MATHS QUEST General Mathematics Preliminary Course

1 COURSE GUIDE MATHS QUEST General Mathematics: Preliminary Course 2

Financial Mathematics 1

FM1: Earning money (Chapter 1) The principal focus of this unit is the range of ways in which individuals earn and manage their money.

Outcomes addressed A student: P1 develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation P2 applies mathematical knowledge and skills to solving problems within familiar contexts P7 determines the degree of accuracy of measurements and calculations P8 models financial situations using appropriate tools P11 justifies his/her response to a given problem using appropriate mathematical terminology

Students learn and acquire the following skills, knowledge and understanding

Skills, knowledge and understanding Exercises/Investigations  calculating monthly, fortnightly, Ex 1A Calculating salary payments (page 3) weekly, daily and hourly payments from salary  calculating wages incorporating Ex 1B Calculating wages (page 7) hourly rates, penalty rates such as Ex 1E Working overtime (page 20) overtime, and special allowances for wet work, confined spaces, toxic substances, heat, heights etc  calculating annual leave loading Ex 1F Additions to and deductions from gross pay (page 26)  calculating earnings based on Ex 1C Commission and royalties (page 13) commissions, piecework, royalties Ex 1D Payment by piece (page 16)  calculating income based on Investigating government payments (page 23) government allowances such as the youth allowance and pensions  determining deductions such as union Ex 1F Additions to and deductions from gross fees, superannuation contributions, pay (page 26) health fund installments and tax installments  calculating and comparing user costs Examining bank fees and taxes (page 29) associated with maintaining accounts with financial institutions  calculating net pay following Ex 1F Additions to and deductions from gross deductions pay (page 26)  creating and managing budgets Ex 1G Budgeting (page 34)  reading information from household Ex 1G Budgeting (page 34) bills, including those for electricity, gas, telephone, council rates and water rates 3

Terminology introduced in this unit

Syllabus/glossary terms Page Annual leave 25 Annual leave loading 25 Deductions 24 Double time 18 Gross pay/gross wage 24 Overtime 18 Royalty 10 Time-and-a-half 18

Further activities Computer application 1: Spreadsheets – calculating simple wages (page 9) Computer application 2: Wages – calculating overtime wages (page 22) 4

Measurement 1

M1: Units of measurement (Chapter 2) In this unit, the principal focus is on metric units of measurement, rates and ratios. The students also learn about making judgments on measurement errors.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P5 represents the relationships between changing quantities in algebraic and graphical form P7 determines the degree of accuracy of measurements and calculations

Skills, knowledge and understanding Exercises/Investigations  determining appropriate units to use Ex 2A Units of measurement (page 47) when measuring physical attributes  converting between common units of Ex 2A Units of measurement (page 47) measurement using standard prefixes  recognising that accurate physical Ex 2B Relative error (page 51) measurement is limited to  of the smallest unit that the measuring instrument is capable of handling  calculating the percentage error in a Ex 2B Relative error (page 51) measurement, eg. if the measured Measuring heights (page 53) height was 155cm  0.5cm (ie. to the nearest cm), the percentage error for this measurement is  0.5  100% 155  determining possible sources of error Ex 2B Relative error (page 51) when measuring Measuring heights (page 53)  repeating and averaging measurements Ex 2B Relative error (page 51) to reduce the likelihood of error  determining the significant figures to Ex 2C Significant figures (page 57) be used when recording measurements, in relation to the accuracy of the measuring instrument being used  using positive and negative powers of Ex 2C Significant figures (page 57) ten to express numbers in scientific notation  calculating rates, eg. pay rates, speeds Ex 2D Rates (page 62) and rates of flow  converting between units for rates, Ex 2D Rates (page 62) eg. km/h to m/s  calculating concentrations expressed as Ex 2D Rates (page 62) weight/weight, weight/volume or volume/volume Note: these calculations have particular applications to nursing and agriculture 5

 determining overall change in a Ex 2E Percentage change (page 65) quantity following repeated percentage changes, eg. an increase of 20% followed by a decrease of 20%  finding the ratio of two quantities in Ex 2F Using ratios (page 69) familiar contexts  dividing quantities in a given ratio Ex 2F Using ratios (page 69)  using a unitary method to solve Ex 2F Using ratios (page 69) problems

