Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
1 MATHS QUEST General Mathematics: Preliminary Course 2 Chapter 1: Earning money FM1: Financial mathematics Section Content Calculating salary payments (page 2) WE 1, 2, 3 Ex 1A Calculating salary payments (page 3) Calculating salary payments Dividing an annual salary from a periodical amount Calculating an annual salary from a periodical amount Calculating wage payments Calculating wages from an hourly rate Including overtime and special allowances Calculating commissions and royalties Calculation as a straight percentage Calculating part commission and part retainer Payment by piece Calculating wages (page 5) WE 4, 5, 6 Ex 1B Calculating wages (page 7) Computer application 1: Spreadsheets (page 9) Commission and royalties (page 10) WE 7, 8, 9, 10 Ex 1C Commission and royalties (page 13) Payment by piece (page 15) WE 11, 12, 13 Ex 1D Payment by piece (page 16) Working overtime (page 18) WE 14, 15, 16, 17 Ex 1E Working overtime (page 20) Computer application 2: Wages (page 22) Fixed incomes (page 23) Investigating government payments (page 23) Additions to and deductions from gross pay (page 24) WE 18, 19, 20, 21 Calculating overtime rates Using time-and-a-half and double time to calculate wages Completing time sheets Calculating income based on government payments Students to investigate government payments Additions and deductions from gross pay Students to calculate net pay given the amount of each deduction from gross pay Prescribed skills, knowledge and understanding calculating monthly, fortnightly, weekly, daily and hourly payments from salary calculating wages; incorporating hourly rates and penalty rates like overtime and special allowances (wet work, confined spaces, toxic substances, heat, heights) calculating earnings based on commission, piecework, royalties calculating earnings based on commission, piecework, royalties calculating wages; incorporating hourly rates and penalty rates like overtime and special allowances (wet work, confined spaces, toxic substances, heat, heights) calculating income based on government allowances such as the youth allowance and pensions determining deductions such as union fees, superannuation contributions, health fund instalments and tax instalments 3 Ex 1F Additions to and deductions from gross pay (page 26) Computer application 3: Wages template (page 28) Costs of banking (page 29) Examining bank fees and taxes (page 29) Budgeting (page 30) WE 22, 23, 24, 25 Ex 1G Budgeting (page 34) Summary (page 39) Chapter review (page 40) Practice examination questions (page 42) Students to calculate the size of various deductions Calculating annual leave loading calculating annual leave loading calculating net pay following deductions Costs of banking Students to investigate the costs associated with maintaining accounts at financial institutions Calculating personal budgets Reading household bills Questions relating to electricity, gas, telephone bills, etc calculating and comparing user costs associated with maintaining accounts at financial institutions creating and managing budgets reading information from household bills, including those for electricity, gas, telephone, council rates and water rates 4 Chapter 2: Units of measurement M1: Measurement Section Content Units of measurement (page 44) WE 1, 2, 3, 4, 5 Ex 2A Units of measurement (page 47) Units of measurement Converting between common units Choosing the most appropriate units to use in practical situations Errors in measurement Determining maximum possible error Calculating relative error in a variety of contexts Determining percentage error Relative error (page 49) WE 6, 7, 8, 9 Ex 2B Relative error (page 51) Measuring heights (page 53) Prescribed skills, knowledge and understanding determining the appropriate units to use when measuring physical attributes converting between commonly used units of measurement using standard prefixes determining possible sources of error recognising that accuracy of physical 1 measurement is limited to of the 2 Significant figures (page 54) WE 10, 11, 12 Ex 2C Significant figures (page 57) Rates (page 59) WE 13, 14, 15, 16, 17, 18, Ex 2D Rates (page 62) Significant figures Explaining the most practical degree of accuracy Rounding off measurements to the required accuracy, including questions involving scientific notation Working with rates Simplifying rates Converting from km/h to m/s and other similar exercises Calculations involving medicine and chemicals in agriculture smallest unit of which the measuring instrument is capable of calculating the percentage error of a measurement repeating and averaging measurements to reduce the likelihood of error determining the significant figures to be used when recording measurements, in