ACMMM146)

#### ACMMM.3.2.4.a.

Calculate the area under a curve (ACMMM132)

#### ACMMM.3.2.4.b.

Use the notation f(x)dx for anti-derivatives or indefinite integrals (ACMMM115)

#### ACMMM.3.2.4.d.

Establish and use the formula (e^x)dx = e^x + c (ACMMM117)

#### ACMMM.3.2.4.f.

Recognise and use linearity of anti-differentiation (ACMMM119)

#### ACMMM.3.3.1.a.

Understand the concepts of a discrete random variable and its associated probability function, and their use in modelling data (ACMMM136)

#### ACMMM.3.3.1.b.

Use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable (ACMMM137)

#### ACMMM.3.3.1.c.

Recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes (ACMMM138)

#### ACMMM.3.3.1.d.

Examine simple examples of non-uniform discrete random variables (ACMMM139)

#### ACMMM.3.3.1.e

Recognise the mean or expected value of a discrete random variable as a measurement of centre, and evaluate it in simple cases (ACMMM140)

#### ACMMM.3.3.1.f.

Recognise the variance and standard deviation of a discrete random variable as a measures of spread, and evaluate them in simple cases (ACMMM141)

#### ACMMM.3.3.1.g.

Use discrete random variables and associated probabilities to solve practical problems. (ACMMM142)

#### ACMMM.3.3.2.a.

Use a Bernoulli random variable as a model for two-outcome situations (ACMMM143)

#### ACMMM.3.3.3.a.

Understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of successes in n independent Bernoulli trials, with the same probability of success p in each trial (ACMMM147)

#### ACMMM.3.3.3.b.

Identify contexts suitable for modelling by binomial random variables (ACMMM148)

#### ACMMM.3.3.3.c.

Determine and use the probabilities P(X=r) = (n r) p^r (1-p)^(n-r) associated with the binomial distribution with parameters n and p; note the mean np and variance np(1-p) of a binomial distribution (ACMMM149)

#### ACMMM.3.3.3.d.

Use binomial distributions and associated probabilities to solve practical problems. (ACMMM150)

#### ACMMM.4.1.1.a.

Define logarithms as indices: a^x = b is equivalent to x = log_a b, i.e. a^(log_a b) = b (ACMMM151)

#### ACMMM.4.1.1.b.

Establish and use the algebraic properties of logarithms (ACMMM152)

#### ACMMM.4.1.1.c.

Recognise the inverse relationship between logarithms and exponentials: y = a^x is equivalent to x = log_a y (ACMMM153)

#### ACMMM.4.1.1.d.

Interpret and use logarithmic scales such as decibels in acoustics, the Richter Scale for earthquake magnitude, octaves in music, pH in chemistry (ACMMM154)

#### ACMMM.4.1.1.e.

Solve equations involving indices using logarithms (ACMMM155)

#### ACMMM.4.1.1.f.

Recognise the qualitative features of the graph of y = log_a x (a &gt; 1) including asymptotes, and of its translations y = log_a x+b and y = log_a (x+c) (ACMMM156)

#### ACMMM.4.1.1.g.

Solve simple equations involving logarithmic functions algebraically and graphically (ACMMM157)

#### ACMMM.4.1.1.h.

Identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems. (ACMMM158)

#### ACMMM.4.1.2.a.

Define the natural logarithm ln x = log_e x (ACMMM159)

#### ACMMM.4.1.2.b.

Recognise and use the inverse relationship of the functions y = e^x and y = ln x (ACMMM160)

#### ACMMM.4.2.1.a.

Use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable (ACMMM164)

#### ACMMM.4.2.1.b.

Understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts (ACMMM165)

#### ACMMM.4.2.1.c.

Recognise the expected value, variance and standard deviation of a continuous random variable and evaluate them in simple cases (ACMMM166)

#### ACMMM.4.2.2.a.

Identify contexts such as naturally occurring variation that are suitable for modelling by normal random variables (ACMMM168)

#### ACMMM.4.2.2.b.

Recognise features of the graph of the probability density function of the normal distribution with mean and standard deviation and the use of the standard normal distribution (ACMMM169)

#### ACMMM.4.2.2.c.

Calculate probabilities and quantiles associated with a given normal distribution using technology, and use these to solve practical problems. (ACMMM170)

#### ACMMM.4.3.1.a.

Understand the concept of a random sample (ACMMM171)