Syllabus/glossary terms Page Concentration 61 Percentage error 50 Prefix 44 Significant figures 54 Unitary method 67 Ratio 66 6

Measurement 2

M2: Applications of area and volume (Chapter 3) The main focus of this unit is calculating and applying area and volume to solving problems.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P6 performs calculations in relation to two-dimensional and three-dimensional figures P7 determines the degree of accuracy of measurements and calculations

Skills, knowledge and understanding Exercises/Investigations  calculating the area of triangles and Ex 3A Review of area (page 78) quadrilaterals (review only)  using a field diagram to calculate the Ex 3B Calculating irregular areas from a area of irregularly-shaped land blocks field diagram (page 84)  classifying polyhedra into prisms Ex 3C Solid shapes (page 88) (named according to their constant cross-section), pyramids or other  constructing nets of solids and Ex 3C Solid shapes (page 88) matching nets to solids  sketching 3D solids using isometric Ex 3C Solid shapes (page 88) paper and vanishing points  using appropriate formulae to calculate Ex 3D Surface area (page 92) the surface area of right prisms, and square and rectangular pyramids  using appropriate formulae to calculate Ex 3E Volume of a prism (page 97) the volume of right prisms, cylinders, Ex 3F Volume of other solids (page 103) pyramids, cones and spheres  applying the relationship between units Ex 3E Volume of a prism (page 97) of capacity and units of volume

Further activities Maximising an area of land (page 81) Land survey (page 83) Exploring the volume of a prism (page 95) 7

Algebraic Modelling 1

AM1: Basic algebraic skills (Chapter 4) The main focus of this unit is to provide a foundation for the basic algebraic skills required for this area of study, and to use these skills to solve real and abstract problems.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P3 develops rules to represent patterns arising from numerical and other sources P7 determines the degree of accuracy of measurements and calculations

Skills, knowledge and understanding Exercises/Investigations  identifying and generalising simple Ex 4A General number patterns (page 114) linear number patterns Ex 4B Number pattern notation (page 119)  adding and subtracting like terms Ex 4C Adding and subtracting like terms (page 123)  evaluating the subject of a formula by Ex 4D Substitution (page 125) substituting numerical values, using a wide variety of formulae such as: 5 D C  (F  32) , v = u + at, S  , 9 T 4 m V  r 3 , A  P(1 r) n , B  , 3 h 2 mA ya kA D  , D  , D  150 (y 12) 70  solving linear equations involving up Ex 4F Solving linear equations (page 134) to three steps (fractions with numerical denominators only)  expanding and simplifying expressions Ex 4E Multiplication and division of such as: algebraic expressions (page 129) 3p(p2), 2 x2(7x) + x(x1)  multiplying algebraic terms such as: 3 Ex 4E Multiplication and division of 9y algebraic expressions (page 129) M 2  5M, 5y 4  dividing single terms (linear, quadratic Ex 4E Multiplication and division of 9L3 algebraic expressions (page 129) and cubic) such as: 3L2  solving equations following Ex 4G Equations arising from substitution substitution and evaluation, eg. find a, (page 137) given that v = 10, u = 5, and s = 8 for: v2 = u2 + 2as 8

Syllabus/glossary terms Page Cubic 128 Linear 128 Power 128 Quadratic 128 9

Data Analysis 1

DA1: Statistics and society (Chapter 5) This unit focuses on the importance of statistical processes and inquiry in society. The material is essential background for the later Data Analysis units and may be integrated with these units if desired.