relation to the accuracy of the measuring instrument being used using positive and negative powers of ten to express numbers in scientific notation calculating rates (pay, speeds, flow rates) converting between units for rates, eg. km/h to m/s calculating concentrations expressed as weight/weight, weight/volume or volume/volume 5 Percentage change (page 64) WE 19 Ex 2E Percentage change (page 65) Using ratios (page 66) WE 20, 21, 22 Ex 2F Using ratios (page 69) Summary (page 71) Chapter review (page 72) Practice examination questions (page 73) Percentage change Exercises on increasing and decreasing a certain quantity by given percentages Calculating repeated discounts Ratios Simplifying ratios Applying ratios to practical situations Dividing a quantity in a given ratio determining overall change in a quantity following repeated percentage changes finding the ratio of two quantities in familiar contexts dividing quantities in a given ratio using a unitary method to solve problems 6 Chapter 3: Applications of area and volume M2: Measurement Section Content Review of area (page 76) WE 1, 2, 3, 4 Ex 3A Review of area (page 78) Maximising an area of land (page 81) Calculating irregular areas from a field diagram (page 82) WE 5 Land survey (page 83) Ex 3B Calculating irregular areas from a field diagram (page 84) Solid shapes (page 86) WE 6, 7, 8 Ex 3C Solid shapes (page 88) Basic areas Review of calculating areas of triangles and quadrilaterals Applying this to practical applications Areas from surveys Calculating areas using traverse surveys leading to field diagrams Performing a survey and using the results to produce a field diagram and calculate an area Solid shapes Naming solids Matching solids with nets Drawing nets of solids Using isometric paper to draw solid shapes Surface area (page 90) WE 9, 10, 11 Ex 3D Surface area (page 92) Volume of a prism (page 95) Exploring the volume of a prism (page 95) WE 12, 13, 14, 15, 16 Ex 3E Volume of a prism (page 97) Areas of solids Calculating surface areas of prisms and pyramids Volume of prisms Calculating volumes of prisms and cylinders Prescribed skills, knowledge and understanding calculating the area of triangles and quadrilaterals (review only) using a field diagram to calculate the area of irregularly-shaped blocks of land classifying polyhedra into prisms (named with respect to their constant crosssection), pyramids or other constructing nets of solids and matching nets to solids sketching 3D solids using isometric paper and vanishing points using appropriate formulae when calculating the surface area of right prisms, square and rectangular pyramids using appropriate formulae when calculating the volume of right prisms and cylinders applying the relationship between units of capacity and units of volume 7 Volume of other solids (page 101) WE 17, 18, 19 Ex 3F Volume of other solids (page 103) Summary (page 106) Chapter review (page 107) Practice examination questions (page 110) Volume of other solids Calculating volumes of pyramids, cones and spheres and composite shapes Calculating capacity after determining volume using appropriate formulae when calculating the volume of pyramids, cones and spheres 8 Chapter 4: Basic algebraic skills AM1: Algebraic modelling Section Content General number patterns (page 112) WE 1, 2, 3 Ex 4A General number patterns (page 114) Number pattern notation (page 116) WE 4, 5, 6, 7 Ex 4B Number pattern notation (page 119) Number patterns Describing patterns in words Calculating terms in a pattern Terms of sequences Calculating terms of a sequence written in algebraic form Writing a sequence in algebraic form Collecting like terms Simplifying algebraic expressions by collecting like terms Adding and subtracting like terms (page 122) WE 8, 9, 10 Ex 4C Adding and subtracting like terms (page 123) Substitution (page 124) WE 11, 12 Ex 4D Substitution (page 125) Multiplication and division of algebraic expressions (page 127) WE 13, 14, 15, 16, 17 Ex 4E Multiplication and division of algebraic expressions (page 129) Solving linear equations (page 131) WE 18, 19, 20, 21, 22 Ex 4F Solving linear equations (page 134) Substitution into formulae Substituting into a variety of formulae to calculate the subject Calculations involving practical applications Algebraic multiplication and division Multiplying algebraic expressions including those with indices Dividing simple algebraic expressions including those with indices Removing brackets and simplifying Simplifying using the three index laws Solving equations Solving linear equations including those with fractions Substituting solutions to verify accuracy Prescribed skills, knowledge and understanding identifying and generalising simple linear number patterns identifying and generalising simple linear