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Mathscape 9: Working Mathematically March 2004 web update Mathscape 9 Syllabus Correlation Grid (Stage 5.1/5.2) Highlight indicates Stage 4 review Text Reference Chapter 1 Rational numbers 1.1 Significant figures 1.2 The calculator 1.3 Estimation Try this: Fermi problem 1.4 Recurring decimals 1.5 Rates Try this: Desert walk 1.6 Solving problems with rates Try this: Passing trains Focus on working mathematically: A number pattern from Galileo 1615 Language link with Macquarie Chapter review Mathscape 9 and Mathscape 9 Extension School CD ROM Substrand Outcome Rational Numbers NS5.2.1 Key Ideas Round numbers to a specified number of significant figures Express recurring decimals as fractions Convert rates from one set of units to another Knowledge and Skills identifying significant figures rounding numbers to a specified number of significant figures using the language of estimation appropriately, including: recognise that calculators show approximations to recurring decimals e.g. 23 displayed as 0.666667 (Communicating) . – rounding justify that 0.9 1 (Reasoning) – approximate – level of accuracy 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 . Working Mathematically . . . 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 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) 1 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 2 Algebra 2.1 Describing simple patterns 2.2 Finite differences Try this: Flags 2.3 Substitution 2.4 Adding and subtracting algebraic expressions 2.5 Multiplying and dividing algebraic expressions Try this: Overhanging the overhang 2.6 The order of operations 2.7 The distributive law 2.8 The highest common factor Try this: Proof 2.9 Adding and subtracting algebraic fractions 2.10 Multiplying and dividing algebraic fractions 2.11 Generalised arithmetic Try this: Railway tickets Focus of working mathematically: Party magic Language link with Macquarie Chapter review March 2004 web update Substrand Outcome Algebraic Techniques PAS5.2.1 Key Ideas Simplify, expand and factorise algebraic expressions including those involving fractions Knowledge and Skills simplifying algebraic expressions involving fractions, such as 2x 2x 5 3 7 a 5a 8 12 2y y 3 6 2ab 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(3 x 2) ( x 1) 3 x 2 (5 x 2 2 xy ) factorising, by determining common factors, algebraic expressions such as 3x 6 x 2 14ab 12a 2 21xy 3x 9 x 2 Working Mathematically 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 xy 2 8xy (3x 2) Mathscape 9 and Mathscape 9 Extension School CD ROM (Reasoning, Communicating) 2 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 3 Consumer arithmetic 3.1 Salaries and wages Try this: Sue’s boutique 3.2 Other methods of payment 3.3 Overtime and other payments 3.4 Wage deductions 3.5 Taxation 3.6 Budgeting Try this: Telephone charges 3.7 Best buys 3.8 Discounts Try this: Progressive discounting 3.9 Profit and loss Focus on working mathematically: Sydney market prices in 1831 Language link with Macquarie Chapter review March 2004 web update Substrand Outcome Consumer Arithmetic NS5.1.2 Key Ideas 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 Knowledge and Skills calculating earnings for various time periods from different sources, including: – wage – salary – commission – 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 Working Mathematically 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) calculating net earnings considering deductions such as taxation and superannuation Mathscape 9 and Mathscape 9 Extension School CD ROM calculating a ‘best buy’ NS5.2.2 Solve consumer arithmetic problems and successive discounts calculating the result of successive discounts 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) 3 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 4 Equations, inequations and formulae 4.1 Inverse operations 4.2 One- and two-step equations 4.3 Equations with pronumerals on both sides 4.4 Equations with grouping symbols 4.5 Equations with one fraction 4.6 Equations with more than one fraction 4.7 Inequations 4.8 Solving worded problems Try this: A prince and a king 4.9 Evaluating the subject of a formula Try this: Arm strength 4.10 Equations arising from substitution Try this: Floodlighting by formula Focus on working mathematically: Bushfires Language link with Macquarie Chapter review Mathscape 9 and Mathscape 9 Extension School CD ROM March 2004 web update Substrand Outcome Algebraic Techniques PAS5.2.2 Key Ideas Solve linear and simple quadratic equations of the form ax 2 c Solve linear inequalities Knowledge and Skills 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 exploring the number of solutions that satisfy simple quadratic equations of the form x 2 c solving simple quadratic equations of the form ax2 c solving equations arising from substitution into formulae Linear Inequalities solving inequalities such as 3x 1 9 2( a 4) 24 t4 3 5 Working Mathematically 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) 4 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update 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 2 y1 x2 x1 1 2 E mv 2 4 V r 3 3 SA 2r 2 2rh m (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) 5 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 5 Measurement 5.1 Length, mass and capacity Try this: Bag of potatoes 5.2 Accuracy and precision 5.3 Time Try this: Overseas call 5.4 Pythagoras’ theorem Try