Outcomes addressed A student: P1 develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation P9 determines an appropriate form of organisation and representation of collected data P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  the importance of analysing data in Why statistical investigation? (page 144) planning and decision-making by governments and businesses  the process of statistical inquiry, Posing questions (page 145) including the following steps: Ex 5A Collecting data (page 147) - posing questions Ex 5B Organising data (page 150) - collecting data Ex 5C Displaying data (page 153) - organising data - summarising and displaying data - analysing data and drawing conclusions - writing a report  the role of statistical methods in quality Ex 5D Quality control (page 156) control in manufacturing industries  issues of privacy and ethics in data Privacy issues (page 157) collection and analysis  organisations that collect and/or use Organisations that use statistics (page 157) statistics, including the Australian Bureau of Statistics (ABS), the United Nations (UN), the World Health Organisation (WHO)

Syllabus/glossary terms Page Data 145 Information 145 Quality control 154 Statistics 144 Statistical inquiry 153 10

Further activities A statistical investigation 1 (page 144) A statistical investigation 2 (page 145) A statistical investigation 3 (page 148) A statistical investigation 4 (page 150) A statistical investigation 5 (page 153) A statistical investigation 6 (page 153) A statistical investigation 7 (page 153) 11

Data Analysis 2

DA2: Data collection and sampling (Chapter 6) The principal focus of this unit is the planning and management of data collection. In some cases, complete data sets may be available. In other cases, a survey may be needed that involves a sample population only. Although the emphasis is on quantitative data, students should be aware of processes related to categorical data.

Outcomes addressed A student: P1 develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation P9 determines an appropriate form of organisation and representation of collected data P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  identifying the target population to be Identifying the target population (page 162) investigated  determining if data for the whole Ex 6A Target populations and sampling population is available or whether (page 166) sampling is necessary Census or sample (page 168)  recognising the purpose of a sample is to Ex 6A Target populations and sampling provide an estimate for a particular (page 166) characteristic when the entire population Census or sample (page 168) cannot be accessed Biased sampling (page 169)  classifying data as: Ex 6B Types of data (page 172) - Quantitative, either discrete or continuous, OR - Categorical eg. gender (male / female) is categorical; height (measured in cm) is quantitative, continuous; quality (poor, average, good, excellent) is categorical; school population (measured in individuals) is quantitative, discrete  distinguishing between the following Ex 6A Target populations and sampling sample types: (page 166) - random, stratified or systematic  determining which of the above sample Ex 6A Target populations and sampling types is appropriate for a given situation (page 166)  relating sample selections to population Ex 6A Target populations and sampling characteristics, eg. if 20% of the (page 166) population is aged under 20, your Choosing a sample (page 177) sample should have 20% of under 20s  generating random numbers with a table Ex 6A Target populations and sampling or calculator to help establish random (page 166) samples 12

 describing and using the ‘capture- Estimating a population (page 174) recapture’ technique for estimating the Ex 6C Estimating populations (page 175) size of populations, eg. fish in a lake  recognising the effect of sample size in Population characteristics (page 176) estimating the nature of the population, eg. using the number of boys and girls in one Year 11 class to estimate the gender ratio in Year 11 across all of NSW  effective questionnaire design rules: Chapter 5: - simple language Posing questions (page 145) - unambiguous questions Ex 5A Collecting data (page 147) - respect for privacy Privacy issues (page 157) - freedom from bias - consideration of the number of choices offered, eg. an even number of good / bad choices may force an opinion in some circumstances when it may be more appropriate to give a neutral option

Syllabus/glossary terms Page Bias 169 Categorical data 170 Census 162 Continuous 171 Discrete 171 Database 162 Poll 162 Population 176 Questionnaire 162 Random sample 163 Sample size 177 Strata 164 Stratified sample 164 Systematic sample 165

Further activities Gallup poll (page 162) 13

Algebraic Modelling 2

AM2: Modelling linear relationships (Chapter 7)

Outcomes addressed A student: P3 develops rules to represent patterns arising from numerical and other sources P4 represents information in symbolic, graphical and tabular forms P5 represents the relationships between changing quantities in algebraic and graphical form