number patterns adding and subtracting like terms evaluating the subject of a formula through substitution of numerical values, using a wide variety of formulae multiplying algebraic terms dividing single terms (linear, quadratic and cubic) expanding and simplifying expressions solving linear equations involving up to three steps (fractions with numerical denominators only) 9 Equations arising from substitution (page 136) WE 23, 24 Ex 4G Equations arising from substitution (page 137) Summary (page 139) Chapter review (page 140) Practice examination questions (page 141) Substituting and solving equations Substituting into a variety of formulae to calculate a pronumeral which is not the subject solving equations following substitution and evaluation 10 Chapter 5: Statistics and society DA1: Data analysis Section Content Analysing data (page 144) Why statistical investigation? (page 144) A statistical investigation – 1 (page 144) Statistical processes (page 145) Posing questions (page 145) A statistical investigation – 2 (page 145) Analysing data Examining applications of data analysis Questions for a statistical investigation Posing questions Examining situations that need prepared questions to conduct an investigation Collecting data (page 146) WE 1 Ex 5A Collecting data (page 147) A statistical investigation - 3 (page 148) Organising data (page 148) WE 2, 3 Ex 5B Organising data (page 150) A statistical investigation – 4 (page 150) Displaying data (page 151) WE 4, 5 Exercise 5C Displaying data (page 153) A statistical investigation - 5 (page 153) A statistical investigation – 6 (page 153) A statistical investigation – 7 (page 153) Collecting information Examining different methods of data collection using external/internal sources Preparatory work in data collection Organising data Organising collected data in tables Selecting the most appropriate type of table for a given set of data Displaying data and writing a report Displaying data using a variety of graphs such as column and sector graphs Examining data, reaching conclusions and making recommendations Writing a report on a statistical inquiry including suitable graphs Quality control Explaining the need for quality control Practical exercises involving calculations to measure whether quality control guidelines are being met or not Quality control (page 154) WE 6, 7 Ex 5D Quality control (page 156) Prescribed skills, knowledge and understanding the importance of analysing data in planning and decision-making by governments and businesses the process of statistical inquiry including – posing questions using the principles of effective questionnaire design the process of statistical inquiry including – collecting data the process of statistical inquiry including – organising data and summarising data the process of statistical inquiry including – displaying data, analysing data and drawing conclusions and writing a report the role of statistical methods in quality control in manufacturing industries 11 Privacy and ethical issues (page 157) Privacy issues (page 157) Statistical organisations (page 157) Organisations that use statistics (page 157) Summary (page 158) Chapter review (page 159) Privacy and ethics in data collection Explanation of some ethical issues surrounding data collection Examining privacy issues Organisations that use statistics Information and an exercise on organisations that require data collection and its analysis issues of privacy and ethics in data collection and analysis using the principles of effective questionnaire design organisations that collect and/or use statistics, including the ABS, the UN, the WHO 12 Chapter 6: Data collection and sampling DA2: Data analysis Section Content Target populations and sampling (page 162) Gallup poll (page 162) Identifying the target population (page 162) WE 1, 2, 3 Ex 6A Target populations and sampling (page 166) Census or sample (page 168) Sampling Identifying sections of the population for a specific statistical investigation Collecting data by census or sample Choosing random, stratified or systematic samples Using a calculator to generate random numbers Bias (page 169) Biased sampling (page 169) Biased sampling Examining types of bias that might occur when sampling Types of data (page 170) WE 4, 5 Ex 6B Types of data (page 172) Estimating populations (page 174) Estimating a population (page 174) WE 6 Ex 6C Estimating populations (page 175) Population characteristics (page 176) Population characteristics (page 176) WE 7 Choosing a sample (page 177) Types of data Examining data to see if it is categorical or quantitative (discrete or continuous) Sampling and releasing Estimating a total population by sampling, releasing and sampling again Choosing a sample Examining how to choose a sample so that