Mathscape 9 Teaching Program Page 1

Stage 5 MATHSCAPE 9

Term Chapter Time

1 1. Rational numbers 2 weeks / 8 hrs 2. Algebra 2 weeks / 8 hrs 3. Consumer arithmetic 2 weeks / 8 hrs 4. Equations, inequations and formulae 2 weeks / 8 hrs

2 5. Measurement 3 weeks / 12 hrs 6. Data representation and analysis 2 weeks / 8 hrs 7. Probability 1 week / 4 hrs 8. Indices 3 weeks / 12 hrs

4 9. Geometry 2 weeks / 8 hrs 10. The linear function 2 weeks / 8 hrs 11. Trigonometry 3 weeks / 12 hrs 12. Co-ordinate geometry 2 weeks / 8 hrs

Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 2

Chapter 1. Rational numbers Text references CD reference Substrand Mathscape 9 Significant figures Rational numbers Chapter 1. Rational Numbers Recurring decimals (pages 1–25) Rates Duration 2 weeks / 8 hours Key ideas Outcomes Round numbers to a specified number of significant figures. NS5.2.1 (page 67): Rounds decimals to a specified number of significant Express recurring decimals as fractions. figures, expresses recurring decimals in fraction form and converts rates from Convert rates from one set of units to another. one set of units to another. Working mathematically Students learn to 2  recognise that calculators show approximations to recurring decimals e.g. 3 displayed as 0.666667 (Communicating) .  justify that 0.9  1(Reasoning)  decide on an appropriate level of accuracy for results of calculations (Applying Strategies)  assess the effect of truncating or rounding during calculations on the accuracy of the results (Reasoning)  appreciate the importance of the number of significant figures in a given measurement (Communicating)  use an appropriate level of accuracy for a given situation or problem solution (Applying Strategies)  solve problems involving rates (Applying Strategies) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  identifying significant figures Fermi Problem (page 10): Estimation problem solving  rounding numbers to a specified number of significant figures Desert Walk (page 15): Problem solving Passing Trains (page 20): Travel graph problem  using the language of estimation appropriately, including:  FOCUS ON WORKING MATHEMATICALLY . rounding Art, Magic Squares and Mathematics (page 20): If you would like to learn . approximate how to make a magic square start with John Webb's article in the June . level of accuracy 2000 journal of nrich, the mathematics enrichment page of the Millenium   using symbols for approximation e.g. Mathematics Project based at the University of Cambridge http://nrich.maths.org/mathsf/journalf/jun00/art2/ There are many sites Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 3

 determining the effect of truncating or rounding during calculations on the which will provide instructions but this is a good one to begin. From accuracy of the results January 2004 the nrich web home page can be found at  writing recurring decimals in fraction form using calculator and non-calculator http://nrich.maths.org/public/viewer.php?obj_id=1376 and the home methods page of the project at http://mmp.maths.org/index.html . . . . The web page http://www-history.mcs.st- e.g. 0.2 , 0.23 , 0.23 and.ac.uk/history/Mathematicians/Durer.html will get you straight to  converting rates from one set of units to another Albrecht Durer. You can scroll through the text to get a look at his e.g. km/h to m/s, interest rate of 6% per annum is 0.5% per month engraving Melancholia which is highlighted in blue. From here you can go to the main index and look for "magic squares" under topics, or check out Leonhard Euler under mathematicians.  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 23)  CHAPTER REVIEW (page 24) a collection of problems to revise the chapter. Technology Significant Figures: this spreadsheet in designed to round off a given number to a desired number of significant figures. To be used with the text. Recurring Decimals: this spreadsheet converts recurring decimals to fractions. Rates: this spreadsheet deals with rates and ratios in units.

Chapter 2. Algebra Text references CD reference Substrand Mathscape 9 Simplify (with fractional indices) Chapter 2. Algebra Expand Algebraic techniques (pages 26–66) Railway tickets Duration 2 weeks / 8 hours Key ideas Outcomes Simplify, expand and factorise algebraic expressions including those involving PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions fractions or with negative and/or fractional indices. involving fractions and negative and fractional indices. Working mathematically Students learn to  describe relationships between the algebraic symbol system and number properties (Reflecting, Communicating)  link algebra with generalised arithmetic e.g. use the distributive property of multiplication over addition to determine that a(b  c)  ab  ac (Reflecting)  determine and justify whether a simplified expression is correct by substituting numbers for pronumerals (Applying Strategies, Reasoning)  generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies)  check expansions and factorisations by performing the reverse process (Reasoning)  interpret statements involving algebraic symbols in other contexts e.g. spreadsheets (Communicating)  explain why an algebraic expansion or factorisation is incorrect e.g. Why is the following incorrect? 24x 2 y  16xy 2  8xy(3x  2) (Reasoning, Communicating) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  simplifying algebraic expressions involving fractions, such as Flags (page 35): Algebraic problem solving Overhanging the overhang (page 42): Practical 2x 2x 7a 5a 2y y 2ab 6     Railway Tickets (page 58): Complete a table and find a rule 5 3 8 12 3 6 3 2b  FOCUS ON WORKING MATHEMATICALLY  expanding, by removing grouping symbols, and collecting like terms where Party Magic (page 59): Teachers may wish to down load the Party Magic possible, algebraic expressions such as with Algebra worksheet in the technology folder for chapter 2 Algebra. This worksheet explores the algebraic structure of the games using technology. The web link http://atschool.eduweb.co.uk/ufa10/tricks.htm at

