Mathscape 9 Extension Syllabus Correlation Grid (Stage 5.1/5.2/5.3)

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1 March 004 web update Mathscape 9 Extension Syllabus Correlation Grid (Stage 5./5./5.) Highlight indicates Stage 4 review Chapter Rational numbers. Significant figures. The calculator. Estimation Try this: Fermi problem.4 Recurring decimals.5 Rates Try this: Desert walk.6 Solving problems with rates Try this: Passing trains Focus on working mathematically: A number pattern from Galileo 65 Mathscape 9 and Mathscape 9 Extension School CD ROM Rational Numbers NS5.. Round numbers to a specified number of significant figures Express recurring decimals as fractions Convert rates from one set of units to another identifying significant figures rounding numbers to a specified number of significant figures using the language of estimation appropriately, including: rounding 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 noncalculator methods.... e.g. 0., 0., 0. 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 recognise that calculators show approximations to recurring decimals e.g. displayed as (Communicating) justify that 0.9. = (Reasoning) decide on an appropriate level of accuracy for results of calculations 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 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

2 March 004 web update Chapter Algebra. Describing simple patterns Try this: Flags. Substitution. Adding and subtracting algebraic expressions.4 Multiplying and dividing algebraic expressions Try this: Overhanging the overhang.5 The order of operations.6 The distributive law.7 The highest common factor.8 Adding and subtracting algebraic fractions.9 Multiplying and dividing algebraic fractions.0 Generalised arithmetic Try this: Railway tickets. Properties of numbers. Generalising solutions to problems using patterns. Binomial products.4 Perfect squares Try this: Proof.5 Difference of two squares.6 Miscellaneous expansions Focus of working mathematically: A number pattern from Blaise Pascal 654 Mathscape 9 and Mathscape 9 Extension School CD ROM Algebraic Techniques PAS5.. Simplify, expand and factorise algebraic expressions including those involving fractions simplifying algebraic expressions involving fractions, such as x x + 5 7a 5a 8 y y 6 ab 6 b expanding, by removing grouping symbols, and collecting like terms where possible, algebraic expressions such as y( y 5) + 4( y 5) 4x(x + ) ( x ) x (5x + xy) factorising, by determining common factors, algebraic expressions such as x 6x 4ab + a xy x + 9x 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 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? 4x y + 6xy = 8xy(x ) (Reasoning, Communicating) Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

3 March 004 web update Chapter Algebra. Describing simple patterns Try this: Flags. Substitution. Adding and subtracting algebraic expressions.4 Multiplying and dividing algebraic expressions Try this: Overhanging the overhang.5 The order of operations.6 The distributive law.7 The highest common factor.8 Adding and subtracting algebraic fractions.9 Multiplying and dividing algebraic fractions.0 Generalised arithmetic Try this: Railway tickets. Properties of numbers. Generalising solutions to problems using patterns. Binomial products.4 Perfect squares Try this: Proof.5 Difference of two squares.6 Miscellaneous expansions Focus of working mathematically: A number pattern from Blaise Pascal 654 Mathscape 9 and Mathscape 9 Extension School CD ROM Algebraic Techniques PAS5.. Use algebraic techniques to simplify expressions, expand binomial products simplifying algebraic expressions, including those involving fractions, such as x + y + 7x 8y + 5 4(x + ) ( x ) 7a 4a b + b x x + 5 expanding binomial products by finding the area of rectangles e.g. x 8 x x 8x x 4 hence ( x + 8)( x + ) = x + 8x + x + 4 = x + x + 4 using algebraic methods to expand a variety of binomial products, such as ( x + )( x ) (y + ) (a )(a + ) recognising and applying the special products (a +b)(a b) =a b (a ± b) = a ± ab +b factorising expressions: common factors describe relationships between the algebraic symbol system and number properties (Reflecting, Communicating) develop facility with the algebraic symbol system in order to apply algebraic techniques to other strands and substrands (Applying Strategies, Communicating) link algebra with generalised arithmetic (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 check expansions and factorisations by performing the reverse process (Reasoning) interpret statements involving algebraic symbols in other contexts e.g. spreadsheets (Communicating) solve problems, such as: find a relationship that describes the number of diagonals in a polygon with n sides Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

4 March 004 web update NS5.. 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 consecutive integers is divisible by (Reasoning) 4 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