all appropriate sections of the population are included Prescribed skills, knowledge and understanding identifying target populations determining if data for the whole population is available or whether sampling is necessary recognising that sampling provides an estimate for a characteristic when the entire population cannot be accessed distinguishing between sample types: random, stratified, systematic determining which of the above sample types is appropriate for a given situation generating random numbers with a table or calculator to assist with sampling recognising that sampling provides an estimate for a population characteristic when the entire population cannot be accessed classifying data as: quantitative (discrete or continuous) or categorical describing and using the ‘capturerecapture’ technique for estimating the size of populations relating sample selection to population characteristics recognising the effect of a sample size in estimating the nature of a population 13 Summary (page 178) Chapter review (page 179) Practice examination questions (page 180) 14 Chapter 7: Modelling linear relationships AM2: Algebraic modelling Section Content Graphing linear functions (page 184) WE 1, 2, 3 Ex 7A Graphing linear functions (page 186) Graph of height versus age (page 187) Graphing linear functions Explaining a function including the terms independent and dependent variables Drawing linear graphs based on real-life situations – recognising that in some cases the model is limited Drawing conversion graphs Constructing a line of best fit Gradient and intercept (page 188) WE 4, 5, 6 Ex 7B Gradient and intercept (page 192) Gradient and y intercept Calculating gradient from a graph The significance of the y intercept and calculating it from a graph Drawing graphs using gradient and intercept (page 194) WE 7, 8, 9, 10 Drawing graphs Drawing graphs of lines when given the gradient and intercept on the vertical axis Prescribed skills, knowledge and understanding sketching graphical representations of quantities that vary over a period of time or in relation to each other identifying independent and dependent variables in practical contexts graphing of linear functions derived from everyday situations (eg. the cost of an excursion = fixed cost + cost per student number of students) by plotting ordered pairs from tables of values recognising the limitations of such models, eg. a person’s height as a function of age may be approximated by a straight line for a limited number of years, but not over a complete lifetime using graphs to convert from one measurement to another ($AUD – Euros) drawing a line of best fit on a graphed set of ordered pairs with a ruler and pencil calculating the gradients of such graphs with a ruler and pencil establishing a meaning for the gradient in the given context establishing a meaning for the intercept on the vertical axis in the given context sketching graphs of linear functions expressed in the form: y = mx + b 15 Ex 7C Drawing graphs using gradient and intercept (page 197) Graphing variations (page 199) WE 11, 12 Ex 7D Graphing variations (page 200) Currency conversions (page 201) Proportional graphs Drawing graphs of direct proportion or variation Finding the equation of proportion by graphing data Step and piecewise functions WE 13, 14 Ex 7E Step and piecewise functions (page 203) Step graphs Drawing step graphs Drawing graphs of piecewise function Simultaneous equations (page 204) WE 15 Ex 7F Simultaneous equations (page 205) Graphing simultaneous equations Drawing graphs of simultaneous linear equations based on practical examples, and using these to solve problems Summary (page 206) Chapter review (page 207) Practice examination questions (page 208) developing a linear graph of the form: y = ax from a description of a situation in which one quantity varies in a direct linear fashion with another, given one ordered pair using the above graph to establish the value of a (the gradient) and to solve problems related to the given variation context interpreting linear functions as models of physical phenomena using graphs to make conversions from one measurement to another (eg. $AUD to Euros) using stepwise and piecewise linear functions to model situations encountered in daily life, eg. parking charges, taxi fares, tax payments, mobile phone bills interpreting the graphical solution of simultaneous linear equations drawn from practical situations 16 Chapter 8: Investing money FM2: Financial mathematics Section Content Calculation of simple interest (page 212) WE 1, 2, 3, 4 Ex 8A Calculation of simple interest (page 215) Simple interest Calculating simple interest on investments over various time periods Finding values other than I in I=Prn Graphing simple interest functions (page 217) WE 