Skills, knowledge and understanding Exercises/Investigations  sketching graphical representations of Ex 7A Graphing linear functions (page 186) quantities that vary over a period of time, or in relation to each other. Note: students should develop an understanding of a function as: input  processing  output (it is not intended that students learn a formal definition of a function)  identifying independent and dependent Ex 7A Graphing linear functions (page 186) variables in practical contexts  graphing linear functions derived from Ex 7A Graphing linear functions (page 186) everyday situations (eg. the cost of a trip = fixed cost + cost per student x number of students) by plotting ordered pairs from tables of values  calculating the gradients of such graphs Ex 7B Gradient and intercept (page 192) with a ruler and pencil  establishing a meaning for the gradient Ex 7B Gradient and intercept (page 192) in the given context  establishing a meaning for the intercept Ex 7C Drawing graphs using gradient and intercept on the vertical axis in the given context (page 197)  sketching graphs of linear functions Ex 7C Drawing graphs using gradient and intercept expressed in the form: y = mx + b (page 197)  developing a linear graph of: y = ax Ex 7D Graphing variations (page 200) from a description of a situation where one quantity varies in a direct linear fashion with another, given one ordered pair  using the graph above to find the value Ex 7D Graphing variations (page 200) of a (the gradient) and solve problems related to the given variation context  interpreting linear functions as models Ex 7D Graphing variations (page 200) of physical phenomena  using stepwise and piecewise linear Ex 7E Step and piecewise functions (page 203) functions to model situations from daily life, eg. parking charges, taxi fares, tax payments, mobile phone bills 14

 recognising the limits of such models, Graph of height versus age (page 187) 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 make conversions from Ex 7A Graphing linear functions (page 186) one measurement to another, eg. $AUD Currency conversions (page 201) to Euros  interpreting the graphical solution of Ex 7F Simultaneous equations (page 205) simultaneous linear equations drawn from practical situations  drawing a line of best fit on a graphed Graph of height versus age (page 187) set of ordered pairs with a ruler and pencil

Syllabus/glossary terms Page Dependent variable 184 Direct linear variation 199 Function 184 Increasing function 189 Interpolate 241 (Ch 8) Proportional to 199 Relation 184 y-intercept 189 Decreasing function 189 Extrapolate 241 (Ch 8) Gradient 188 Independent variable 184 Piecewise linear function* 202 Rate of change 188 Step function* 201

* in practical contexts only 15

FM2: Investing money (Chapter 8) The principal focus of this unit is to use formulae and tables to perform calculations related to the value of investments over a period of time.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P3 develops rules to represent patterns arising from numerical and other sources P7 determines the degree of accuracy of measurements and calculations P8 models financial situations using appropriate tools P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  calculating simple interest using: Ex 8A Calculation of simple interest I=Prn, where P = principal, r = the (page 215) percentage interest rate per period expressed as a decimal (eg. if the rate is 8.2% then r = 0.082), and n = the number of periods  for fixed values of P, using tables of Ex 8B Graphing simple interest functions values and hence drawing and (page 219) describing graphs of I against n for differing values of r Note: these are linear graphs whose gradient is determined by the value of r (see AM2, AM4)  calculating monthly, quarterly and six- Ex 8A Calculation of simple interest monthly interest rates based on quoted (page 215) rates per annum (pa)  using formulae to calculate future Ex 8C Calculation of compound interest value, compound interest and present (page 225) value with pen and paper: A  P(1 r) n , where A (amount) = final balance (future value), P (principal) = initial quantity (present value), n = number of compounding periods, and r = interest rate per compounding period Note: in the financial world, the compound interest formula quoted above is generally presented as: FV  PV 1 rn , where FV = future value and PV = present value  for fixed values of P, using tables of Ex 8E Graphing compound interest values and hence drawing and functions (page 236) describing graphs of A against n for differing values of r 16

Note: these are examples of exponential growth (see AM3, AM4)  calculating the dividend paid on a share Ex 8F Share dividends (page 239) holding and the dividend yield, excluding franked dividends  extrapolating from the information Ex 8G Graphing share performance shown on a prepared graph of share (page 243) performance to suggest possible future Researching share prices (page 244) movement  calculating the future and present value Ex 8D Calculating compound interest from a of an investment from prepared tables table of compounded values (page 232)  calculating the price of goods Ex 8H Inflation and appreciation (page 247) following inflation  calculating the appreciated value of Ex 8H Inflation and appreciation (page 247) items such as stamp collections and memorabilia