2y(y  5)  4(y  5) Birmingham in England has great resources for students and teachers. 4x(3x  2)  (x 1) The web page http://www.umassmed.edu/bsrc/tricks.cfm has good links  3x2 (5x2  2xy) and lots of activites to show that maths really can be fun. For the addicted to fun and games check out Martin Gardner's books at  factorising, by determining common factors, algebraic expressions such as http://thinks.com/books/gardner.htm 3x2  6x  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING 14ab 12a2 (page 61) 21xy  3x  9x2  CHAPTER REVIEW (page 62) a collection of problems to revise the chapter. Technology Simplify (with fractional indices): algebraic program that simplifies algebraic terms. To be used with the worksheets. Also to be used with the Focus on Working mathematically section. Expand: this program will expand a given algebraic expression. Railway Tickets: worksheet to use with the “Try This” problem on page 58.

Chapter 3. Consumer arithmetic Text references CD reference Substrand Mathscape 9 Money Consumer arithmetic Chapter 3. Consumer Arithmetic (pages 67–107) Duration 2 weeks / 8 hours Key ideas Outcomes Solve simple consumer problems including those involving earning and spending NS5.1.2 (page 70): Solves consumer arithmetic problems involving earning money. and spending money. Calculate simple interest and find compound interest using a calculator and tables of NS5.2.2 (page 71): Solves Consumer arithmetic problems involving compound values. interest, depreciation, and successive discounts. Use compound interest formula. Solve consumer arithmetic problems involving compound interest, depreciation and successive discounts. Working mathematically Students learn to  read and interpret pay slips from part-time jobs when questioning the details of their own employment (Questioning, Communicating)  prepare a budget for a given income, considering such expenses as rent, food, transport etc (Applying Strategies)  interpret the different ways of indicating wages or salary in newspaper ‘positions vacant’ advertisements e.g. $20K (Communicating)  compare employment conditions for different careers where information is gathered from a variety of mediums including the Internet e.g. employment rates, payment (Applying Strategies)  explain why, for example, a discount of 10% following a discount of 15% is not the same as a discount of 25% (Applying Strategies, Communicating, Reasoning) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  calculating earnings for various time periods from different sources, including: Sue’s Boutique (page 72): Problem Solving - wage Telephone Charges (page 92): Problem Solving - salary Progressive Discounting (page 98): Investigation - commission  FOCUS ON WORKING MATHEMATICALLY - piecework Sydney Market prices in 1831 (page 102): The purpose of the learning

- overtime activities is for students to think about the cost of living in Australia today - bonuses using market prices in 1831 as a starting point. As an extension students - holiday loadings are given opportunity to explore inflation and how the consumer price - interest on investments index (CPI) is calculated. An invitation to a member of the Economics  calculating income earned in casual and part-time jobs, considering agreed rates staff to your class could be stimulating. Teachers should note that the and special rates for Sundays and public holidays further apart the years being compared, the less valid it is to use the relative prices of goods in those years to measure the standard of living.  calculating weekly, fortnightly, monthly and yearly incomes This point is well made in the article by Nell Ingalls published on the web  calculating net earnings considering deductions such as taxation and site http://www.sls.lib.il.us/reference/por/features/98/money.html. This superannuation is a useful source of information on the value of money.  calculating a ‘best buy’ A good summary of how the CPI is calculated in Australia can be found at  calculating the result of successive discounts http://www.aph.gov.au/library/pubs/mesi/features/cpi.htm  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 105)  CHAPTER REVIEW (page 106) a collection of problems to revise the chapter. Technology Money: series of worksheets to use with spreadsheets to explore Commission, Net Income, Piece Work, Salaries, Wages and a Weekly Budget.

Chapter 4. Equations, inequations and formulae Text references CD reference Substrand Mathscape 9 Evaluating Algebraic techniques Chapter 4: Equations, inequations and Floodlighting formulae (pages 108–40) Duration 2 weeks / 8 hours Key ideas Outcomes Solve linear and simple quadratic equations of the form ax2  c PAS5.2.2 (page 90): Solves linear and simple quadratic equations, solves Solve linear inequalities linear inequalities and solves simultaneous equations using graphical and analytical methods.