5, 6 Ex 8B Graphing simple interest functions (page 219) Computer application 1: Simple interest spreadsheets (page 221) Calculation of compound interest (page 222) WE 7, 8, 9 Ex 8C Calculation of compound interest (page 225) Computer application 2: Compound interest spreadsheets (page 227) Calculating compound interest from a table of compounded values (page 229) WE 10, 11, 12 Ex 8D Calculating compound interest from a table of compounded values (page 232) Graphing compound interest functions (p 234) WE 13, 14 Ex 8E Graphing compound interest functions (page 236) Simple interest graphs Completing simple interest tables Using these tables to graph interest and investment values against time Using a simple interest spreadsheet Prescribed skills, knowledge and understanding calculating simple interest using I = Prn, where P = principal, r = percentage interest rate per period expressed as a decimal (eg. rate of 8.2% means r = 0.082), and n = the number of periods calculating monthly, quarterly and six-monthly interest rates based on quoted rates per annum (pa) for fixed values of P, using tables of values and hence drawing and describing graphs of A against n for differing values of r Compound interest Calculating Future Value when interest is compounded over various time periods Calculating the interest on investments Using a compound interest spreadsheet using formulae to calculate future value, compound interest and present value Compound interest table Using a table to solve compound interest problems Finding Present Value using the formula using formulae to calculate future value, compound interest and present value calculating future and present values of an investment from prepared tables Compound interest graphs Completing compound interest tables Using these tables to draw and describe graphs of FV against time for fixed values of P, using tables of values and hence drawing and describing graphs of A against n for differing values of r 17 Share dividends (page 238) WE 15, 16 Ex 8F Share dividends (page 239) Share dividends Calculating dividend yield for shares Calculating dividends on shares when given the company profits calculating the dividend paid on a share holding and the dividend yield, excluding franked dividends Graphing share performance (page 241) WE 17, 18 Ex 8G Graphing share performance (page 243) Researching share prices (page 244) Graphing shares Graphing the value of a share over a certain time period and making predictions from such graphs Graphing the performance of a given share over a 6 month period Inflation and appreciation Calculating the effect of inflation over different time periods Using the compound interest formula in inflation questions Calculating appreciation extrapolating from the information shown on a prepared graph of share performance to suggest possible future stockmarket movement calculating the price of goods following inflation calculating the appreciated value of items such as stamp collections and memorabilia Inflation and appreciation (page 245) WE 19, 20, 21 Ex 8H Inflation and appreciation (page 247) Summary (page 248) Chapter review (page 249) Practice examination questions (page 251) 18 Chapter 9: Displaying single data sets DA3: Data analysis Section Content Frequency tables (page 254) WE 1, 2 Ex 9A Frequency tables (page 257) Types of graphs (page 259) WE 3, 4, 5, 6, 7 Ex 9B Types of graphs (page 262) Choice of graph (page 264) Producing graphs using technology (page 264) Frequency tables Entering grouped and ungrouped data into frequency tables Statistical graphs Creating dot plots, sector graphs, line charts, bar and column graphs and radar charts Advantages and disadvantages of various graphs Graphing with a spreadsheet or graphics calculator Statistical graphs (page 265) WE 8, 9, 10 Ex 9C Statistical graphs (page 268) Statistical graphs Creating frequency histograms and polygons and cumulative frequency graphs Working with grouped and ungrouped data Prescribed skills, knowledge and understanding creating tally charts and frequency tables to organise undergrouped and grouped data creating dot plots, sector graphs (pie charts), bar graphs, histograms and line graphs, with attention being paid to the scale on each axis selecting a suitable scale for graph axes drawing a radar chart to display data such as sales figures, temperature or rainfall readings (see example at end of unit) describing the strengths and weaknesses of sector graphs, bar graphs, histograms, frequency polygons and radar charts; including suitability for data represented noting the capacity of statistical displays for misrepresentation, particularly in the selection of the scale used on the axes linking types of data with displays, eg. continuous quantitative data is best represented by a