Syllabus/glossary terms Page Compounding period 223 Dividend 238 Dividend yield 238 Future value 222 Present value 229 Shares 238

Further activities Computer application 1: Simple interest spreadsheets (page 221) Computer application 2: Compound interest spreadsheets (page 227) 17

Data Analysis 3

DA3: Displaying single data sets (Chapter 9) In this unit, students prepare a variety of data displays and consider the appropriateness of each for the stated purpose. The power of statistical displays to both inform and misinform should be emphasised.

Outcomes addressed A student: P1 develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation P4 represents information in symbolic, graphical and tabular forms P7 determines the degree of accuracy of measurements and calculations P9 determines an appropriate form of organisation and representation of collected data P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  creating tally charts and frequency Ex 9A Frequency tables (page 257) tables to organise undergrouped and grouped data  creating dot plots, sector graphs (pie Ex 9B Types of graphs (page 262) charts), bar graphs, histograms and line graphs, with attention being paid to the scale on each axis  selecting a suitable scale for each axis Ex 9B Types of graphs (page 262) of a graph  noting the capacity for the Theory (page 261) misinterpretation of statistical displays, Choice of graph (page 264) particularly in the selection of the scale used on the axes  creating a stem-and-leaf plot to Ex 9E Stem-and-leaf plots (page 286) illustrate a small data set  drawing a radar chart to display data Ex 9B Types of graphs (page 262) such as sales figures, temperatures or rainfall readings  dividing data into deciles and quartiles Ex 9D Range and interquartile range (page 277)  determining the range and interquartile Ex 9D Range and interquartile range range as measures of the spread of a (page 277) data set  creating frequency graphs and Ex 9D Range and interquartile range cumulative frequency graphs (ogives) (page 277)  determining the median and upper and Ex 9D Range and interquartile range lower quartiles of a data set from a (page 277) cumulative frequency polygon  establishing a five number summary Ex 9F Five-number summaries (page 291) for a data set (lower extreme, lower quartile, median, upper quartile, upper extreme) 18

 developing a box-and-whisker plot Ex 9F Five-number summaries (page 291) from a five number summary  linking types of data with appropriate Ex 9B Types of graphs (page 262) displays, eg. continuous quantitative data is best represented by a histogram; categorical data is best represented by a bar graph or sector graph (pie chart)  describing the strengths and/or Choice of graph (page 264) weaknesses of sector graphs, bar graphs, histograms, frequency polygons and radar charts, including suitability for the data represented

Syllabus/glossary terms Page Bar graph 260 Column graph 260 Cumulative frequency 266 Dot plot 259 Frequency table 254 Grouped data 254 Histogram 265 Line graph 260 Ogive 266 Sector graph 259 Polygon 265 Relative frequency 454 (Ch 15) Box-and-whisker plot 289 Five number summary 288 Range 273 Interquartile range 274 Decile 279 Upper/lower quartile 274 Quartile 274 Upper/lower extreme 289 Stem-and-leaf plot 283 Radar chart 261

Further activities Producing graphs using technology (page 264) 19

Data Analysis 4

DA4: Summary statistics (Chapter 10) The key focus is the calculation of summary statistics for single data sets and their use in interpretation.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P4 represents information in symbolic, graphical and tabular forms P7 determines the degree of accuracy of measurements and calculations P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  calculating the mean of small data sets Ex 10A Calculating the mean (page 304) using the formulae:

  x   fx  x  , x , where x , represents n  f the mean of the sample  determining the mean for larger data Ex 10A Calculating the mean (page 304) sets (of either grouped or ungrouped data) using the statistical functions of a calculator  calculating the means of a range of Ex 10A Calculating the mean (page 304) samples from a population  informal description of standard Standard deviation theory (page 309) deviation, as a measure of the spread of data, in relation to the mean  determining the population standard Ex 10B Standard deviation (page 312)