this: Pythagorean proof by Perigal 5.5 Solving problems with Pythagoras’ theorem Try this: The box and the wall 5.6 Perimeter 5.7 Circumference Try this: Command module 5.8 Converting units of area 5.9 Calculating area Try this: The area of a circle 5.10 Area of a circle 5.11 Composite areas Try this: Area 5.12 Problem involving area Focus on working mathematically: The Melbourne Cup Language link with Macquarie Chapter review March 2004 web update Substrand Outcome Algebraic Techniques MS5.1.1 Key Ideas Uses formulae to calculate the area of quadrilaterals and finds areas and perimeters of simple composite figures Knowledge and Skills Working Mathematically developing and using formulae to find the area of quadrilaterals: – for a kite or rhombus, Area 1 xy 2 where x and y are the lengths of the diagonals; – for a trapezium, Area 1 2 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 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) calculating the perimeter of simple composite figures consisting of two shapes including quadrants and semicircles Mathscape 9 and Mathscape 9 Extension School CD ROM 6 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update MS5.2.1 Finds areas and perimeters of composite figures calculating the area and perimeter of sectors calculating the perimeter and area of composite figures by dissection into triangles, special quadrilaterals, semicircles and sectors 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) 7 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update Text Reference Substrand Chapter 6 Data representation and analysis 6.1 Graphs 6.2 Organising data 6.3 Analysing data 6.4 Problems involving the mean Try this: The English language 6.5 Cumulative frequency 6.6 Grouped data Try this: Earthquakes Focus on working mathematically: World health Language link with Macquarie Chapter review Data Representati on and Analysis Outcome DS5.1.1 Key Ideas Construct frequency tables for grouped data Find mean and modal class for grouped data Knowledge and Skills 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 Determine cumulative frequency grouping data into class intervals Find median using a cumulative frequency table or polygon constructing a histogram for grouped data constructing a frequency table for grouped data finding the mean using the class centre finding the modal class Mathscape 9 and Mathscape 9 Extension School CD ROM Working Mathematically 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) 8 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 7 Probability 7.1 Probability and its language 7.2 Experimental probability Try this: Two-up 7.3 Computer simulations Try this: The game of craps 7.4 Theoretical probability Try this: Winning chances Focus on working mathematically: Getting through traffic lights Language link with Macquarie Chapter review Mathscape 9 and Mathscape 9 Extension School CD ROM March 2004 web update Substrand Probability Outcome NS5.1.3 Key Ideas Knowledge and Skills Determine relative frequencies to estimate probabilities repeating an experiment a number of times to determine the relative frequency of an event Determine theoretical probabilities 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 P( A) = number of favourable outcomes n where n is the total number of outcomes in the sample space using the formula to calculate probabilities for simple events simulating probability experiments using random number generators Working Mathematically 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) 9 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 8 Indices 8.1 Index notation Try this: Power pulse graphs 8.2 Simplifying numerical expressions using the index laws 8.3 The index law for multiplication 8.4 The index law for division 8.5 The index law for further powers 8.6 Miscellaneous questions on the index laws 8.7 The zero index Try this: Smallest to largest 8.8 The negative index 8.9 Products and quotients with negative indices Try this: Digit patterns 8.10 The fraction index 8.11 Scientific notation 8.12 Scientific notation on the calculator Focus on working mathematically: Mathematics is the heart of science Language link with Macquarie Chapter review Mathscape 9 and Mathscape 9 Extension School CD ROM March 2004 web update Substrand Rational Numbers Outcome NS5.1.1 Key Ideas Knowledge and Skills Working Mathematically Define and use zero index and negative integral indices describing numbers written in index form using terms such as base, power, index, exponent solve numerical problems involving indices (Applying Strategies) Develop the index laws arithmetically evaluating numbers expressed as powers of positive whole numbers Use index notation for square and cube roots establishing the meaning of the zero index and negative indices e.g. by patterns explain the incorrect use of index laws e.g. why 32 34 96 (Communicating, Reasoning) Express numbers in scientific notation (positive and negative powers of 10) 32 31 30 3 1 9 3 1 1 3 verify the index laws by using a calculator e.g. to compare the values of 32 1 9 1 32 writing reciprocals of powers using negative indices 1 1 e.g. 34 4 81 3 translating numbers to index form (integral indices) and vice versa 5 2 2 1 , 5 2 and 5 (Reasoning) communicate and interpret technical information using scientific notation (Communicating) developing index laws arithmetically by expressing each term in expanded form e.g. explain the difference between numerical expressions such