Working mathematically Students learn to  compare and contrast different methods of solving linear equations and justify a choice for a particular case (Applying Strategies, Reasoning)  use a number of strategies to solve unfamiliar problems, including: - using a table - drawing a diagram - looking for patterns - working backwards - simplifying the problem and - trial and error (Applying Strategies, Communicating)  solve non-routine problems using algebraic methods (Communicating, Applying Strategies)  explain why a particular value could not be a solution to an equation (Applying Strategies, Communicating, Reasoning)  create equations to solve a variety of problems and check solutions (Communicating, Applying Strategies, Reasoning)  write formulae for spreadsheets (Applying Strategies, Communicating)  solve and interpret solutions to equations arising from substitution into formulae used in other strands of the syllabus and in other subjects. Formulae such as the following could be used: y  y m  2 1 x2  x1 1 E  mv2 2 4 V  r3 3 SA  2r 2  2rh (Applying Strategies, Communicating, Reflecting)  explain why quadratic equations could be expected to have two solutions (Communicating, Reasoning)  justify a range of solutions to an inequality (Applying Strategies, Communicating, Reasoning) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS Linear and Quadratic Equations A Prince and a King (page 129): Two Ancient Problems Arm Strength (page 132): Formulae Investigation

 solving linear equations such as Floodlighting by formula (page 136): Formulae Investigation x x  FOCUS ON WORKING MATHEMATICALLY   5 Bushfires (page 137): Teachers may wish to use a spreadsheet to evaluate 2 3 F given C and vice versa. Go to the Evaluating the subject.xls 2y  3  2 spreadsheet in the technology folder for Chapter 4. There is also a useful 3 worksheet. Extension students could discuss whether F = 9C/5 + 32 is a z  3 formula or an equation and what constitutes the difference -- see page 136  6  1 2 on Floodlighting for example. For newspaper reports of the fires try the Sydney Morning herald web site http://www.smh.com.au/. If you type 3(a  2)  2(a  5)  10 'Sydney bushfires' into a search engine you will get a range of options. 3(2t  5)  2t  5 http://www.gi.alaska.edu/ScienceForum/ASF13/1317.html will give 3r 1 2r  4 you a short account how the two men Daniel Fahrenheit and Anders  4 5 Celcius constructed their scales. This will be very useful link with the study of science.  solving word problems that result in equations  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING  exploring the number of solutions that satisfy simple quadratic equations of the (page 138) form x2  c  CHAPTER REVIEW (page 139) a collection of problems to revise the 2 chapter.  solving simple quadratic equations of the form ax  c  solving equations arising from substitution into formulae

Linear Inequalities  solving inequalities such as 3x 1  9 2(a  4)  24 t  4  3 5 Technology Evaluating: students analyse a spreadsheet and then design their own. Floodlighting: activity to complement the “Try This” problem on page 136.

Chapter 5. Measurement Text references CD reference Substrand Mathscape 9 Perigal Algebraic techniques Chapter 5:.Measurement Measuring plane shapes (pages 141–205) Circle measuring Duration 3 weeks / 12 hours Key ideas Outcomes Develop formulae and use to find the area of rhombuses, trapeziums and kites. MS5.1.1 (page 126): Use formulae to calculate the area of quadrilaterals and Find the area and perimeter of simple composite figures consisting of two shapes find areas and perimeters of simple composite figures. including quadrants and semicircles. MS5.2.1 (page 127): Find areas and perimeters of composite figures. Find area and perimeter of more complex composite figures. Working mathematically Students learn to  identify the perpendicular height of a trapezium in different orientations (Communicating)  select and use the appropriate formula to calculate the area of a quadrilateral (Applying Strategies)  dissect composite shapes into simpler shapes (Applying Strategies)  solve practical problems involving area of quadrilaterals and simple composite figures (Applying Strategies)  solve problems involving perimeter and area of composite shapes (Applying Strategies)  calculate the area of an annulus (Applying Strategies)  apply formulae and properties of geometrical shapes to find perimeters and areas e.g. find the perimeter of a rhombus given the lengths of the diagonals (Applying Strategies)  identify different possible dissections for a given composite figure and select an appropriate dissection to facilitate calculation of the area (Applying Strategies, Reasoning) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  developing and using formulae to find the area of quadrilaterals: Bags of potatoes (page 147): Problem Solving 1 Overseas Call (page 154): Problem Solving - for a kite or rhombus, Area  2 xy where x and y are the lengths of the diagonals; Pythagorean Proof by Perigal (page 160): Proof The box and the wall (page 163): Problem Solving 1 - for a trapezium, Area  2 h(a  b) where h is the perpendicular height and a Command Module (page 174): Investigation of Apollo 11 and b the lengths of the parallel sides The area of a circle (page 185): Archimedes method  calculating the area of simple composite figures consisting of two shapes Area (page 195): Challenge Problem Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 12