5 March 004 web update Chapter Consumer arithmetic. Salaries and wages. Other methods of payment. Overtime and other payments.4 Wage deductions.5 Taxation.6 Budgeting Try this: Telephone charges.7 Best buys.8 Discounts Try this: Progressive discounting.9 Profit and loss Focus on working mathematically: Sydney market prices in 8 Mathscape 9 and Mathscape 9 Extension School CD ROM Consumer Arithmetic NS5.. 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 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 calculating net earnings considering deductions such as taxation and superannuation 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 interpret the different ways of indicating wages or salary in newspaper positions vacant advertisements e.g. $0K (Communicating) compare employment conditions for different careers where information is gathered from a variety of mediums including the Internet e.g. employment rates, payment calculating a best buy NS5.. Solve consumer arithmetic problems and successive discounts calculating the result of successive discounts explain why, for example, a discount of 0% following a discount of 5% is not the same as a discount of 5% (Applying Strategies, Communicating, Reasoning) 5 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

6 March 004 web update Chapter 4 Equations, inequations and formulae 4. One- and two-step equations 4. Equations with pronumerals on both sides 4. Equations with grouping symbols 4.4 Equations with one fraction 4.5 Equations with more than one fraction 4.6 Inequations 4.7 Solving worded problems Try this: A prince and a king 4.8 Evaluating the subject of a formula 4.9 Equations arising from substitution Try this: Floodlighting by formula 4.0 Changing the subject of a formula Focus on working mathematically: Splitting the atom Mathscape 9 and Mathscape 9 Extension School CD ROM Algebraic Techniques PAS5.. Solve linear and simple quadratic equations of the form ax = c Solve linear inequalities Linear and Quadratic Equations solving linear equations such as x x + = 5 y = z + 6 = ( a + ) + ( a 5) = 0 (t 5) = t + 5 r + r 4 = 4 5 solving word problems that result in equations exploring the number of solutions that satisfy simple quadratic equations of the form x = c solving simple quadratic equations of the form ax = c solving equations arising from substitution into formulae Linear Inequalities solving inequalities such as x < 9 ( a + 4) 4 t + 4 > 5 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) 6 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

7 March 004 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 y m = x x E = mv 4 V = πr SA= πr + πrh (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) 7 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

8 March 004 web update Chapter 4 Equations, inequations and formulae 4. One- and two-step equations 4. Equations with pronumerals on both sides 4. Equations with grouping symbols 4.4 Equations with one fraction 4.5 Equations with more than one fraction 4.6 Inequations 4.7 Solving worded problems Try this: A prince and a king 4.8 Evaluating the subject of a formula 4.9 Equations arising from substitution Try this: Floodlighting by formula 4.0 Changing the subject of a formula Focus on working mathematically: Splitting the atom Algebraic Techniques PAS5.. Solves a range of linear equations 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 (a 6) = 5 ( a + ) x 5 x + 7 = 0 5 y y + = 4 solving problems involving linear equations 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 Mathscape 9 and Mathscape 9 Extension School CD ROM 8 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

9 March 004 web update Chapter 5 Measurement 5. Length, mass, capacity and time 5. Accuracy and precision 5. Pythagoras theorem Try this: Pythagorean proof by Perigal 5.4 Perimeter 5.5 Circumference Try this: Command module 5.6 Converting units of area 5.7 Calculating area Try this: The area of a circle 5.8 Area of a circle 5.9 Composite areas Try this: Area 5.0 Problems involving area Focus on working mathematically: The solar system Algebraic Techniques MS5.. Uses formulae to calculate the area of quadrilaterals and finds areas and perimeters of simple composite figures developing and using formulae to find the area of quadrilaterals for a kite or rhombus, Area = xy where x and y are the lengths of the diagonals; for a trapezium, Area = 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 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 solve practical problems involving area of quadrilaterals and simple composite figures (Applying Strategies) Mathscape 9 and Mathscape 9 Extension School CD ROM 9 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

10 March 004 web update MS5.. 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 calculate the area of an annulus 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 identify different possible dissections for a given composite figure and select an appropriate dissection to facilitate calculation of the area (Applying Strategies, Reasoning) 0 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

11 March 004 web update Chapter 6 Data representation and analysis 6. Graphs 6. Organising data 6. Analysing data 6.4 Problems involving the mean Try this: The English language 6.5 Cumulative frequency 6.6 Grouped data Focus on working mathematically: World health Mathscape 9 and Mathscape 9 Extension School CD ROM Data Representation and Analysis DS5.. 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 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 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) Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