as 2104 and 2 4 , particularly with reference to calculator displays (Communicating, Reasoning) 32 34 (3 3) (3 3 3 3) 324 36 3 3 3 3 3 35 32 352 33 3 3 solve problems involving scientific notation (Applying Strategies) 3 3 3 3 3 3 3 3 3 3 2 4 24 38 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 1 9 92 10 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 8 Indices 8.1 Index notation Try this: Power pulse graphs 8.2 Simplifying numerical expressions using the index laws 8.3 The index law for multiplication 8.4 The index law for division 8.5 The index law for further powers 8.6 Miscellaneous questions on the index laws 8.7 The zero index Try this: Smallest to largest 8.8 The negative index 8.9 Products and quotients with negative indices Try this: Digit patterns 8.10 The fraction index 8.11 Scientific notation 8.12 Scientific notation on the calculator Focus on working mathematically: Mathematics is the heart of science Language link with Macquarie Chapter review March 2004 web update Substrand Rational Numbers Outcome NS5.1.1 Key Ideas Knowledge and Skills Working Mathematically writing square roots and cube roots in index form 1 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 Mathscape 9 and Mathscape 9 Extension School CD ROM 11 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 8 Indices 8.1 Index notation Try this: Power pulse graphs 8.2 Simplifying numerical expressions using the index laws 8.3 The index law for multiplication 8.4 The index law for division 8.5 The index law for further powers 8.6 Miscellaneous questions on the index laws 8.7 The zero index Try this: Smallest to largest 8.8 The negative index 8.9 Products and quotients with negative indices Try this: Digit patterns 8.10 The fraction index 8.11 Scientific notation 8.12 Scientific notation on the calculator Focus on working mathematically: Mathematics is the heart of science Language link with Macquarie Chapter review March 2004 web update Substrand Outcome Algebraic Techniques PAS5.1.1 Key Ideas Apply the index laws to simplify algebraic expressions (positive integral indices only) Knowledge and Skills Working Mathematically using the index laws previously established for numbers to develop the index laws in algebraic form verify the index laws using a calculator e.g. use a calculator to compare the values of (34 ) 2 e.g. 2 2 2 3 2 2 3 2 5 a m a n a mn 25 2 2 252 23 a m a n a mn 2 2 (a m ) n a mn 2 3 6 establishing that a0 1 using the index laws a3 a3 a33 a0 e.g. and a3 a3 1 a0 1 simplifying algebraic expressions that include index notation e.g. 5x 0 3 8 2 x 2 3x 3 6 x 5 12a 6 3a 2 4a 4 2m 3 (m 2 3) 2m 5 6m 3 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 a 3 a 2 a 6 is incorrect (Communicating, Reasoning) 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) Mathscape 9 and Mathscape 9 Extension School CD ROM 12 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update Text Reference Substrand Outcome Chapter 8 Indices 8.1 Index notation Try this: Power pulse graphs 8.2 Simplifying numerical expressions using the index laws 8.3 The index law for multiplication 8.4 The index law for division 8.5 The index law for further powers 8.6 Miscellaneous questions on the index laws 8.7 The zero index Try this: Smallest to largest 8.8 The negative index 8.9 Products and quotients with negative indices Try this: Digit patterns 8.10 The fraction index 8.11 Scientific notation 8.12 Scientific notation on the calculator Focus on working mathematically: Mathematics is the heart of science Language link with Macquarie Chapter review Algebraic Techniques PAS5.2.1 Key Ideas Apply the index laws to simplify algebraic expressions involving negative and fractional indices Knowledge and Skills applying the index laws to simplify expressions involving pronumerals establishing that a 2 a a a a a2 a using index laws to assist with the definition of the fractional index for square root given a a 2 2 a 12 a and Working Mathematically 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? 1 a a2 then 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 a 1 1 1 1 , a 2 2 , a 3 3 , … a a a applying the index laws to simplify algebraic expressions such as (3 y 2 ) 3 4b 5 8b 3 9 x 4 3x 3 1 1 3 x 2 5 x 2 1 5x 0 1 9 x 5 3x 5 3x a5 a7 a 2 1 2c 4 4 2c (Applying Strategies, Reasoning, Communicating) explain the difference between particular pairs of algebraic expressions, such as x 2 and 2 x (Reasoning, Communicating) 1 6 y 3 4 y 3 Mathscape 9 and Mathscape 9 Extension School CD ROM 13 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update Text Reference Substrand Chapter 9 Geometry 9.1 Angles 9.2 Parallel lines 9.3 Triangles Try this: The badge of the Pythagoreans 9.4 Angle sum of a quadrilateral 9.5 Special quadrilaterals Try this: Five shapes 9.6 Polygons Try this: How many diagonals in a polygon? Focus on working mathematically: A surprising finding Language link with Macquarie Chapter review Properties of Geometrical Figures Outcome SGS5.2.1 Key Ideas Establish sum of exterior angles result and sum of interior angles result for polygons Knowledge and Skills Working Mathematically 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 express in algebraic terms the interior angle sum of a polygon with n sides e.g. (n–2) 180 (Communicating) defining the exterior angle of a convex polygon find the size of the interior and exterior angles of regular polygons with 5,6,7,8, … sides (Applying Strategies) establishing that the sum of the