including quadrants and semicircles  FOCUS ON WORKING MATHEMATICALLY  calculating the perimeter of simple composite figures consisting of two shapes The Melbourne Cup (page 198): These activities focus on units of including quadrants and semicircles measurement linked to the Melbourne cup. The history of the cup provides  calculating the area and perimeter of sectors insight into the dramatic changes in the Australian way of life since the  calculating the perimeter and area of composite figures by dissection into race began in 1861. Students also explore the origin of the words used to triangles, special quadrilaterals, semicircles and sectors describe the units. The web page http://www.unc.edu/~rowlett/units/ is a dictionary of unusual units you will find fascinating. If you type in 'Melbourne cup' into a search engine, you will have lots of choice. Try the VRC web page http://home.vicnet.net.au/~basiced3/cup/history.html The web page http://www.equine-world.co.uk/about_horses/height.htm will show you nice diagram on the way the heights of horses are measured.  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 199)  CHAPTER REVIEW (page 201) a collection of problems to revise the chapter. Technology Perigal: Cabri Geometry interactive worksheet on the Pythagorean proof by Perigal. Measuring Plane Shapes: this file contains hyperlinks to a number of interactive geometric diagrams. Circle Measuring: a set of Cabri Geometry interactive worksheets that are used for students to explore the parts and use of circles.

Chapter 6. Data representation and analysis Text references CD reference Substrand Mathscape 9 Data analysis Data representation and analysis Chapter 6. Data Representation and Cumulative analysis Analysis (pages 206–53) Duration 2 weeks / 8 hours Key ideas Outcomes Construct frequency tables for grouped data. DS5.1.1 (page 116): Groups data to aid analysis and constructs frequency and Find mean and modal class for grouped data. cumulative frequency tables and graphs. Determine cumulative frequency. Find median using a cumulative frequency table or polygon Working mathematically Students learn to  construct frequency tables and graphs from data obtained from different sources (e.g. the Internet) and discuss ethical issues that may arise from the data (Applying Strategies, Communicating, Reflecting)  read and interpret information from a cumulative frequency table or graph (Communicating)  compare the effects of different ways of grouping the same data (Reasoning)  use spreadsheets, databases, statistics packages, or other technology, to analyse collected data, present graphical displays, and discuss ethical issues that may arise from the data (Applying Strategies, Communicating, Reflecting) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  constructing a cumulative frequency table for ungrouped data The English Language (page 232): Investigation Earthquakes (page 246): Can we predict the number of Earthquakes there  constructing a cumulative frequency histogram and polygon (ogive) will be in a year?  using a cumulative frequency polygon to find the median  FOCUS ON WORKING MATHEMATICALLY  grouping data into class intervals World Health (page 246): This investigation provides an opportunity for  constructing a frequency table for grouped data students to analyse two indicators of world public health and to apply their skills in Working mathematically. The objective is to show how statistical  constructing a histogram for grouped data evidence can play a role in arguing a case for the development of  finding the mean using the class centre programs to support global health. There is an excellent opportunity for  finding the modal class class discussion about the sort of data governments need in order to make sensible policy decisions for global health. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 14

A good international web site is http://www.globalhealth.gov/worldhealthstatistics.shtml The frequently asked questions page at http://www.globalhealth.gov/faq.shtml provides useful background information for teachers  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 249)  CHAPTER REVIEW (page 250) a collection of problems to revise the chapter. Technology Data Analysis: students Analyse data with the help of a spreadsheet. Cumulative Analysis: students use the spreadsheet to calculate the median using the cumulative frequency

Chapter 7. Probability Text reference CD reference Substrand Mathscape 9 Probability Probability Chapter 8. Probability (pages 254–80) Craps simulation Weighted dice Duration 1 week / 4 hours Key ideas Outcomes Determine relative frequencies to estimate probabilities. NS5.1.3 (page 75): Determines relative frequencies and theoretical Determine theoretical probabilities. probabilities. Working mathematically Students learn to  recognise and explain differences between relative frequency and theoretical probability in a simple experiment (Communicating, Reasoning)  apply relative frequency to predict future experimental outcomes (Applying Strategies)  design a device to produce a specified relative frequency e.g. a four-coloured circular spinner (Applying Strategies)  recognise that probability estimates become more stable as the number of trials increases (Reasoning)  recognise randomness in chance situations (Communicating)  apply the formula for calculating probabilities to problems related to card, dice and other games (Applying Strategies) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  repeating an experiment a number of times to determine the relative frequency of Two-Up (page 265): Experiment an event The game of Craps (page 270): Simulation Winning Chances (page 274): Problem Solving  estimating the probability of an event from experimental data using relative  FOCUS ON WORKING MATHEMATICALLY frequencies Getting through traffic lights (page 275): This activity is designed to  expressing the probability of an event A given a finite number of equally likely introduce students to the idea of a simulation. It is designed for all students outcomes as to enjoy. Teachers should carry out the simulation first using the number of favourable outcomes technology they wish to use in class. P(A) = n The Maths Online web site at http://www.mathsonline.co.uk/nonmembers/resource/prob/ is a great where n is the total number of outcomes in the sample space help to teachers looking for lesson plans to simulate real life probability  using the formula to calculate probabilities for simple events problems. Includes on line flash movies which will draw graphs directly Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 16