#### ACMMM.4.3.1.b.

Discuss sources of bias in samples, and procedures to ensure randomness (ACMMM172)

#### ACMMM.4.3.1.c.

Use graphical displays of simulated data to investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli. (ACMMM173)

#### ACMMM.4.3.2.a.

Understand the concept of the sample proportion p as a random variable whose value varies between samples, and the formulas for the mean p and standard deviation _(p(1-p)/n) of the sample proportion p (ACMMM174)

#### ACMMM.4.3.2.b.

Examine the approximate normality of the distribution of p for large samples (ACMMM175)

#### ACMMM.4.3.2.c.

Simulate repeated random sampling, for a variety of values of p and a range of sample sizes, to illustrate the distribution of p and the approximate standard normality p-p/_(p(1-p)/n) of where the closeness of the approximation depends on both n and p. (ACMMM176)

#### ACMMM.4.3.3.a.

The concept of an interval estimate for a parameter associated with a random variable (ACMMM177)

#### ACMMM.4.3.3.b.

Use the approximate confidence interval (p-z_((p(1-p)/n), p+z_(p(1-p)/n)), as an interval estimate for , where is the appropriate quantile for the standard normal distribution (ACMMM178)

#### ACMMM.4.3.3.c.

Define the approximate margin of error E=z_(p(1-p)/n) and understand the trade-off between margin of error and level of confidence (ACMMM179)

#### ACMMM.4.3.3.d.

Use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain p. (ACMMM180)

#### ACMSM.1.1.1.a.

Solve problems involving permutations (ACMSM001)

#### ACMSM.1.1.1.b.

Use the multiplication principle (ACMSM002)

#### ACMSM.1.1.1.c.

Use factorial notation (ACMSM003)

#### ACMSM.1.1.1.d.

Solve problems involving permutations and restrictions with or without repeated objects. (ACMSM004)

#### ACMSM.1.1.2.a.

Determine and use the formulas for finding the number of elements in the union of two and the union of three sets. (ACMSM005)

#### ACMSM.1.1.3.a.

Solve problems and prove results using the pigeon-hole principle. (ACMSM006)

#### ACMSM.1.2.1.a.

Examine examples of vectors including displacement and velocity (ACMSM010)

#### ACMSM.1.2.1.b.

Define and use the magnitude and direction of a vector (ACMSM011)

#### ACMSM.1.2.1.c.

Represent a scalar multiple of a vector (ACMSM012)

#### ACMSM.1.2.1.d.

Use the triangle rule to find the sum and difference of two vectors. (ACMSM013)

#### ACMSM.1.2.2.a.

Use ordered pair notation and column vector notation to represent a vector (ACMSM014)

#### ACMSM.1.2.2.b.

Define and use unit vectors and the perpendicular unit vectors i and j (ACMSM015)

#### ACMSM.1.2.2.c.

Express a vector in component form using the unit vectors i and j (ACMSM016)

#### ACMSM.1.2.2.d.

Examine and use addition and subtraction of vectors in component form (ACMSM017)

#### ACMSM.1.2.2.e.

Define and use multiplication by a scalar of a vector in component form (ACMSM018)

#### ACMSM.1.2.2.f.

Define and use scalar (dot) product (ACMSM019)

#### ACMSM.1.2.2.g.

Apply the scalar product to vectors expressed in component form (ACMSM020)

#### ACMSM.1.2.2.h.

Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021)

#### ACMSM.1.2.2.i.

Define and use projections of vectors (ACMSM022)

#### ACMSM.1.2.2.j.

Solve problems involving displacement, force and velocity involving the above concepts. (ACMSM023)

#### ACMSM.1.3.1.a.

Use implication, converse, equivalence, negation, contrapositive (ACMSM024)

#### ACMSM.1.3.1.c.

Use the symbols for implication (), equivalence (), and equality (=) (ACMSM026)

#### ACMSM.1.3.1.d.

Use the quantifiers for all and there exists (ACMSM027)

#### ACMSM.1.3.2.a.

An angle in a semicircle is a right angle (ACMSM029)

#### ACMSM.1.3.2.b.

The angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc (ACMSM030)

#### ACMSM.1.3.2.c.

Angles at the circumference of a circle subtended by the same arc are equal (ACMSM031)