exterior angles of any convex polygon is 360 applying angle sum results to find unknown angles solve problems using angle sum of polygon results (Applying Strategies) Mathscape 9 and Mathscape 9 Extension School CD ROM 14 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update Text Reference Substrand Outcome Chapter 10 The linear function 10.1 The number plane 10.2 Graphing straight lines Try this: Size 8 10.3 Horizontal and vertical lines 10.4 Gradient of a line Try this: Hanging around 10.5 The linear equation Try this: Latitude and temperature Mathscape 9 prelims page iv Tuesday 10.6 The intersection of straight lines Focus on working mathematically: Paper sizes in the printing industry Language link with Macquarie Chapter review Coordinate Geometry PAS5.1.2 Key Ideas Graph linear and simple nonlinear relationships from equations Knowledge and Skills 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 Working Mathematically 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) 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 Mathscape 9 and Mathscape 9 Extension School CD ROM 15 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 10 The linear function 10.1 The number plane 10.2 Graphing straight lines Try this: Size 8 10.3 Horizontal and vertical lines 10.4 Gradient of a line Try this: Hanging around 10.5 The linear equation Try this: Latitude and temperature Mathscape 9 prelims page iv Tuesday 10.6 The intersection of straight lines Focus on working mathematically: Paper sizes in the printing industry Language link with Macquarie Chapter review Mathscape 9 and Mathscape 9 Extension School CD ROM March 2004 web update Substrand Outcome PAS5.1.2 Key Ideas Graph linear and simple non-linear relationships from equations Knowledge and Skills 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 Working Mathematically 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) determining whether a point lies on a line by substituting into the equation of the line 16 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update Text Reference Substrand Chapter 11 Trigonometry 11.1 Side ratios in right-angled triangles 11.2 The trigonometric ratios Try this: Height to base ratio 11.3 Trigonometric ratios using a calculator 11.4 Finding the length of a side 11.5 Problems involving finding sides Try this: Make a hypsometer 11.6 Finding the size of an angle 11.7 Problems involving finding angles 11.8 Angles of elevation and depression Try this: Pilot instructions Focus on working mathematically: Finding your latitude from the sun Language link with Macquarie Chapter review Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression Mathscape 9 and Mathscape 9 Extension School CD ROM Outcome MS5.1.2 Key Ideas Use trigonometry to find sides and angles in right-angled triangles Solve problems involving angles of elevation and angles of depression from diagrams Knowledge and Skills 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 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 Working Mathematically 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) 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 17 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 12 Co-ordinate geometry 12.1 The distance between two points 12.2 The distance formula 12.3 The midpoint of an interval 12.4 The gradient formula 12.5 General form of the equation of a line Try this: Car hire 12.6 Parallel lines Try this: Temperature rising Focus on working mathematically: Finding the gradient of a ski run Language link with Macquarie Chapter review March 2004 web update Substrand Coordinate Geometry Outcome PAS5.1.2 Key Ideas Use a diagram to determine midpoint, length and gradient of an interval joining two points on the number plane Knowledge and Skills 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 Working Mathematically 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) Mathscape 9 and Mathscape 9 Extension School CD ROM 18 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically Text Reference Chapter 12 Co-ordinate geometry 12.1 The distance between two points 12.2 The distance formula 12.3 The midpoint of an interval 12.4 The gradient formula 12.5 General form of the equation of a line Try this: Car hire 12.6 Parallel lines Try this: Temperature rising Focus on working mathematically: Finding the gradient of a ski run Language link with Macquarie Chapter review Mathscape 9 and Mathscape 9 Extension School CD ROM March 2004 web update Substrand Outcome Coordinate Geometry PAS5.2.3 Key Ideas Use midpoint, distance and gradient formulae Knowledge and Skills 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 Working Mathematically 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 lefthand column d ( x2 x1 ) 2 ( y2 y1 ) 2 d ( x1 x2 ) 2 ( y1 y2 ) 2 using the formula to find the distance between two points on the number plane and using the relationship m rise run to establish the formula for the gradient, m, of an interval joining two points x1, y1 and x2 , y2 on the gradient y1 y2 x1 x2 (Reasoning, Communicating) number plane m y2 y1 x2 x1 using the formula to find the gradient of an interval joining two points on the number plane 19 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia. Mathscape 9: Working Mathematically March 2004 web update 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 20 Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004. Published by Macmillan Education Australia.