 simulating probability experiments using random number generators from your input. For a good reference text with a CD ROM to simulate probability problems using a graphics calculator try Winter MJ and Carlson RJ (2001) Probability Simulations, Key Curriculum Press, Emeryville, California. Barry Kissane's web page http://wwwstaff.murdoch.edu.au/ %7Ekissane/graphicscalcs.htm is invaluable for CASIO users. The Open University Centre for Mathematics Education at http://mcs.open.ac.uk/cme/TIcourses/timain.html has some excellent courses for teachers with little experience in using a TI graphics calculator.  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 276)  CHAPTER REVIEW (page 278) a collection of problems to revise the chapter. Technology Probability: the spreadsheet simulates the drawing of different coloured balls from a bag with replacement. Craps Simulation: this spreadsheet explores the probabilities of winning and losing a game of craps. Weighted Dice: dice simulation spreadsheet.

Chapter 8. Indices Text references CD reference Substrands Mathscape 9 Simplify (with fractional indices) Rational numbers Chapter 8. Indices (pages 281–312) Expand Algebraic techniques Duration 3 weeks / 12 hours Key ideas Outcomes Define and use zero index and negative integral indices. NS5.1.1 (page 65): Applies index laws to simplify and evaluate arithmetic Develop the index laws arithmetically. expressions and uses scientific notation to write large and small numbers. Use index notation for square and cube roots. PAS5.1.1 (page 87): Applies the index laws to simplify algebraic expressions. Express numbers in scientific notation (positive and negative powers of 10) PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions Apply the index laws to simplify algebraic expressions (positive integral indices involving fractions and negative and fractional indices. only). Simplify, expand and factorise algebraic expressions including those involving fractions or with negative and/or fractional indices. Working mathematically Students learn to  solve numerical problems involving indices (Applying Strategies)  explain the incorrect use of index laws e.g. why 32  34  96 (Communicating, Reasoning)

1 2  verify the index laws by using a calculator e.g. to compare the values of  52 , 5 2  and 5 (Reasoning)    communicate and interpret technical information using scientific notation (Communicating)  explain the difference between numerical expressions such as 2104 and 2 4 , particularly with reference to calculator displays (Communicating, Reasoning)  solve problems involving scientific notation (Applying Strategies)  verify the index laws using a calculator e.g. use a calculator to compare the values of (34 )2 and 38 (Reasoning)  explain why x0  1 (Applying Strategies, Reasoning, Communicating)  link use of indices in Number with use of indices in Algebra (Reflecting)  explain why a particular algebraic sentence is incorrect e.g. explain why a3  a2  a6 is incorrect (Communicating, Reasoning)

 examine and discuss the difference between expressions such as 3a2  5a and 3a2  5a by substituting values for a (Reasoning, Applying Strategies, Communicating)  explain why finding the square root of an expression is the same as raising the expression to the power of a half (Communicating, Reasoning)  state whether particular equivalences are true or false and give reasons e.g. Are the following true or false? Why? 5x0  1 9x5  3x5  3x a5  a7  a2 1 2c4  2c4 (Applying Strategies, Reasoning, Communicating)  explain the difference between particular pairs of algebraic expressions, such as x 2 and  2x (Reasoning, Communicating) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  describing numbers written in index form using terms such as base, power, index, Power Pulse Graphs (page 283): Investigation exponent Smallest to Largest (page 295): Problem Solving  evaluating numbers expressed as powers of positive whole numbers Digit Patterns (page 300): Investigation  FOCUS ON WORKING MATHEMATICALLY  establishing the meaning of the zero index and negative indices e.g. by patterns Mathematics is at the heart of Science (page 308): The Powers of 10 web 32 31 30 31 32 site http://www.powersof10.com/ should be explored before starting this 9 3 1 1 1 1 Working mathematically activity. There are excellent pictures and ideas 3 9  2 3 for creating absorbing lessons. The learning activities are suitable for 1 1  writing reciprocals of powers using negative indices e.g. 34   students working in pairs. Calculators are recommended. In particular try 34 81 the patterns section at  translating numbers to index form (integral indices) and vice versa http://www.powersof10.com/powers/patterns/patterns.html  developing index laws arithmetically by expressing each term in expanded form The ABC web site http://www.abc.net.au/science has a wealth of ideas to e.g. 32  34  (3 3)  (3 3 3 3)  324  36 enable students to see how mathematics lies at the heart of science. The Dr Karl page has a live Q & A opportunity. The class could formulate a 3 3 3 3 3 35  32   352  33 question, send it in and listen to the answer on radio or online. There is 3 3 also a news page which provides great ideas for lesson starters. Teachers 32 4 3 3 3 3 3 3 3 3  324  38 are encouraged to liaise with science staff for further information and to Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 19

 using index laws to simplify expressions invite them to the lesson.  using index laws to define fractional indices for square and cube roots  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING 2 (page 309) 2 1 1  CHAPTER REVIEW (page 311) a collection of problems to revise the  2  e.g.  9  9 and 9   9 , hence 9  9 2   chapter.