histogram; categorical data with a bar graph or sector graph etc creating frequency graphs and cumulative frequency graphs (ogives) 19 Range and interquartile range (page 273) WE 11, 12, 13, 14, 15 Ex 9D Range and interquartile range (page 277) Stem-and-leaf plots (page 282) WE 16, 17, 18 Ex 9E Stem-and-leaf plots Five-number summaries (page 288) WE 19, 20, 21, 22 Ex 9F Five-number summaries (page 291) Summary (page 294) Chapter review (page 295) Practice examination questions (page 297) Some statistical measures Calculating range, median, quartiles and interquartile range Dividing data into deciles Stem-and-leaf plots Creating stem-and-leaf plots Calculating quartiles and the median from a stem-and-leaf plot Five-number summaries Calculating the five-number summary for a set of scores Using an ogive to develop five-number summaries Creating box-and-whisker plots based on these summaries determining the range and interquartile range as measures of a data set spread dividing data into deciles and quartiles creating a stem-and-leaf plot to illustrate a small data set establishing a five-number summary for a data set (lower extreme, lower quartile, median, upper quartile, upper extreme) developing a box-and-whisker plot from a five-number summary determining the median and upper and lower quartiles of a data set from a cumulative frequency polygon 20 Chapter 10: Summary statistics DA4: Data analysis Section Content Prescribed skills, knowledge and understanding calculating the mean of small data sets, x fx using the formulae: x , , x n f where x is the mean of the sample determining the mean for larger data sets of grouped or ungrouped data using the statistical functions of a calculator calculating the means of a range of samples from a population an informal description of standard deviation as a measure of the spread of data in relation to the mean determining the population standard deviation using the n button of a calculator and the sample standard deviation as an estimate of the population measure, using the n 1 button Calculating the mean (page 300) Average – what does it mean? (page 300) WE 1, 2, 3, 4 Ex 10A Calculating the mean (page 304) The mean Calculating the mean of small sets of data and tabulated data using formulae Using a calculator in statistics mode to calculate the mean Standard deviation (page 309) WE 5, 6, 7, 8 Ex 10B Standard deviation (page 312) Standard deviation Calculating population and sample standard deviation – small sets and tabulated data Median and mode (page 316) WE 9, 10, 11, 12, 13 Ex 10C Median and mode (page 319) Best summary statistics (page 324) WE 14 Ex 10D Best summary statistics (page 325) Wage rise (page 328) Best summary statistics and comparison of samples (page 328) Median and mode Calculating median and mode Mean, median and mode Calculating and comparing relevance of mean, median and mode Examining statistics to decide on the most appropriate summary statistic determining the median and mode(s) of a data set, either from a list or from a frequency table selecting and using the appropriate statistic (mean, median or mode) to describe features of a data set, eg. median house prices, modal shirt size etc comparing the summary statistics of various samples from the same population 21 Summary (page 329) Chapter review (page 330) Practice examination questions (page 335) 22 Chapter 11: Similarity of two-dimensional figures M3: Measurement Section Content Similar figures and scale factors (page 338) WE 1, 2, 3, 4 Ex 11A Similar figures and scale factors (page 340) Enlarging a figure (page 342) Investigating scale factors (page 342) Solving problems using similar figures (page 343) WE 5, 6, 7 Ex 11B Solving problems using similar figures (page 344) Scale drawing of the classroom (page 345) House plans (page 346) WE 8, 9 Ex 11C House plans (page 348) House plans (page 350) Similar figures Testing figures for similarity Calculating scale factors of similar figures Drawing similar figures using an enlargement factor Using similar figures Solving practical problems using similarity Drawing a scale diagram Summary (page 351) Chapter review (page 352) Practice examination questions (page 354) Working with house plans Calculating dimensions and areas from scale drawings of land and buildings Interpreting house plans Calculations from elevations Prescribed skills, knowledge and understanding establishing properties of similar figures recognising similarity in everyday life finding scale factors of similar figures recognising that similar figures related by a scale factor of 1 are said to be congruent using relevant enlargement or reduction factors to calculate actual dimensions developing scale drawings of objects and images using scale factor to solve problems involving similar figures