#### ACMSM.1.3.2.d.

The opposite angles of a cyclic quadrilateral are supplementary (ACMSM032)

#### ACMSM.1.3.2.e.

Chords of equal length subtend equal angles at the centre and conversely chords subtending equal angles at the centre of a circle have the same length (ACMSM033)

#### ACMSM.1.3.2.f.

The alternate segment theorem (ACMSM034)

#### ACMSM.1.3.2.g.

When two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the intervals on the other chord (ACMSM035)

#### ACMSM.1.3.2.h.

When a secant (meeting the circle at A and B) and a tangent (meeting the circle at T) are drawn to a circle from an external point M, the square of the length of the tangent equals the product of the lengths to the circle on the secant. (AM x BM = TM^2) (ACMSM036)

#### ACMSM.1.3.2.i.

Suitable converses of some of the above results (ACMSM037)

#### ACMSM.1.3.2.j.

Solve problems finding unknown angles and lengths and prove further results using the results listed above. (ACMSM038)

#### ACMSM.1.3.3.a.

The diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039)

#### ACMSM.1.3.3.b.

Midpoints of the sides of a quadrilateral join to form a parallelogram (ACMSM040)

#### ACMSM.1.3.3.c.

The sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides. (ACMSM041)

#### ACMSM.2.1.1.a.

Find all solutions of f(a(x-b))=c where f is one of sin, cos or tan (ACMSM042)

#### ACMSM.2.1.1.b.

Graph functions with rules of the form y=f(a(x-b)) where f is one of sin, cos or tan. (ACMSM043)

#### ACMSM.2.1.2.a.

Prove and apply the angle sum, difference and double angle identities. (ACMSM044)

#### ACMSM.2.1.3.a.

Define the reciprocal trigonometric functions, sketch their graphs, and graph simple transformations of them. (ACMSM045)

#### ACMSM.2.1.4.a.

Prove and apply the Pythagorean identities (ACMSM046)

#### ACMSM.2.1.4.b.

Prove and apply the identities for products of sines and cosines expressed as sums and differences (ACMSM047)

#### ACMSM.2.1.4.c.

Convert sums a cos x+b sin x to R cos (x) or R sin(x) and apply these to sketch graphs, solve equations of the form a cos x + b sin x = c and solve problems (ACMSM048)

#### ACMSM.2.1.4.d.

Prove and apply other trigonometric identities such as cos 3x = 4 cos^3 x-3 cos x. (ACMSM049)

#### ACMSM.2.1.5.a.

Model periodic motion using sine and cosine functions and understand the relevance of the period and amplitude of these functions in the model. (ACMSM050)

#### ACMSM.2.2.1.a.

Understand the matrix definition and notation (ACMSM051)

#### ACMSM.2.2.1.b.

Define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication, multiplicative identity and inverse (ACMSM052)

#### ACMSM.2.2.1.c.

Calculate the determinant and inverse of 2x2 matrices and solve matrix equations of the form AX=B, where A is a 2x2 matrix and X and B are column vectors. (ACMSM053)

#### ACMSM.2.3.2.a.

Express rational numbers as terminating or eventually recurring decimals and vice versa (ACMSM062)

#### ACMSM.2.3.4.a.

Define the imaginary number as a root of the equation x^2 = -1 (ACMSM067)

#### ACMSM.2.3.4.b.

Use complex numbers in the form a+bi where a and b are the real and imaginary parts (ACMSM068)

#### ACMSM.2.3.4.c.

Determine and use complex conjugates (ACMSM069)

#### ACMSM.2.3.4.d.

Perform complex-number arithmetic: addition, subtraction, multiplication and division. (ACMSM070)

#### ACMSM.2.3.5.a.

Consider complex numbers as points in a plane with real and imaginary parts as Cartesian coordinates (ACMSM071)

#### ACMSM.2.3.5.b.

Examine addition of complex numbers as vector addition in the complex plane (ACMSM072)

#### ACMSM.2.3.5.c.

Understand and use location of complex conjugates in the complex plane. (ACMSM073)

#### ACMSM.2.3.6.a.

Use the general solution of real quadratic equations (ACMSM074)

#### ACMSM.2.3.6.b.

Determine complex conjugate solutions of real quadratic equations (ACMSM075)

#### ACMSM.2.3.6.c.

Determine linear factors of real quadratic polynomials. (ACMSM076)