1  writing square roots and cube roots in index form e.g. 8 3  3 8 2  recognising the need for a notation to express very large or very small numbers  expressing numbers in scientific notation  entering and reading scientific notation on a calculator  using index laws to make order of magnitude checks for numbers in scientific notation e.g.  3.12104  4.2106   121010  1.21011  converting numbers expressed in scientific notation to decimal form  ordering numbers expressed in scientific notation  using the index laws previously established for numbers to develop the index laws in algebraic form e.g. 22  23  223  25 am  an  amn 25  22  252  23 am  an  amn 22 3 26 (am )n  amn  establishing that a0  1 using the index laws e.g. a3  a3  a33  a0 and a3  a3  1  a0  1  simplifying algebraic expressions that include index notation e.g. 5x0  3  8 2x2  3x3  6x5 12a6  3a2  4a4 2m3(m2  3)  2m5  6m3  applying the index laws to simplify expressions involving pronumerals

2  establishing that  a   a  a  a  a  a2  a  using index laws to assist with the definition of the fractional index for square root 2 given  a   a

1 2 and a 2   a   1 then a  a 2  using index laws to assist with the definition of the fractional index for cube root  using index notation and the index laws to establish that 1 1 1 a1  , a2  , a3  , … a a2 a3  applying the index laws to simplify algebraic expressions such as (3y2 )3 4b5 8b3 9x4  3x3 1 1 3x 2 5x 2 1 1 6y 3 4y 3 Technology Simplify (with fractional indices): algebraic program that simplifies algebraic terms. To be used with the worksheet. Also to be used with the Focus on Working mathematically section. Worksheet included. Expand: this program will expand a given algebraic expression.

Chapter 9. Geometry Text references CD reference Substrand Mathscape 9 N polygon Properties of geometric figures Chapter 9. Geometry (pages 313–63) Exterior angle Duration Euler line 2 weeks / 8 hours Key idea Outcomes Establish sum of exterior angles result and sum of interior angles result for polygons. SGS5.2.1 (page 157): Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon. Working mathematically Students learn to  express in algebraic terms the interior angle sum of a polygon with n sides e.g. (n–2)  180 (Communicating)  find the size of the interior and exterior angles of regular polygons with 5,6,7,8, … sides (Applying Strategies)  solve problems using angle sum of polygon results (Applying Strategies) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS  applying the result for the interior angle sum of a triangle to find, by dissection, The badge of the Pythagoreans (page 337): Historical Problem the interior angle sum of polygons with 4,5,6,7,8, … sides Five Shapes (page 348): Problem Solving  defining the exterior angle of a convex polygon How many diagonals in a polygon? (page 353): Investigation  establishing that the sum of the exterior angles of any convex polygon is 360  FOCUS ON WORKING MATHEMATICALLY  applying angle sum results to find unknown angles A surprising finding (page 354): In this activity we arrive at Pythagoras' theorem from a cutting and pasting activity with hexagons of equal area. It is designed as a fun activity for all students. However there is a deeper mathematical idea which is really for teachers but could be used as an extension. The 4 hexagons drawn on pages 355–356 each tessellate the plane. See http://www.cut-the-knot.org/pythagoras/index.shtml for details -- read proof 16 and 38 of Pythagoras' theorem.  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 356)  CHAPTER REVIEW (page 357): A collection of problems to revise the chapter. Technology Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 22

N Polygon: this geometry program draws regular polygons at speed and displays their diagonals. Explores a curious geometrical pattern that would be time consuming if drawn by hand. Exterior Angle: this learning activity makes use of the exterior angle property of a triangle. Students have the opportunity to apply the reasoning to solve a problem in geometry. Euler Line: the Euler line of a triangle is a line that passes through three special points of a triangle. Investigative exercise.