obtaining measurements from plans of buildings and rooms calculating lengths and areas from a floor plan interpreting commonly used symbols on house plans 23 Chapter 12: Taxation FM3: Financial mathematics Section Content Calculating allowable deductions (page 356) WE 1, 2, 3, 4 Ex 12A Calculating allowable deductions (page 359) Taxable income (page 361) WE 5, 6 Ex 12B Taxable income (page 363) Computer application 1: Calculating taxable income (page 364) Tax deductions calculating a variety of tax deductions including exercises involving repeated depreciation Calculating taxable income Calculating gross incomes, deductions and then taxable income Using a prepared spreadsheet to calculate taxable income Medicare levy (page 367) WE 7 Ex 12C Medicare levy (page 367) Medicare levy (page 367) Calculating tax (page 368) WE 8, 9, 10, 11 Ex 12D Calculating tax (page 372) Computer application 2: Tax calculation (page 374) Calculating GST and VAT (page 375) WE 12, 13, 14, 15 Ex 12E Calculating GST and VAT (page 377) Graphing tax functions (page 379) Ex 12F Graphing tax functions Prescribed skills, knowledge and understanding calculating the amount of allowable deductions from gross income calculating taxable income Medicare levy Calculating the Medicare levy calculating the Medicare levy (basic levy only – see Tax Pack for details) Calculating income tax Calculating income tax using a tax table Calculating refunds or tax owing Using a prepared spreadsheet to calculate tax Calculating GST Calculating a new price after adding VAT or GST Calculating the pre-tax price calculating the PAYE (Pay As You Earn) tax payable or refund owing, using current tax scales Graphing tax functions Drawing piecewise linear graphs to show tax payable on different incomes calculate the Value Added Tax (VAT) payable on a range of goods and services, given the tax rates of various countries calculating the goods and services tax (GST) payable on a range of goods and services creating graphs to illustrate and describe different tax rates 24 Summary (page 380) Chapter review (page 381) Practice examination questions (page 383) 25 Chapter 13: Right-angled triangles M4: Measurement Section Content History of Mathematics – Pythagoras of Samos (page 386) Pythagoras’ theorem (page 387) WE 1, 2, 3, 4, 5 Ex 13A Pythagoras’ theorem (page 390) Pythagoras Historical information on Pythagoras Using the theorem of Pythagoras using Pythagoras’ theorem to find an unknown side in a right-angled triangle Calculating sides in right-angled triangles Testing if a triangle is right-angled applying Pythagoras’ theorem to: - determine whether or not a triangle is Applying Pythagoras to practical right-angled problems - solve problems based on single right-angled triangles - calculate perimeters of irregularly shaped blocks of land Introductory trigonometry defining sine, cosine and tangent ratios Using a calculator to find trigonometric ratios of acute angles, including degrees and minutes Finding an acute angle using trigonometry Calculating trigonometric ratios (page 392) Looking at the tangent ratio (page 392) WE 6 Looking at the sine ratio (page 394) WE 7 Looking at the cosine ratio (page 395) WE 8, 9, 10 Ex 13B Calculating trigonometric ratios (page 397) Finding an unknown side (page 399) WE 11, 12, 13, 14 Ex 13C Finding an unknown side (page 402) Finding angles (page 406) WE 15, 16, 17 Ex 13D Finding angles (page 408) Using trigonometry to find sides Finding the sides of right triangles using trigonometry Solving practical problems using trig. Using trigonometry to find angles Finding angles in right triangles using trigonometry and applying this to practical problems Prescribed skills, knowledge and understanding using trigonometric ratios to find the length of an unknown side in a right-angled triangle using trigonometric ratios to find the size of an unknown angle in a right-angled triangle and using a calculator to find this angle to the nearest minute 26 Angles of elevation and depression (page 411) WE 18, 19, 20, 21 Ex 13E Angles of elevation and depression (page 414) Calculation of heights (page 415) Proportional diagrams (page 416) Checking with a proportional diagram (page 416) Using proportional diagrams (page 416) Summary (page 417) Chapter review (page 418) Practice examination questions (page 420) Elevation and depression Solving practical problems involving depression and elevation Calculating heights of objects around the school Scale diagrams in trigonometry Checking accuracy using scale diagrams solving problems involving angles of elevation and depression, when given the appropriate diagram determining whether an answer seems reasonable by using a diagram drawn roughly in proportion 27 Chapter 14: The language of chance P1: Probability Section