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Mathscape 9 Extension Teaching Program Page 1 Stage 5 MATHSCAPE 9 EXTENSION Term Chapter 1 2 3 4 Time 1. Rational numbers 2 weeks / 8 hrs 2. Algebra 3 weeks / 12 hrs 3. Consumer arithmetic 2 weeks / 8 hrs 4. Equations, inequations and formulae 2 weeks / 8 hrs 5. Measurement 2 weeks / 8 hrs 6. Data representation and analysis 2 weeks / 8 hrs 7. Probability 1 week / 4 hrs 8. Surds 2 weeks / 8 hrs 9. Indices 2 weeks / 8 hrs 10. Geometry 2 weeks / 8 hrs 11. The linear function 2 weeks / 8 hrs 12. Trigonometry 2 weeks / 8 hrs 13. Simultaneous equations 1 week / 4 hrs 14. Co-ordinate geometry 2 weeks / 8 hrs Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 2 Chapter 1. Rational numbers Text references CD reference Substrand Mathscape 9 Extension Chapter 1. Rational Numbers (pages 1–24) Significant figures Recurring decimals Rates Rational numbers Duration 2 weeks / 8 hours Key ideas Outcomes Round numbers to a specified number of significant figures. Express recurring decimals as fractions. Convert rates from one set of units to another. NS5.2.1 (page 67): Rounds decimals to a specified number of significant figures, expresses recurring decimals in fraction form and converts rates from one set of units to another. Working mathematically Students learn to recognise that calculators show approximations to recurring decimals e.g. 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) 2 3 displayed as 0.666667 (Communicating) . Knowledge and skills Teaching, learning and assessment Students learn about identifying significant figures rounding numbers to a specified number of significant figures using the language of estimation appropriately, including: rounding approximate level of accuracy TRY THIS Fermi Problem (page 10): Estimation problem solving Desert Walk (page 15): Problem solving Passing Trains (page 19): Travel graph problem FOCUS ON WORKING MATHEMATICALLY A number pattern from Galileo (page 20): This is an activity designed for students to enjoy mathematics for its own sake. It illustrates the power of Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program using symbols for approximation e.g. determining the effect of truncating or rounding during calculations on the accuracy of the results writing recurring decimals in fraction form using calculator and non-calculator methods . Page 3 . . . e.g. 0. 2 , 0. 2 3 , 0.2 3 converting rates from one set of units to another e.g. km/h to m/s, interest rate of 6% per annum is 0.5% per month inductive reasoning. Teachers are advised to be ready to discuss the differences in inductive and deductive reasoning. It also provides an opportunity for students to see how a famous scientist loved mathematics for its own sake too. The web site at http://www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Galileo.html describes the major contributions Galileo made to mathematics and science. The NASA site http://www.jpl.nasa.gov/galileo/countdown/ has wonderful information about the Galileo mission to Jupiter which ended in September 2003, including fly bys of the moons of Jupiter. Other References: Wood, L. and Perrett, G. (1997) Advanced Mathematical Discourse, University of Technology Sydney. The chapter by Peter and Dubravka Petocz on pattern and proof is very helpful for teachers. Nelsen, R.B. (1993) Proof Without Words, The Mathematical Association of America, Washington DC page 115 has a nice visual explanation for the result explored in this activity on page 115. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 21) CHAPTER REVIEW (page 23) 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: rates are converted in this interactive program. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 4 Chapter 2. Algebra Text references CD reference Substrand Mathscape 9 Extension Chapter 2. Algebra (pages 25–77) Simplify (with fractional indices) Binomial Products Perfect squares Railway tickets Algebraic techniques Duration 3 weeks / 12 hours Key ideas Outcomes Simplify, expand and factorise algebraic expressions including those involving fractions or with negative and/or fractional indices. Use algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions. PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices. PAS5.3.1 (page 92): Uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions. NS5.3.1 (page 68): Performs operations with surds and 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? 24 x 2 y 16 xy2 8 xy(3x 2) (Reasoning, Communicating) develop facility with the algebraic symbol system in order to apply algebraic techniques to other strands and substrands (Applying Strategies, Communicating) use factorising techniques to solve quadratic equations and draw graphs of parabolas (Applying Strategies, Communicating) solve problems, such as: find a relationship that describes the number of diagonals in a polygon with n sides (Applying Strategies) prove some general properties of numbers such as - the sum of two odd integers is even - the product of an odd and even integer is even - the sum of 3 consecutive integers is divisible by 3 (Reasoning) Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 5 Knowledge and skills Teaching, learning and assessment Students learn about simplifying algebraic expressions involving fractions, such as 2 x 2 x 7 a 5a 2 y y 2ab 6 5 3 8 12 3 6 3 2b expanding, by removing grouping symbols, and collecting like terms where possible, algebraic expressions such as 2 y( y 5) 4( y 5) 4 x(3x 2) ( x 1) 3x 2 (5 x 2 2 xy) factorising, by determining common factors, algebraic expressions such as 3x 2 6 x 14 ab 12 a 2 21xy 3x 9 x 2 simplifying algebraic expressions, including those involving fractions, such as 11x 2 y 7 x 8 y 5 4(3 x 2) ( x 1) 2 7a 4a 3b b 3 2 x x 1 3 5 expanding binomial products by finding the area of rectangles x e.g. 8 hence x x 2 8x 3 3x 24 TRY THIS Flags (page 31): Algebraic problem solving Overhanging the overhang (page 38): Practical Railway Tickets (page 53): Complete a table and find a rule Proof (page 66): Challenge FOCUS ON WORKING MATHEMATICALLY A number pattern from Blaise Pascal 1654 (page 71): In contrast with the pure number pattern activity in chapter 1, this investigation provides a challenge for students to apply their new knowledge to solve a problem. To give them something to aim at I have given the specific answer for 19 flavours in question 1 and the general solution for n flavours in question 5. The web site http://mathforum.org/library/drmath/view/59185.html "Ask Dr Math" has a good explanation of the ice cream problem. The intention is to encourage students to have a go, make a guess, think intuitively, be Pascal like! Teachers are encouraged to use the Reflecting exercise on page 73 as a way of deepening student understanding of the individual preferences in which individual mathematicians like to work. A useful book reference is Kline, Morris (1972) Mathematical Thought from Ancient to Modern Times, Oxford University Press pages 295–7 for general information on Pascal's love of intuition and the contributions he made to mathematics. Go to http://ptri1.tripod.com/ for a good overview of the number patterns in Pascal's triangle. Note that this site is in the process of changing as the book goes to press. You may wish to make a connection between Pascal's triangle and the Sierpinski fractal, by colouring the odd numbers black and leaving the even numbers white. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 72) CHAPTER REVIEW (page 74) a collection of problems to revise the chapter. x 8x 3 x2 8x 3x 24 x 2 11x 24 Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 6 using algebraic methods to expand a variety of binomial products, such as ( x 2)( x 3) (2 y 1) 2 (3a 1)(3a 1) recognising and applying the special products (a b)(a b) a 2 b 2 (a b) 2 a 2 2ab b 2 factorising expressions: - common factors 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. Binomial Products: a worksheet using the executable “Expand”. Perfect Squares: shows students how squaring expressions in Algebra can be represented by drawing. To be used with page 63. Railway Tickets: worksheet to use with the “Try This” problem on page 58. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 7 Chapter 3. Consumer arithmetic Text references CD reference Substrand Mathscape 9 Extension Chapter 3. Consumer Arithmetic (pages 78–116) Money Consumer arithmetic Duration 2 weeks / 8 hours Key ideas Outcomes Solve simple consumer problems including those involving earning and spending money. Calculate simple interest and find compound interest using a calculator and tables of values. Use compound interest formula. Solve consumer arithmetic problems involving compound interest, depreciation and successive discounts. NS5.1.2 (page 70): Solves consumer arithmetic problems involving earning and spending money. NS5.2.2 (page 71): Solves 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 calculating earnings for various time periods from different sources, including: - wage - salary - commission TRY THIS Telephone Charges (page 101): Problem Solving Progressive Discounting (page 107): Investigation FOCUS ON WORKING MATHEMATICALLY (page 111) Sydney Market prices in 1831 Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 8 - piecework - overtime - bonuses - holiday loadings - interest on investments calculating income earned in casual and part-time jobs, considering agreed rates and special rates for Sundays and public holidays calculating weekly, fortnightly, monthly and yearly incomes calculating net earnings considering deductions such as taxation and superannuation calculating a ‘best buy’ calculating the result of successive discounts The purpose of the learning activities is for students to think about the cost of living in Australia today using market prices in 1831 as a starting point. Students are given opportunity to explore inflation and how the consumer price index (CPI) is calculated. An invitation to a member of the Economics staff to your class could be stimulating, and to get help from the web sites provided in the teacher support material. A good summary of how the CPI is calculated in Australia can be found at http://www.aph.gov.au/library/pubs/mesi/features/cpi.htm Teachers should note that the 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. This point is well made in the article by Nell Ingalls published on the web site http://www.sls.lib.il.us/reference/por/features/98/money.html This is a useful source of information on the value of money. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 112) CHAPTER REVIEW (page 114) 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. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 9 Chapter 4. Equations, inequations and formulae Text references CD reference Substrand Mathscape 9 Extension Chapter 4. Equations, inequations and formulae (pages 117–53) Evaluating Floodlighting Algebraic techniques Duration 2 weeks / 8 hours Key ideas Outcomes Solve linear and simple quadratic equations of the form ax c Solve linear inequalities Solves a range of linear equations 2 PAS5.2.2 (page 90): Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods. PAS5.3.2 (page 94): Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations. 