Chapter 10. The linear function Text references CD reference Substrand Mathscape 9 Line equation Co-ordinate geometry Chapter 10. The Linear Function Intersecting lines (pages 364–99) Duration 2 weeks / 8 hours Key ideas Outcomes Use a diagram to determine midpoint, length and gradient of an interval joining two PAS5.1.2 (page 97): Determines the midpoint, length and gradient of an points on the number plane. interval joining two points on the number plane and graphs linear and simple Graph linear and simple non-linear relationships from equations. non-linear relationships from equations. Working mathematically Students learn to  explain the meaning of gradient and how it can be found for a line joining two points (Communicating, Applying Strategies)  distinguish between positive and negative gradients from a graph (Communicating)  describe horizontal and vertical lines in general terms (Communicating)  explain why the x -axis has equation y = 0 (Reasoning, Communicating)  explain why the y -axis has equation x = 0 (Reasoning, Communicating)  determine the difference between equations of lines that have a negative gradient and those that have a positive gradient (Reasoning)  use a graphics calculator and spreadsheet software to graph, compare and describe a range of linear and simple non-linear relationships(Applying Strategies, Communicating)  apply ethical considerations when using hardware and software (Reflecting) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS Size 8 (page 374): Problem Solving Midpoint, Length and Gradient Hanging around (page 383): Problem Solving  using the right-angled triangle drawn between two points on the number plane and Latitude and Temperature (page 389): Investigation the relationship  FOCUS ON WORKING MATHEMATICALLY rise Paper Sizes in the printing industry (page 394): The web link gradient  http://www.cl.cam.ac.uk/~mgk25/iso-paper.html is a good overview of run the length to breadth relationship of A4 to A3. to find the gradient of the interval joining two points The web link http://www.twics.com/~eds/paper/papersize.html Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 24

 determining whether a line has a positive or negative slope by following the line provides you with more information about international paper sizes. Note from left to right – if the line goes up it has a positive slope and if it goes down it that the B series is about half way between two A sizes. It is intended as an has a negative slope alternative to the A sizes, and used primarily for books, posters and wall  finding the gradient of a straight line from the graph by drawing a right-angled charts. Note that the ratio of length to breadth for the B series is also √2 : 1 triangle after joining two points on the line and the ratio of their areas 2 : 1.  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING Graphs of Relationships (page 395)  constructing tables of values and using coordinates to graph vertical and  CHAPTER REVIEW (page 396) a collection of problems to revise the horizontal lines such as chapter. x  3, x  1 y  2, y  3  identifying the x - and y -intercepts of graphs  identifying the x -axis as the line y = 0  identifying the y -axis as the line x = 0  graphing a variety of linear relationships on the number plane by constructing a table of values and plotting coordinates using an appropriate scale e.g. graph the following: y  3  x x 1 y  2 x  y  5 x  y  2 2 y  x 3  determining whether a point lies on a line by substituting into the equation of the line Technology Line Equation: interactive program with accompanying worksheet. Intersecting Lines: interactive program with accompanying worksheet.

Chapter 11. Trigonometry Text references CD reference Substrand Mathscape 9 Sine Cosine Trigonometry Chapter 11. Trigonometry SOHCAHTOA (pages 400–38) Duration 3 weeks / 12 hours Key ideas Outcomes Use trigonometry to find sides and angles in right-angled triangles. MS5.1.2 (page 139): Applies trigonometry to solve problems (diagrams given) Solve problems involving angles of elevation and angles of depression from diagrams including those involving angles of elevation and depression. Working mathematically Students learn to  label sides of right-angled triangles in different orientations in relation to a given angle (Applying Strategies, Communicating)  explain why the ratio of matching sides in similar right-angle triangles is constant for equal angles (Communicating, Reasoning)  solve problems in practical situations involving right-angled triangles e.g. finding the pitch of a roof (Applying Strategies)  interpret diagrams in questions involving angles of elevation and depression (Communicating)  relate the tangent ratio to gradient of a line (Reflecting) Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS Trigonometric Ratios of Acute Angles Height to Base Ratio (page 408): Investigation Make a Hypsometer (page 421): Practical  identifying the hypotenuse, adjacent and opposite sides with respect to a given Pilot Instructions (page 432): Problem Solving angle in a right-angled triangle in any orientation  FOCUS ON WORKING MATHEMATICALLY: (page 433) Finding your  labelling the side lengths of a right-angled triangle in relation to a given angle e.g. latitude from the sun the side c is opposite angle C This is designed as a fun outdoor activity. Teachers need to prepare well in  recognising that the ratio of matching sides in similar right-angled triangles is advance because of the restricted days of the equinox. However the constant for equal angles activity could be carried out on a day close to the equinox if it happens to  defining the sine, cosine and tangent ratios for angles in right-angled triangles be cloudy. An explanation of the difference can be found in Mathscape 9  using trigonometric notation e.g. sin A Extension page 481. The geometry should be discussed carefully in class  using a calculator to find approximations of the trigonometric ratios of a given before the outdoor lesson. angle measured in degrees See what a sailor does to determine latitude using an astrolabe at  using a calculator to find an angle correct to the nearest degree, given one of the http://www.ruf.rice.edu/~feegi/measure.html trigonometric ratios of the angle A great site to look at navigation in the 15th century is

Trigonometry of Right-Angled Triangles http://www.ruf.rice.edu/~feegi/site_map.html  selecting and using appropriate trigonometric ratios in right-angled triangles to Read about advances in navigational technology from the Astrolabe to find unknown sides, including the hypotenuse today's Global Positioning System at  selecting and using appropriate trigonometric ratios in right-angled triangles to http://www.canadiangeographic.ca/Magazine/ND01/ find unknown angles correct to the nearest degree findingourway.html  identifying angles of elevation and depression  EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 435)  solving problems involving angles of elevation and depression when given a diagram  CHAPTER REVIEW (page 436) a collection of problems to revise the chapter. Technology Sine Cosine: explores the range of Trig graphs. SOHCAHTOA: investigation of the tan ratio.