#### ACMSM.3.1.1.a.

Review real and imaginary parts Re(z) and Im(z) of a complex number z (ACMSM077)

#### ACMSM.3.1.1.c.

Review complex arithmetic using Cartesian forms. (ACMSM079)

#### ACMSM.3.1.5.a.

Prove and apply the factor theorem and the remainder theorem for polynomials (ACMSM089)

#### ACMSM.3.1.5.b.

Consider conjugate roots for polynomials with real coefficients (ACMSM090)

#### ACMSM.3.1.5.c.

Solve simple polynomial equations. (ACMSM091)

#### ACMSM.3.2.1.a.

Determine when the composition of two functions is defined (ACMSM092)

#### ACMSM.3.2.1.b.

Find the composition of two functions (ACMSM093)

#### ACMSM.3.2.1.c.

Determine if a function is one-to-one (ACMSM094)

#### ACMSM.3.2.1.d.

Consider inverses of one-to-one function (ACMSM095)

#### ACMSM.3.2.1.e.

Examine the reflection property of the graph of a function and the graph of its inverse. (ACMSM096)

#### ACMSM.3.2.1.f.

Sketching graphs: (ACMSM097)

#### ACMSM.3.2.1.g.

Use and apply the notation |x| for the absolute value for the real number x and the graph of y=|x| (ACMSM098)

#### ACMSM.3.2.1.i.

Sketch the graphs of simple rational functions where the numerator and denominator are polynomials of low degree. (ACMSM100)

#### ACMSM.3.3.2.b.

Examine the three cases for solutions of systems of equations a unique solution, no solution, and infinitely many solutions and the geometric interpretation of a solution of a system of equations with three variables. (ACMSM110)

#### ACMSM.3.3.3.a.

Recognise the general form of a system of linear equations in several variables, and use elementary techniques of elimination to solve a system of linear equations (ACMSM109)

#### ACMSM.3.3.4.a.

Consider position of vectors as a function of time (ACMSM111)

#### ACMSM.3.3.4.b.

Derive the Cartesian equation of a path given as a vector equation in two dimensions including ellipses and hyperbolas (ACMSM112)

#### ACMSM.3.3.4.c.

Differentiate and integrate a vector function with respect to time (ACMSM113)

#### ACMSM.4.1.1.c.

Establish and use the formula (1/x)dx = ln |x| + c, for x 0 (ACMSM118)

#### ACMSM.4.1.2.a.

Calculate areas between curves determined by functions (ACMSM124)

#### ACMSM.4.1.2.c.

Use numerical integration using technology (ACMSM126)

#### ACMSM.4.1.2.d.

Use and apply the probability density function, f(t) = e^(-t) for t0, of the exponential random variable with parameter &gt;0 and use the exponential random variables and associated probabilities and quantiles to model data and solve practical problems. (ACMSM127)

#### ACMSM.4.2.1.c.

Solve simple first-order differential equations of the form dy/dx = f(x), differential equations of the form dy/dx = g(y) and, in general, differential equations of the form dy/dx = f(x)g(y) using separation of variables (ACMSM130)

#### ACMSM.4.3.1.a.

Examine the concept of the sample mean X as a random variable whose value varies between samples where X is a random variable with mean _ and the standard deviation (ACMSM137)

#### ACMSM.4.3.1.b.

Simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate properties of the distribution of x across samples of a fixed size n, including its mean _, its standard deviation /_n (where _ and are the mean and standard deviation of X), and its approximate normality if n is large (ACMSM138)

#### ACMSM.4.3.1.c.

Simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate the approximate standard normality of x-_/(s/_n) for large samples (n 30), where s is the sample standard deviation. (ACMSM139)

#### ACMSM.4.3.2.a.

Understand the concept of an interval estimate for a parameter associated with a random variable (ACMSM140)

#### ACMSM.4.3.2.b.

Examine the approximate confidence interval (x-(zs/_n), x+(zs/_n)), as an interval estimate for _, the population mean, where z is the appropriate quantile for the standard normal distribution (ACMSM141)

#### ACMSM.4.3.2.c.

Use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain _ (ACMSM142)

#### ACMSM.4.3.2.d.

Use x and s to estimate _ and , to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for _ (ACMSM143)

#### ACMSM.4.3.2.e.

Collect data and construct an approximate confidence interval to estimate a mean and to report on survey procedures and data quality. (ACMSM144)