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: Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 10 m y2 y1 x2 x1 1 2 mv 2 4 V r 3 3 SA 2r 2 2rh E (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) solve non-routine problems using algebraic techniques (Applying Strategies, Communicating) create equations to solve a variety of problems and check solutions (Communicating, Applying Strategies, Reasoning) explain why a particular value could not be a solution to an equation (Applying Strategies, Communicating, Reasoning) Knowledge and skills Teaching, learning and assessment Students learn about Linear and Quadratic Equations solving linear equations such as x x 5 2 3 2y 3 2 3 z 3 6 1 2 3(a 2) 2(a 5) 10 3(2t 5) 2t 5 3r 1 2r 4 4 5 solving word problems that result in equations TRY THIS A Prince and a King (page 1137): Two Ancient Problems Floodlighting by formula (page 143): Formulae Investigation FOCUS ON WORKING MATHEMATICALLY Splitting the Atom (page 149): In this activity students learn about the relationship between mass and energy, stated in the form of a famous equation. Teachers are advised to work through the example first noting the units for mass, speed and kinetic energy. It will be a good idea to complete the Challenge questions before using them in class. The increase in mass is small, just over a tenth of a milligram. This everyday implications of this should be discussed. In general the web links I have looked at are rather hard to read for students. The link http://www.phys.virginia.edu/classes/109N/lectures/mass_increase.html contains good background information for teachers. I recommend the following excellent book reference for students and teachers wanting to read further. Guillen, M. (2000) Five Equations That Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 11 exploring the number of solutions that satisfy simple quadratic equations of the form x 2 c solving simple quadratic equations of the form ax 2 c solving equations arising from substitution into formulae Changed the World, Abacus Books, London. pages 215–266. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 150) CHAPTER REVIEW (page 152) a collection of problems to revise the chapter. Linear Inequalities solving inequalities such as 3x 1 9 2(a 4) 24 t4 3 5 Linear, Quadratic and Simultaneous Equations using analytical and graphical methods to solve a range of linear equations, including equations that involve brackets and fractions such as 3(2a 6) 5 (a 2) 2x 5 x 7 0 3 5 y 1 2y 3 1 4 3 2 solving problems involving linear equations Technology Evaluating: students analyse a spreadsheet and then design their own. Floodlighting: activity to complement the “Try This” problem on page 143. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 12 Chapter 5. Measurement Text references CD reference Substrand Mathscape 9 Extension Chapter 5. Measurement (pages 155–210) Perigal Measuring plane shapes Circle measuring Algebraic techniques Duration 2 weeks / 8 hours Key ideas Outcomes Develop formulae and use to find the area of rhombuses, trapeziums and kites. Find the area and perimeter of simple composite figures consisting of two shapes including quadrants and semicircles. Find area and perimeter of more complex composite figures. MS5.1.1 (page 126): Use formulae to calculate the area of quadrilaterals and find areas and perimeters of simple composite figures. MS5.2.1 (page 127): Find areas and perimeters of 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 developing and using formulae to find the area of quadrilaterals: - for a kite or rhombus, Area 12 xy where x and y are the lengths of the diagonals; TRY THIS Pythagorean Proof by Perigal (page 170): Proof Command Module (page 180): Investigation of Apollo 11 The area of a circle (page 191): Archimedes method Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 13 for a trapezium, Area 12 h(a b) where h is the perpendicular height and a and b the lengths of the parallel sides calculating the area of simple composite figures consisting of two shapes including quadrants and semicircles calculating the perimeter of simple composite figures consisting of two shapes including quadrants and semicircles calculating the area and perimeter of sectors calculating the perimeter and area of composite figures by dissection into triangles, special quadrilaterals, semicircles and sectors - Area (page 200): Challenge Problem FOCUS ON WORKING MATHEMATICALLY The Solar System (page 203): For good pictures try http://www.the-solarsystem.net/ At http://www.nineplanets.org/ you can take a tour of the solar system and find out lots about each planet. Highly recommended. There are lots of interesting things to see at the NASA web site http://www.nasa.gov/ including the recent exploration of the red planet Mars. In August 2003 Mars was also at its closest point to Earth in 70 000 years. A drawing of the orbits of the two planets at their closest point will help students grasp the periodic nature of this phenomenon. On the web link http://www.exploratorium.edu/ronh/solar_system/ you can build a model of the solar system on the spot. Try it out. There is also a universe explorer called Celestia available for download from http://www.shatters.net/celestia/. Worth a look. Very powerful. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 205) CHAPTER REVIEW (page 206) a collection of problems to revise the chapter. Technology Pythagoras Theorem: students use the worksheet and the program to discover how to use and prove Pythagoras’ theorem. 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. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 14 Chapter 6. Data representation and analysis Text references CD reference Substrand Mathscape 9 Extension Chapter 6. Data representation and analysis (pages 211–55) Data analysis Cumulative analysis Data representation and analysis Duration 2 weeks / 8 hours Key ideas Outcomes Construct frequency tables for grouped data. Find mean and modal class for grouped data. Determine cumulative frequency. Find median using a cumulative frequency table or polygon DS5.1.1 (page 116): Groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs. 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 constructing a cumulative frequency table for ungrouped data constructing a cumulative frequency histogram and polygon (ogive) using a cumulative frequency polygon to find the median grouping data into class intervals constructing a frequency table for grouped data constructing a histogram for grouped data finding the mean using the class centre finding the modal class TRY THIS The English Language (page 236): Investigation FOCUS ON WORKING MATHEMATICALLY World Health (page 248): This investigation provides an opportunity for students to analyse two indicators of world public health and to apply their skills in Working mathematically. Teachers are encouraged to work through the activities first. The objective is to show how statistical evidence can play a role in arguing a case for the development of programs to support global health. There is an excellent opportunity for class discussion about the sort of data Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 15 governments need in order to make sensible policy decisions for global health. 