Chapter 12. Co-ordinate geometry Text references CD reference Substrand Mathscape 9 Intersecting lines Co-ordinate geometry Chapter 12. Co-ordinate Geometry (pages 439–70) Crow flying

Duration 2 weeks / 8 hours Key ideas Outcomes Use a diagram to determine midpoint, length and gradient of an PAS5.1.2 (page 97): Determines the midpoint, length and gradient of an interval joining two interval joining two points on the number plane. points on the number plane and graphs linear and simple non-linear relationships from equations. Graph linear and simple non-linear relationships from equations PAS5.2.3 (page 99): Uses formulae to find midpoint, distance and gradient and applies the Use midpoint, distance and gradient formulae. gradient/intercept form to interpret and graph straight lines. Apply the gradient/intercept form to interpret and graph straight lines. Working mathematically Students learn to  describe the meaning of the midpoint of an interval and how it can be found (Communicating)  describe how the length of an interval joining two points can be calculated using Pythagoras’ theorem (Communicating, Reasoning)  relate the concept of gradient to the tangent ratio in trigonometry for lines with positive gradients (Reflecting)  explain the meaning of each of the pronumerals in the formulae for midpoint, distance and gradient (Communicating)  use the appropriate formulae to solve problems on the number plane (Applying Strategies)  use gradient and distance formulae to determine the type of triangle three points will form or the type of quadrilateral four points will form and justify the answer (Applying Strategies, Reasoning)  explain why the following formulae give the same solutions as those in the left-hand column y  y 2 2 m  1 2 d  (x1  x2 )  (y1  y2 ) and (Reasoning, Communicating) x1  x2 Knowledge and skills Teaching, learning and assessment Students learn about  TRY THIS Car Hire (page 459): Problem Solving Midpoint, Length and Gradient Temperature Rising (page 462): Problem Solving  determining the midpoint of an interval from a diagram  FOCUS ON WORKING MATHEMATICALLY Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Teaching Program Page 28

 graphing two points to form an interval on the number plane and Finding the gradient of a ski run (page 463) forming a right-angled triangle by drawing a vertical side from The resource book Kleeman, G. and Peters A. (2002) Skills in Australian Geography, the higher point and a horizontal side from the lower point Cambridge University Press, Cambridge is your best guide for this activity. Try the Social  using the right-angled triangle drawn between two points on the Science department for a copy or your school library. number plane and Pythagoras’ theorem to determine the length For a good model of calculating gradient from contour maps go to of the interval joining the two points http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/map_sample_answer2. html. However measurements are calculated in feet which are still used in the USA. Midpoint, Distance and Gradient Formulae A good site written for scouts which looks at gradients, contours and features of ordinance  using the average concept to establish the formula for the survey maps is http://www.scoutingresources.org.uk/mapping_contour.html Note that the Sun moves from east to west through the northern sky in our (southern) midpoint, M, of the interval joining two points x , y  and 1 1 hemisphere. This means the sun will shine on the northern and eastern slopes during the day. x2, y2  on the number plane Hence the preference for these slopes. Just what we need to enjoy skiing.  x  x y  y   EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 466) M (x, y)   1 2 , 1 2   2 2   CHAPTER REVIEW (page 468) a collection of problems to revise the chapter.  using the formula to find the midpoint of the interval joining two points on the number plane  using Pythagoras’ theorem to establish the formula for the

distance, d, between two points x1, y1 and x2, y2  on the number plane

2 2 d  (x2  x1)  (y2  y1)  using the formula to find the distance between two points on the number plane  using the relationship rise gradient  run to establish the formula for the gradient, m, of an interval

joining two points x1, y1 and x2, y2  on the number plane y  y m  2 1 x2  x1  using the formula to find the gradient of an interval joining two points on the number plane

Gradient/Intercept Form  rearranging an equation in general form (ax + by + c = 0) to the gradient/intercept form  determining that two lines are parallel if their gradients are equal Technology Intersecting Lines: interactive program with accompanying worksheet. Crow Flying: students use Pythagoras’ Theorem to investigate how much distance they would save if they could fly in a straight line (as the crow flies) across city blocks. students create their own spreadsheet and investigate when the saving is greatest. Midpoint: interactive geometry worksheet.