12 March 004 web update Chapter 7 Probability 7. Probability and its language 7. Experimental probability Try this: Two-up 7. Computer simulations Try this: The game of craps 7.4 Theoretical probability Try this: Winning chances Focus on working mathematically: Getting through traffic lights Mathscape 9 and Mathscape 9 Extension School CD ROM Probability NS5.. Determine relative frequencies to estimate probabilities Determine theoretical probabilities 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 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 recognise and explain differences between relative frequency and theoretical probability in a simple experiment (Communicating, Reasoning) apply relative frequency to predict future experimental outcomes design a device to produce a specified relative frequency e.g. a four-coloured circular spinner 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 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

13 March 004 web update Chapter 8 Surds 8. Rational and irrational numbers 8. Simplifying surds Try this: Greater number 8. Addition and subtraction of surds 8.4 Multiplication and division of surds Try this: Imaginary numbers 8.5 Binomial products with surds 8.6 Rationalising the denominator Try this: Exact values Focus on working mathematically: Fibonacci numbers and the golden mean Performs operations with surds and indices NS5.. Define the system of real numbers distinguishing between rational and irrational numbers Perform operations with surds defining a rational number: A rational number is the ratio b a of two integers where b 0. 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 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) xy= = x = x. y x 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) explain why a particular sentence is incorrect e.g (Communicating, Reasoning) x y = x y using the four operations of addition, subtraction, multiplication and division to simplify expressions involving surds expanding expressions involving surds such as ( 5) + or ( )( + ) rationalising the denominators of surds of the form a c b d Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

14 March 004 web update Chapter 9 Indices 9. Index notation 9. Simplifying numerical expressions using the index laws 9. The index laws 9.4 Miscellaneous questions on the index laws 9.5 The zero index Try this: Smallest to largest 9.6 The negative index 9.7 Products and quotients with negative indices Try this: Digit patterns 9.8 The fraction index 9.9 Scientific notation 9.0 Scientific notation on the calculator Focus on working mathematically: Mathematics is at the heart of science Mathscape 9 and Mathscape 9 Extension School CD ROM Rational Numbers NS5.. 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 0) 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 0 9 = 9 writing reciprocals of powers using negative indices 4 e.g. = = 4 8 translating numbers to index form (integral indices) and vice versa developing index laws arithmetically by expressing each term in expanded form e.g = ( ) ( ) = = 5 5 = = = ( ) = ( ) ( ) ( ) ( ) = = using index laws to simplify expressions using index laws to define fractional indices for square and cube roots solve numerical problems involving indices explain the incorrect use of index laws e.g. why (Communicating, Reasoning) verify the index laws by using a calculator e.g. to compare the values of ( 5 ), 5 and 5 (Reasoning) communicate and interpret technical information using scientific notation (Communicating) explain the difference between numerical expressions such as and, particularly with reference to calculator displays (Communicating, Reasoning) solve problems involving scientific notation e.g. ( 9) 9 = and 9 = 9, hence 9 = 9 4 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

15 March 004 web update Chapter 9 Indices 9. Index notation 9. Simplifying numerical expressions using the index laws 9. The index laws 9.4 Miscellaneous questions on the index laws 9.5 The zero index Try this: Smallest to largest 9.6 The negative index 9.7 Products and quotients with negative indices Try this: Digit patterns 9.8 The fraction index 9.9 Scientific notation 9.0 Scientific notation on the calculator Focus on working mathematically: Mathematics is at the heart of science Rational Numbers NS5.. writing square roots and cube roots in index form e.g. 8 = 8 = 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 ( ) ( ) converting numbers expressed in scientific notation to decimal form ordering numbers expressed in scientific notation Rational Numbers Mathscape 9 and Mathscape 9 Extension School CD ROM 5 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