Content Informal description of chance (page 424) WE 1, 2, 3, 4, 5 Ex 14A Informal description of chance (page 427) Common descriptions of chance (page 429) Sample space (page 429) WE 6, 7, 8 Ex 14B Sample space (page 431) Matching actual and expected results (page 432) Tree diagrams (page 434) WE 9, 10, 11 Ex 14C Tree diagrams (page 437) Two stage experiments (page 438) Language of chance Appropriate probability terms and their use in ordering everyday events Equally likely outcomes (page 439) WE 12, 13 Ex 14D Equally likely outcomes (page 441) Using the fundamental counting principle (page 443) WE 14, 15, 16 Ex 14E Using the fundamental counting principle (page 446) Summary (page 448) Chapter review (page 449) Practice examination questions (page 451) Prescribed skills, knowledge and understanding ordering everyday events from the very unlikely to the almost certain Recording sample spaces Listing sample spaces Listing favourable outcomes using a list or table to identify the sample space (set of all possible outcomes) of a simple experiment or game Drawing and using tree diagrams Listing sample spaces using tree diagrams Determining some possible combinations using tree diagrams Equally likely outcomes Calculating if outcomes are equally likely Multi-stage events Calculating the number of different ways separate events can occur using a list or table to identify the sample space (set of all possible outcomes) of a simple experiment or game using systematic lists to verify the total number of outcomes for simple multistage events performing experiments and determining whether the outcomes are equally likely or not determining the number of outcomes for a multi-stage event by multiplying the number of choices at each stage using systematic lists to verify the total number of outcomes for simple multistage events 28 Chapter 15: Relative frequency and probability P2: Probability Section Content Relative frequency (page 454) WE 1, 2, 3 Ex 15A Relative frequency (page 456) Researching relative frequencies (page 458) Using relative frequencies Calculating relative frequencies Using relative frequencies to draw conclusions Obtaining relative frequencies from recorded information Calculating simple probabilities Calculating probabilities of simple single-event experiments Comparing results in an activity with expected results Single event probability (page 459) WE 4, 5, 6, 7, 8 Ex 15B Single event probability (page 461) Comparing probabilities with actual results (page 464) Prescribed skills, knowledge and understanding estimating the relative frequency of events from recorded data performing simple experiments to obtain relative frequencies from recorded results using relative frequencies to obtain approximate probabilities using the probability formula of an event where outcomes are equally likely: P(event) Writing probabilities as decimals and percentages (page 466) WE 9, 10 Ex 15C Writing probabilities as decimals and percentages (page 467) Range of probabilities (page 469) WE 11, 12, 13 Ex 15D Range of probabilities (page 471) Graphing results (page 473) Probabilities written in different forms Determining probabilities and expressing answers as fractions, decimals and percentages Complementary events (page 474) WE 14, 15, 16 Ex 15E Complementary events (page 476) Complementary events Calculating the probabilities of complements of events Range of probabilities Matching answers to probability questions with their informal descriptions number of favourable outcomes total number of outcomes comparing calculated probabilities with experimental results calculating probabilities in terms of the fractional, decimal, or percentage chance demonstrating the range of possible probabilities, 0 P(E) 1, by examining a variety of results illustrating the results of experiments through statistical graphs and displays defining and using the relationship between complementary events P(an event does not occur) = 1 P(the event does occur) 29 Summary (page 479) Chapter review (page 480) Practice examination questions (page 481) Glossary (page 483) SKILLSHEETS Skillsheet 1 – Converting fractions to decimals (page 489) Skillsheet 2 – Converting decimals to fractions (page 490) Skillsheet 3 – Converting a fraction to a percentage (page 491) Skillsheet 4 – Converting percentages to fractions (page 491) Skillsheet 5 – Converting decimals to percentages (page 492) Skillsheet 6 – Converting percentages to decimals (page 492) Skillsheet 7 – Finding a percentage of a quantity (page 493) Skillsheet 8 – Increase or decrease by a percentage (page 494) Skillsheet 9 – Writing one quantity as a percentage of another (page 495) Skillsheet 10 – Unitary method of percentages (page 496) Skillsheet 11 – Simplifying fractions (page 497) Answers (page 499 – 533)