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  determining the median and mode(s) of Ex 10C Median and mode (page 319) a data set - either from a list or from a frequency table  selecting and using the appropriate Ex 10D Best summary statistics (page 325) statistic (mean, median or mode) to Wage rise (page 328) describe features of a data set, eg. median house prices, modal shirt size  comparing summary statistics of Best summary statistics and comparison of various samples from the same samples (page 328) population 20

Syllabus/glossary terms Page Mean 300 Median 316 Mode 318 Standard deviation 309 Summary statistic 324

Further activities Average – what does it mean? (page 300) 21

Measurement 3

M3: Similarity of two-dimensional figures (Chapter 11) The principal focus of this unit is to apply similarity properties to problems in everyday life.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P6 performs calculations in relation to two-dimensional and three-dimensional figures P7 determines the degree of accuracy of measurements and calculations

Skills, knowledge and understanding Exercises/Investigations  establishing properties of similar Ex 11A Similar figures and scale factors figures (page 340)  recognising similarity in daily life Ex 11A Similar figures and scale factors (page 340)  finding scale factors of similar figures Ex 11A Similar figures and scale factors (page 340)  recognising that similar figures related Theory (page 339) by a scale factor of 1 are congruent  using enlargement or reduction factors Ex 11B Solving problems using similar to calculate actual dimensions figures (page 344)  developing scale drawings of objects Scale drawing of the classroom (page 345) and images  using scale factors to solve problems Ex 11B Solving problems using similar involving similar figures figures (page 344)  transferring measurements between House plans (page 350) floor plans and elevations  obtaining measurements from plans of Ex 11C House plans (page 348) buildings and rooms  calculating lengths and areas from a Ex 11C House plans (page 348) floor plan  interpreting commonly used symbols House plans (page 350) on house plans

Syllabus/glossary terms Page Congruent 339 Elevation 347 Enlargement 342 Reduction 338 Scale factor 339 Similar 338

Further activities Enlarging a figure (page 342) Investigating scale factors (page 342) 22

FM3: Taxation (Chapter 12) The principal focus of this unit is the calculation of tax payable on income and goods and services.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P5 represents the relationships between changing quantities in algebraic and graphical form P7 determines the degree of accuracy of measurements and calculations P8 models financial situations using appropriate tools P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  calculating the amount of allowable Ex 12A Calculating allowable deductions deductions from gross income (page 359)  calculating taxable income Ex 12B Taxable income (page 363)  calculating the Medicare levy (basic Ex 12C Medicare levy (page 367) levy only – see Tax Pack for details) Medicare levy (page 367)  calculating PAYE (Pay As You Earn) Ex 12D Calculating tax (page 372) tax payable or refund owing, using current tax scales  calculating the Value Added Tax Ex 12E Calculating GST and VAT (VAT) payable on a range of goods and (page 377) services, once given the tax rates for a range of countries  creating graphs to illustrate and Ex 12F Graphing tax functions (page 379) describe different tax rates Note: a graph of tax paid against taxable income is a piecewise linear function (see AM2)

Syllabus/glossary terms Page Goods and Services Tax (GST) 375 Group certificate 370 Income yield 356 PAYE 356 Medicare levy 367 Tax deduction 356 Taxable income 356 Value Added Tax (VAT) 376

Further activities Computer application 1: Calculating taxable income (page 364) Computer application 2: Tax calculation (page 374) 23