16 March 004 web update Chapter 9 Indices 9. Index notation 9. Simplifying numerical expressions using the index laws 9. The index laws 9.4 Miscellaneous questions on the index laws 9.5 The zero index Try this: Smallest to largest 9.6 The negative index 9.7 Products and quotients with negative indices Try this: Digit patterns 9.8 The fraction index 9.9 Scientific notation 9.0 Scientific notation on the calculator Focus on working mathematically: Mathematics is at the heart of science Algebraic Techniques PAS5.. Apply the index laws to simplify algebraic expressions (positive integral indices only) using the index laws previously established for numbers to develop the index laws in algebraic form e.g. 5 = = 5 + m n m+ n 5 = = 6 ( ) a a m a a n = a = a m n mn = ( a ) = a m n establishing that a 0 = using the index laws e.g. and a a = a 0 = a a = a = a simplifying algebraic expressions that include index notation e.g. a x 6 m ( m 0 5x a x + = 8 = 4a 5 = 6x ) = m + 6m 0 verify the index laws using a calculator e.g. use a calculator to compare the values of and 8 (Reasoning) ( 4 ) explain why x 0 = (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 6 e.g. explain why a a = a is incorrect (Communicating, Reasoning) examine and discuss the difference between expressions such as a 5a and a + 5a by substituting values for a (Reasoning, Applying Strategies, Communicating) Mathscape 9 and Mathscape 9 Extension School CD ROM 6 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

17 March 004 web update Text Reference Substrand Outcome Key Ideas Knowledge and Skills Working Mathematically Chapter 9 Indices 9. Index notation 9. Simplifying numerical expressions using the index laws 9. The index laws 9.4 Miscellaneous questions on the index laws 9.5 The zero index Try this: Smallest to largest 9.6 The negative index 9.7 Products and quotients with negative indices Try this: Digit patterns 9.8 The fraction index 9.9 Scientific notation 9.0 Scientific notation on the calculator Focus on working mathematically: Mathematics is at the heart of science Mathscape 9 and Mathscape 9 Extension School CD ROM Algebraic Techniques PAS5.. Apply the index laws to simplify algebraic expressions involving negative and fractional indices applying the index laws to simplify expressions involving pronumerals establishing that ( a ) = a a = a a = a = a using index laws to assist with the definition of the fractional index for square root given ( a ) = a and a = a then a = a 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 = a, a =, a = a a, applying the index laws to simplify algebraic expressions such as 4b 9x 5 4 x 6y (y ) 8b 5x x 4y 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? 5 9x 5 0 5x 5 x 7 = = x a a = a 4 c = 4 c (Applying Strategies, Reasoning, Communicating) explain the difference between particular pairs of algebraic expressions, such as x and x (Reasoning, Communicating) 7 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

18 March 004 web update Real Numbers NS5.. Use integers and fractions for index notation Convert between surd and index form using the index laws to demonstrate the reasonableness of the definitions for fractional indices x x n m n = = translating expressions in surd form to expressions in index form and vice versa evaluating numerical expressions involving fractional indices e.g. 7 n n x x m solve numerical problems involving surds and/or fractional indices using the y x key on a calculator evaluating a fraction raised to the power of, leading to a b = b a 8 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

19 March 004 web update Chapter 0 Geometry 0. Angles 0. Parallel lines 0. Triangles Try this: The badge of the Pythagoreans 0.4 Angle sum of a quadrilateral 0.5 Special quadrilaterals Try this: Five shapes 0.6 Polygons Try this: How many diagonals in a polygon? Properties of Geometrical Figures SGS5.. Establish sum of exterior angles result and sum of interior angles result for polygons 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 60 applying angle sum results to find unknown angles express in algebraic terms the interior angle sum of a polygon with n sides e.g. (n ) 80 (Communicating) find the size of the interior and exterior angles of regular polygons with 5,6,7,8, sides solve problems using angle sum of polygon results (Applying Strategies) Try this: An investigation of triangles 0.7 Tests for congruent triangles 0.8 Congruent proofs Try this: Triangle angles 0.9 Deductive reasoning and congruent triangles Focus on working mathematically: Does a triangle have a centre? Properties of Geometric Figures SGS5.. Apply tests for congruent triangles Use simple deductive reasoning in numerical and nonnumerical problems Congruent Triangles determining what information is needed to show that two triangles are congruent using the 4 tests: SSS, SAS, AAS, RHS applying the congruency tests to justify that two triangles are congruent applying the 4 triangle congruency tests in numerical exercises to find unknown sides and angles apply the properties of congruent triangles to solve problems justifying the results apply simple deductive reasoning in solving numerical and nonnumerical problems 9 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

20 March 004 web update Chapter The linear function. The number plane. Graphing straight lines Try this: Size 8. Graphing straight lines.4 Gradient of a line Try this: Hanging around.5 The linear equation y=mx+b Try this: Latitude and temperature Focus on working mathematically: Television advertising Coordinate Geometry PAS5.. Graph linear and simple non-linear relationships from equations 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 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 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 0 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