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 CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 250) CHAPTER REVIEW (page 252) 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 Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 16 Chapter 7. Probability Text references CD reference Substrand Mathscape 9 Extension Chapter 7. Probability (pages 256–81) Probability Craps simulation Weighted dice Probability Duration 1 week / 4 hours Key ideas Outcomes Determine relative frequencies to estimate probabilities. Determine theoretical probabilities. NS5.1.3 (page 75): Determines relative frequencies and theoretical 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 repeating an experiment a number of times to determine the relative frequency of an event estimating the probability of an event from experimental data using relative frequencies expressing the probability of an event A given a finite number of equally likely outcomes as number of favourable outcomes P( A) = n where n is the total number of outcomes in the sample space TRY THIS Two-Up (page 266): Experiment The game of Craps (page 271): Simulation Winning Chances (page 275): Problem Solving FOCUS ON WORKING MATHEMATICALLY A Party Game: Roll a six and eat the chocolate (page 276): This activity applies probability concepts in a game which could actually be played in class with an adventurous teacher! It is designed for students to enjoy. Teachers should carry out the simulation in the Challenge questions first using the technology they wish to use in class. A spreadsheet demonstration in a lab or the use of a Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 17 using the formula to calculate probabilities for simple events simulating probability experiments using random number generators set of graphics calculators (see pages 266–8) would be appropriate. Note that the activity is a binomial experiment. Choose questions appropriate for your class. Some questions could be revisited at a later time in stage 5. A simple introduction to the Chevalier's famous gambling problem can be found at http: //www.ga.k12.pa.us/academics/us/Math/Geometry/stwk98/CASINO/History.htm For a good example of a high school student's project on why you should not gamble go to http://www.ga.k12.pa.us/academics/us/Math/Geometry/stwk98/CASINO /index.htm Barry Kissane's web page http://wwwstaff.murdoch.edu.au/%7Ekissane /graphicscalcs.htm is invaluable for CASIO users. The Maths Online web site at http://www.mathsonline.co.uk/nonmembers/resource /prob/ is a great help to teachers looking for lesson plans to simulate real life probability problems. Includes on line flash movies which will draw graphs directly 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. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 277) CHAPTER REVIEW (page 279) 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. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 18 Chapter 8. Surds Text reference CD reference Substrand Mathscape 9 Extension Chapter 8. Surds (pages 282–310) Simplify (with fractional indices) Iterative square root Real numbers Duration 2 weeks / 8 hours Key ideas Outcomes Define the system of real numbers distinguishing between rational and irrational numbers. Perform operations with surds. Use integers and fractions for index notation. Convert between surd and index form NS5.3.1 (page 68): Performs operations with surds and indices. Working mathematically Students learn to explain why all integers and recurring decimals are rational numbers (Communicating, Reasoning) explain why rational numbers can be expressed in decimal form (Communicating, Reasoning) demonstrate that not all real numbers are rational (Communicating, Applying Strategies, Reasoning) solve numerical problems involving surds and/or fractional indices (Applying Strategies) explain why a particular sentence is incorrect e.g. 3 5 8 (Communicating, Reasoning) Knowledge and skills Teaching, learning and assessment Students learn about defining a rational number: a of two integers where b ≠ 0. b distinguishing between rational and irrational numbers using a pair of compasses and a straight edge to construct simple rationals and surds on the number line A rational number is the ratio TRY THIS Greater Number (page 291): Problem Solving Imaginary Numbers (page 297): Introduction to i Exact Values (page 304): Challenge FOCUS ON WORKING MATHEMATICALLY Fibonacci Numbers and the Golden Mean (page 305): This activity is the study of a very important surd known as the golden mean. It appears in fictional writing such as the Dan Brown's The Da Vinci Code for example, which could be used to show that it not just mathematicians who find it interesting. The link with the work of Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 19 defining real numbers: Real numbers are represented by points on the number line. Irrational numbers are real numbers that are not rational. demonstrating that x is undefined for x < 0, x 0 for x = 0, and x is the positive square root of x when x 0 using the following results for x, y > 0: x 2 x xy x. y x y x x2 Leonardo da Vinci is of great interest. The web link http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html will give you lots of information about Fibonacci Numbers and Nature. The web link http://galaxy.cau.edu/tsmith/KW/golden.html is a good start to explore the Golden Number . The web link http://www-groups.dcs.st-andrews.ac.uk/~history /Mathematicians/Fibonacci.html provides information on the life of Fibonacci and his contribution to mathematics. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 307) CHAPTER REVIEW (page 309) a collection of problems to revise the chapter. y using the four operations of addition, subtraction, multiplication and division to simplify expressions involving surds expanding expressions involving surds such as 3 5 or 2 3 2 3 2 rationalising the denominators of surds of the form a b c d 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. Iterative Square Root: spreadsheet activity finds the square root by iteration. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 20 Chapter 9. Indices Text references CD reference Substrands Mathscape 9 Extension Chapter 9. Indices (pages 311–42) Simplify (with fractional indices) Rational numbers Algebraic Techniques Duration 2 weeks / 8 hours Key ideas Outcomes Define and use zero index and negative integral indices. Develop the index laws arithmetically. Use index notation for square and cube roots. Express numbers in scientific notation (positive and negative powers of 10) Apply the index laws to simplify algebraic expressions (positive integral indices only). Simplify, expand and factorise algebraic expressions including those involving fractions or with negative and/or fractional indices. NS5.1.1 (page 65): Applies index laws to simplify and evaluate arithmetic expressions and uses scientific notation to write large and small numbers. PAS5.1.1 (page 87): Applies the index laws to simplify algebraic expressions. PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions involving fractions and negative and 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) verify the index laws by using a calculator e.g. to compare the values of 5 2 1 , 5 2 and 5 (Reasoning) communicate and interpret technical information using scientific notation (Communicating) explain the difference between numerical expressions such as 2 10 4 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 x 0 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 a 3 a 2 a 6 is incorrect (Communicating, Reasoning) 2 Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 21 examine and discuss the difference between expressions such as 3a 2 5a and 3a 2 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 9 x5 3x5 3x a5 a7 a 2 2c 4 1 2c 4 (Applying Strategies, Reasoning, Communicating) explain the difference between particular pairs of algebraic expressions, such as x 2 and 2 x (Reasoning, Communicating) Knowledge and skills Teaching, learning and assessment Students learn about describing numbers written in index form using terms such as base, power, index, exponent evaluating numbers expressed as powers of positive whole numbers establishing the meaning of the zero index and negative indices e.g. by patterns 32 31 30 3 1 9 3 1 1 3 3 2 1 9 312 writing reciprocals of powers using negative indices 1 1 e.g. 3 4 4 81 3 translating numbers to index form (integral indices) and vice versa developing index laws arithmetically by expressing each term in expanded form e.g. 32 34 (3 3) (3 3 3 3) 32 4 36 3 3 3 3 3 35 32 35 2 33 3 3 32 4 3 3 3 3 3 3 3 3 324 38 TRY THIS Smallest to Largest (page 323): Problem Solving Digit Patterns (page 328): Investigation FOCUS ON WORKING MATHEMATICALLY Mathematics is at the heart of Science (page 338): The Powers of 10 web site http://www.powersof10.com/ should be explored before starting this Working mathematically activity. There are excellent pictures and ideas for creating absorbing lessons. The learning activities are suitable for students working in pairs. Calculators are recommended. In particular try the patterns section at http://www.powersof10.com/powers/patterns/patterns.html The ABC web site http://www.abc.net.au/science has a wealth of ideas to 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 question, send it in and listen to the answer on radio or online. There is also a news page which provides great ideas for lesson starters. Teachers are encouraged to liaise with science staff for further information and to invite them to the lesson. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 339) Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 22 using index laws to simplify expressions using index laws to define fractional indices for square and cube roots e.g. 9 2 2 1 9 and 9 2 9 , hence CHAPTER REVIEW (page 340) a collection of problems to revise the chapter. 1 9 92 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 e.g. 22 23 223 25 am an amn 25 22 25 2 23 am an amn 2 2 2 3 6 (a m ) n a mn using the index laws previously established for numbers to develop the index laws in algebraic form establishing that a 0 1 using the index laws e.g. a 3 a 3 a 3 3 a 0 and a3 a3 1 a 0 1 simplifying algebraic expressions that include index notation e.g. 5x0 3 8 2 x 2 3x3 6 x5 12 a 6 3a 2 4a 4 2m3 ( m 2 3) 2m5 6m3 applying the index laws to simplify expressions involving pronumerals Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program a 2 Page 23 establishing that using index laws to assist with the definition of the fractional index for square root 2 1 a and a 2 a 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 2 , a 3 3 , … a a a applying the index laws to simplify algebraic expressions such as given a a a a a a2 a 2 1 (3 y 2 )3 4b 5 8b 3 9 x 4 3x3 1 1 3 x 2 5 x 2 1 1 3 6 y 4 y 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. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 24 Chapter 10. Geometry Text references CD reference Substrand Mathscape 9 Extension Chapter 10. Geometry (pages 343–407) nPolygon Triangle angles Fermat point Exterior angle Euler line Properties of geometric figures Duration 2 weeks / 8 hours Key ideas Outcomes Establish sum of exterior angles result and sum of interior angles result for polygons. Identify similar triangles and describe their properties. Apply tests for congruent triangles. Use simple deductive reasoning in numerical and non-numerical problems. Verify the properties of special quadrilaterals using congruent triangles. SGS5.2.1 (page 157): Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon. SGS5.2.2 (page 158): Develops and applies results for proving that triangles are congruent or similar. 