Measurement 4

M4: Right-angled triangles (Chapter 13) In this unit, students learn to solve practical mathematical problems involving right-angled triangles.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P3 develops rules to represent patterns arising from numerical and other sources P6 performs calculations in relation to two-dimensional and three-dimensional figures P7 determines the degree of accuracy of measurements and calculations P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  using Pythagoras’ theorem to find an Ex 13A Pythagoras’ theorem (page 390) unknown side in a right-angled triangle  applying Pythagoras’ theorem to: Ex 13A Pythagoras’ theorem (page 390) - determine whether or not a triangle is right-angled - solve problems based on single, right-angled triangles - calculate perimeters of irregularly shaped blocks of land  defining sine, cosine and tangent ratios Looking at the tangent ratio (page 392) Looking at the sine ratio (page 394) Looking at the cosine ratio (page 395) Ex 13B Calculating trigonometric ratios (page 397)  using trigonometric ratios to find the Ex 13C Finding an unknown side (page 402) length of an unknown side in a right- angled triangle  using trigonometric ratios to find the Ex 13D Finding angles (page 408) size of an unknown angle in a right- angled triangle, using a calculator to approximate the angle to the nearest minute  solving problems involving angles of Ex 13E Angles of elevation and depression elevation and depression, given the (page 414) appropriate diagram  determining whether an answer seems Checking with a proportional diagram reasonable or not by using a diagram (page 416) drawn roughly in proportion Using proportional diagrams (page 416) 24

Syllabus/glossary terms Page Adjacent 392 Angle of depression 411 Angle of elevation 411 Cosine 395 Hypotenuse 387 Opposite 392 Sine 394 Tangent 392 Trigonometry 392

Further activities History of Mathematics: Pythagoras of Samos (page 386) Calculation of heights (page 415) 25

Probability 1

PB1: The language of chance (Chapter 14) In this unit, students learn to use the language of probability, count outcomes, and describe the sample space of an event.

Outcomes addressed A student: P1 develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation P3 develops rules to represent patterns arising from numerical and other sources P10 performs simple calculations in relation to the likelihood of familiar events P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  ordering everyday events from the very Ex 14A Informal description of chance unlikely to the almost certain (page 427)  using a list or table to identify the Ex 14B Sample space (page 431) sample space (set of all possible outcomes) of a simple experiment  performing experiments and Matching actual and expected results determining whether or not the (page 432) outcomes are equally likely Two stage experiments (page 438)  determining the number of outcomes Ex 14E Using the fundamental counting for a multi-stage event by multiplying principle (page 446) the number of choices at each stage, eg. the total number of ways to place three different letters in three envelopes is: 3  2  1  using systematic lists to verify the total Ex 14C Tree diagrams (page 437) number of outcomes for simple multi- stage events (factorial notation not req)

Syllabus/glossary terms Page Equally likely 439 Event 434 Frequency 425 Multi-stage event 434 Outcomes 430 Sample space 429

Further activities Common descriptions of chance (page 429) 26

Probability 2

PB2: Relative frequency and probability (Chapter 15) The main focus of this unit is to compare relative frequency and calculated probability.

Outcomes addressed A student: P2 applies mathematical knowledge and skills to solving problems within familiar contexts P4 represents information in symbolic, graphical and tabular forms P10 performs simple calculations in relation to the likelihood of familiar events P11 justifies his/her response to a given problem using appropriate mathematical terminology

Skills, knowledge and understanding Exercises/Investigations  estimating the relative frequencies of Ex 15A Relative frequency (page 456) events from recorded data  performing simple experiments to Researching relative frequencies (page 458) obtain relative frequencies from recorded results  using relative frequencies to obtain approximate probabilities  using the following definition of Ex 15B Single event probability (page 461) probability of an event where outcomes are equally likely: number of favourable outcomes P(event)  total number of outcomes  calculating probabilities in terms of Ex 15C Writing probabilities as decimals their fractional, decimal or percentage and percentages (page 467) chance  demonstrating the range of Ex 15D Range of probabilities (page 471) probabilities, 0  P(E)  1, by examining a variety of results  comparing calculated probabilities with Comparing probabilities with actual results experimental results (page 464)  illustrating the results of experiments Graphing results (page 473) through statistical graphs and displays (see DA3)  defining and using the relationship Ex 15E Complementary events (page 476) between complementary events P (an event does not occur) = 1 – P (the event does occur) 27

Syllabus/glossary terms Page Complementary events 474 Percentage chance 466 Probability 459 Relative frequency 454