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) apply the properties of congruent and similar triangles to solve problems, justifying the results (Applying Strategies, Reasoning) apply simple deductive reasoning in solving numerical and non-numerical problems (Applying Strategies, Reasoning) Knowledge and skills Teaching, learning and assessment Students learn about applying the result for the interior angle sum of a triangle to find, by dissection, the interior angle sum of polygons with 4,5,6,7,8, … sides defining the exterior angle of a convex polygon establishing that the sum of the exterior angles of any convex polygon is 360 applying angle sum results to find unknown angles determining what information is needed to show that two triangles are congruent If three sides of one triangle are respectively equal to three sides of another TRY THIS The badge of the Pythagoreans (page 363): Historical Problem Five Shapes (page 374): Problem Solving How many diagonals in a polygon? (page 379): Investigation An Investigation of Triangles (page 360): Investigation Triangle Angles (page 392): Investigation FOCUS ON WORKING MATHEMATICALLY Does a triangle have a centre? (page 397): It is suggested that teachers do Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 25 triangle, then the two triangles are congruent (SSS). If two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, then the two triangles are congruent (SAS). If two angles and one side of one triangle are respectively equal to two angles and the matching side of another triangle, then the two triangles are congruent (AAS). If the hypotenuse and a second side of one right-angled triangle are respectively equal to the hypotenuse and a second side of another right-angled triangle, then the two triangles are congruent (RHS). applying the congruency tests to justify that two triangles are congruent applying the four triangle congruency tests in numerical exercises to find unknown sides and angles some lead up construction work with the circumcentre, orthocentre, centroid and incentre of a triangle before starting this activity. The site http://www.punahou.edu/acad/sanders/CenterTriangle.html has some lovely applications for the centroid, circumcentre and incentre, the teacher may wish to use one of these problems as a starter for the lesson. The Fermat point P is the solution to the optimisation problem which requires the sum of the distances from P to each vertex to be a minimum. A teaching suggestion is to find where to build a new electricity sub station from three isolated towns so that the cost of laying cable (related directly to the length required) to each town is a minimum. You could work from a map of western NSW for example. The site http://www2.evansville.edu/ck6/tcenters/ illustrates the classical and more recent "centres" in some detail which will excite the mathematical curiosity of students. Higgins, P.M (2002) Mathematics for the Imagination, Oxford University Press, pp 82-100 has excellent background material for this activity. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 400) CHAPTER REVIEW (page 402) a collection of problems to revise the chapter. Technology nPolygon: 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. Triangle Angles: this program models the drawing of isosceles triangles. Fermat Point: this program is designed to give students more experience in deductive reasoning in geometry. 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. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 26 Chapter 11. The linear function Text references CD reference Substrand Mathscape 9 Extension Chapter 11. The Linear Function (pages 408–39) Line equation Intersecting lines Co-ordinate geometry Duration 2 weeks / 8 hours Key ideas Use a diagram to determine midpoint, length and gradient of an interval joining two points on the number plane. Graph linear and simple non-linear relationships from equations. Outcomes PAS5.1.2 (page 97): Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple 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 Students learn about Teaching, learning and assessment Midpoint, Length and Gradient using the right-angled triangle drawn between two points on the number plane and the relationship rise gradient run to find the gradient of the interval joining two points TRY THIS Size 8 (page 417): Problem Solving Hanging around (page 427): Problem Solving Latitude and Temperature (page 433): Investigation FOCUS ON WORKING MATHEMATICALLY Television Advertising (page 433): A good reference text is Lowe, I. (1991) Mathematics at work: Modelling your world volume 1, Australian Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 27 determining whether a line has a positive or negative slope by following the line from left to right – if the line goes up it has a positive slope and if it goes down it has a negative slope finding the gradient of a straight line from the graph by drawing a right-angled triangle after joining two points on the line Graphs of Relationships constructing tables of values and using coordinates to graph vertical and horizontal lines such as 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 Academy of Mathematics, pages 393–5. Advanced students do not need to know the mathematical basis of correlation to appreciate that it simply measures goodness of fit. The correlation of x with y in the table is 0.83 which indicates strong linear relationship. Such statistics are valuable in interpretation of the linear model to the problem at hand. A good web site which lists indexes used for financial forecasting is http://www.neatideas.com/nasd.htm Teachers might like to discuss forecasting of the money market as a very important business investment strategy. For example TV news bulletins give stock indexes like the NASDAQ and Dow Jones, money rates every day. Correlation coefficients show how well trends in predicted values follow trends in actual values in the past. It is important for students to note that a strong correlation between two variables does not necessarily imply a causal relationship. It is simply the strength of the linear model which is indicated. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 435) CHAPTER REVIEW (page 436) a collection of problems to revise the chapter. Technology Line Equation: interactive program with accompanying worksheet. Intersecting Lines: interactive program with accompanying worksheet. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 28 Chapter 12. Trigonometry Text references CD reference Substrand Mathscape 9 Extension Chapter 12. Trigonometry (pages 440–87) Sine cosine SOHCAHTOA Trigonometry Duration 2 weeks / 8 hours Key ideas Outcomes Use trigonometry to find sides and angles in right-angled triangles. Solve problems involving angles of elevation and angles of depression from diagrams. Solve further trigonometry problems including those involving three-figure bearings. MS5.1.2 (page 139): Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression. MS5.2.3 (page 140): Applies trigonometry to solve problems including those involving bearings. 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) solve simple problems involving three-figure bearings (Applying Strategies, Communicating) interpret directions given as bearings (Communicating) Knowledge and skills Teaching, learning and assessment Students learn about Trigonometric Ratios of Acute Angles identifying the hypotenuse, adjacent and opposite sides with respect to a given angle in a right-angled triangle in any orientation labelling the side lengths of a right-angled triangle in relation to a given angle e.g. the side c is opposite angle C recognising that the ratio of matching sides in similar right-angled triangles is constant for equal angles TRY THIS Height to Base Ratio (page 448): Investigation Make a Hypsometer (page 460): Practical Pilot Instructions (page 470): Problem Solving The Sine Rule (page 478): Investigation FOCUS ON WORKING MATHEMATICALLY Finding your latitude from the sun (page 479): This is designed as a fun outdoor activity. Teachers need to prepare well in advance and study the Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 29 defining the sine, cosine and tangent ratios for angles in right-angled triangles using trigonometric notation e.g. sin A using a calculator to find approximations of the trigonometric ratios of a given angle measured in degrees using a calculator to find an angle correct to the nearest degree, given one of the trigonometric ratios of the angle Trigonometry of Right-Angled Triangles selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown sides, including the hypotenuse selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest degree identifying angles of elevation and depression solving problems involving angles of elevation and depression when given a diagram using three-figure bearings (e.g. 035º, 225º) and compass bearings e.g. SSW drawing diagrams and using them to solve word problems which involve bearings or angles of elevation and depression diagrams carefully. The activity can be carried out on any sunny day and an adjustment to the observed angle made for the number of days since the last equinox. The geometry should be discussed carefully in class before the outdoor measurement s are taken. See what a sailor does to determine latitude using an astrolabe at http://www.ruf.rice.edu/~feegi/measure.html A great site to look at navigation in the 15th century is http://www.ruf.rice.edu/~feegi/site_map.html Read about advances in navigational technology from the Astrolabe to today's Global Positioning System at http: //www.canadiangeographic.ca/Magazine/ND01/findingourway.html CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 461) CHAPTER REVIEW (page 484) a collection of problems to revise the chapter. Technology Sine Cosine: explores the range of Trig graphs. SOHCAHTOA: investigation of the tan ratio. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 30 Chapter 13. Simultaneous equations Text references CD reference Substrand Mathscape 9 Extension Chapter 13. Simultaneous Equations (pages 488–510) Intersecting lines Porous rocks Algebraic techniques Duration 1 week / 4 hours Key ideas Outcomes Solve simultaneous equations using graphical and analytical methods for simple examples. PAS5.2.2 (page 90): Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods Working mathematically Students learn to use graphics calculators and spreadsheet software to plot pairs of lines and read off the point of intersection (Applying Strategies) solve linear simultaneous equations resulting from problems and interpret the results (Applying Strategies, Communicating) Knowledge and skills Teaching, learning and assessment Students learn about Simultaneous Equations solving simultaneous equations using non-algebraic methods, such as ‘guess and check’, setting up tables of values or looking for patterns solving linear simultaneous equations by finding the point of intersection of their graphs solving simple linear simultaneous equations using an analytical method e.g. solve the following 3a b 17 2a b 8 generating simultaneous equations from simple word problems TRY THIS Find the Values (page 498): Problem Solving A Pythagorean Problem (page 502): Problem Solving FOCUS ON WORKING MATHEMATICALLY Exploring for water, oil and gas—The density of air-filled porous rock (page 506): This activity is an application of simultaneous equations in earth science. It has been adapted for school use. Teachers will note that the equations to be solved are not linear. They can be made so by setting x = 1/V in equations (1) and (2) on page 507. Students can then solve for D and x and finally D and V. The graphs are straight lines and the solution easily verified. The web link http://www.earthsci.ucl.ac.uk/undergrad/geomaths/rev/simnb/sim9.htm sets out the context of the porous rocks problem clearly but with of course different symbolism. This example comes from a set of revision exercises in undergraduate geomaths at the Dept of Earth Sciences at University College London. The introduction to simultaneous equations is found at Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 31 http://www.earthsci.ucl.ac.uk/undergrad/geomaths/rev/simnb/MHsimnb.htm The web link http://www.blackgold.ab.ca/leduc1/blkgld1.htm discusses the particular case of oil in porous rocks. On Mars today, low temperature and pressure limit the stability of liquid water. The effects of pore sizes and atmospheric pressure on liquid water in rocks were studied to examine the possibility of liquid water existing on Mars inside rocks and in pore spaces in the soil. Read about it at http://www.asgsb.org/programs/2000/24.html CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 507) CHAPTER REVIEW (page 509) A collection of problems to revise the chapter. Technology Intersecting Lines: interactive program with accompanying worksheet. Porous Rocks: worksheet to accompany the Focus on Working mathematically section. Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 32 Chapter 14. Co-ordinate geometry Text references CD reference Substrand Mathscape 9 Extension Chapter 14. Co-ordinate Geometry (pages 511–62) Intersecting lines Crow flying Co-ordinate geometry Duration 2 weeks / 8 hours Key ideas Outcomes Use and apply various standard forms of the equation of a straight line, and graph regions on the number plane. PAS5.3.3 (page 102): Uses various standard forms of the equation of a straight line and graphs regions on the number plane. Working mathematically Students learn to recognise from a list of equations those that result in straight line graphs (Communicating) describe conditions for lines to be parallel or perpendicular (Reasoning, Communicating) show that if two lines are perpendicular then the product of their gradients is -1 (Applying Strategies, Reasoning, Communicating) discuss the equations of graphs that can be mapped onto each other by a translation or by reflection in the y-axis e.g. consider the graphs y 2 x, y 2 x, y 2 x 1 and describe the transformation that would map one graph onto the other (Communicating) describe the conditions for a line to have a negative gradient (Reasoning, Communicating) prove that a particular triangle drawn on the number plane is right-angled (Applying Strategies, Reasoning) use a graphics calculator and spreadsheet software to graph, compare and describe a range of linear relationships (Applying Strategies, Communicating) apply ethical considerations when using hardware and software (Reflecting) find areas of shapes enclosed within a set of lines on the number plane e.g. find the area of the triangle enclosed by the lines y = 0, y = 2x, x + y = 6 (Applying Strategies) describe a region from a graph by identifying the boundary lines and determining the appropriate inequalities for describing the enclosed region (Applying Strategies, Communicating) derive the formula for the distance between two points (Applying Strategies, Reasoning) show that two intervals with equal gradients and a common point form a straight line and use this to show that three points are collinear (Applying Strategies, Reasoning) use coordinate geometry to investigate and describe the properties of triangles and quadrilaterals (Applying Strategies, Reasoning, Communicating) Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 33 use coordinate geometry to investigate the intersection of the perpendicular bisectors of the sides of acute-angled triangles (Applying Strategies, Reasoning, Communicating) show that four specified points form the vertices of particular quadrilaterals (Applying Strategies, Reasoning, Communicating) Knowledge and skills Students learn about Teaching, learning and assessment Midpoint, Length and Gradient determining the midpoint of an interval from a diagram graphing two points to form an interval on the number plane and forming a right-angled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point using the right-angled triangle drawn between two points on the number plane and Pythagoras’ theorem to determine the length of the interval joining the two points Midpoint, Distance and Gradient Formulae using the average concept to establish the formula for the midpoint, M, of the interval joining two points x1 , y1 and x2 , y2 on the number plane x x y y2 M ( x, y) 1 2 , 1 2 2 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 TRY THIS A line with no integer co-ordinates (page 525): Investigation Car Hire (page 536): Problem Solving Temperature Rising (page 540): Problem Solving FOCUS ON WORKING MATHEMATICALLY Finding the gradient of a ski run (page 554): The resource book Kleeman, G. and Peters A. (2002) Skills in Australian Geography, Cambridge University Press, Cambridge is your best guide for this activity. Try the Social Science department for a copy or your school library. For a good model of calculating gradient from contour maps go to 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. A good site written for scouts which looks at gradients, contours and features of ordinance 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) hemisphere. This means the sun will shine on the eastern and northern slopes during the day. Hence the preference for these slopes. Just what we need to enjoy skiing. CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 557) CHAPTER REVIEW (page 559) a collection of problems to revise the chapter. d ( x2 x1 )2 ( y2 y1 )2 using the formula to find the distance between two points on the number plane Published by Macmillan Education Australia. © Macmillan Education Australia 2004. Mathscape 9 Extension Teaching Program Page 34 using the relationship rise run to establish the formula for the gradient, m, of an interval joining two points x1 , y1 and x2 , y2 on the number plane gradient m y2 y1 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. Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