21 March 004 web update Graphs of Relationships constructing tables of values and using coordinates to graph vertical and horizontal lines such as x =, x = y =, y = 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 = x x + y = x + y = 5 x y = y = x determining whether a point lies on a line by substituting into the equation of the line 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) Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

22 March 004 web update Chapter Trigonometry 440. Side ratios in right-angled triangles. The trigonometric ratios Try this: Height to base ratio. Trigonometric ratios using a calculator.4 Finding the length of a side.5 Problems involving finding sides Try this: Make a hypsometer.6 Finding the size of an angle.7 Problems involving finding angles.8 Angles of elevation and depression Try this: Pilot instructions.9 Bearings Try this: The sine rule Focus on working mathematically: Finding your latitude from the Sun Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression MS5.. Use trigonometry to find sides and angles in rightangled triangles Solve problems involving angles of elevation and angles of depression from diagrams 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 rightangled 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 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 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 interpret diagrams in questions involving angles of elevation and depression (Communicating) relate the tangent ratio to gradient of a line (Reflecting) Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

23 March 004 web update Applies trig to solve problems including those involving bearings MS5.. Further Trigonometry of Right -Angled Triangles using three-figure bearings and compass bearings drawing diagrams and using them to solve word problems which involve bearings or angles of elevation and depression. Solve simple problems involving three-figure bearings (Applying Strategies, Communication) Interpret directions given as bearings (Communication) Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

24 March 004 web update Chapter Simultaneous equations. Equations with two unknowns. The graphical method. The substitution method Try this: Find the values.4 The elimination method Try this: A Pythagorean problem.5 Solving problems using simultaneous equations Focus on working mathematically: Exploring for water, oil and gas the density of air-filled porous rock Algebraic Techniques PAS5.. Solve simultaneous equations graph simultaneous equations using nonalgebraic methods, such as guess and check, setting up tables of values and 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 a + b = 7 a b = 8 generating simultaneous equations from simple word problems using graphics calculators and spreadsheet software to plot pairs of lines and read off the point of intersection solve linear simultaneous equations resulting from problems and interpret the results 4 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

25 March 004 web update Chapter 4 Co-ordinate geometry 4. The distance between two points 4. The midpoint of an interval 4. The gradient formula Try this: A line with no integer coordinates 4.4 General form of the equation of a line 4.5 The equation of a line given the gradient and a point 4.6 The equation of a line given two points Try this: Car hire 4.7 Parallel lines Try this: Temperature rising 4.8 Perpendicular lines 4.9 Regions in the number plane 4.0 Co-ordinate geometry problems Focus on working mathematically: Finding the gradient of a ski run Coordinate Geometry PAS5.. Use and apply various standard forms of the equation of a straight line, and graph regions on the number plane describing the equation of a line as the relationship between the x - and y -coordinates of any point on the line finding the equation of a line passing through a point ( x, y ),with a given gradient m, using: y y = m( x ) x y = mx+ b finding the equation of a line passing through two points recognising and finding the equation of a line in the general form: ax + by + c = 0 rearranging equations from the general form to the gradient/intercept form and hence graphing the line rearranging equations in the gradient-intercept form to the general form sketching the graph of a line by finding the x -and y -intercepts from its equation demonstrating that two lines are perpendicular if the product of their gradients is finding the equation of a line that is parallel or perpendicular to a given line recognise from a list of equations those that result in straight line graphs (Communicating) describe the conditions for a line to have a negative gradient (Reasoning, 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 (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 = x, y = x, y = x + and describe the transformation that would map one graph onto the other (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) 5 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

26 March 004 web update 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 = x, x + y = 6 describe a region from a graph by identifying the boundary lines and determining the appropriate inequalities for describing the enclosed region (Applying Strategies, Communicating) 6 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.

27 March 004 web update Chapter 4 Co-ordinate geometry 4. The distance between two points 4. The midpoint of an interval 4. The gradient formula Try this: A line with no integer coordinates 4.4 General form of the equation of a line 4.5 The equation of a line given the gradient and a point 4.6 The equation of a line given two points Try this: Car hire 4.7 Parallel lines Try this: Temperature rising 4.8 Perpendicular lines 4.9 Regions in the number plane 4.0 Co-ordinate geometry problems Focus on working mathematically: Finding the gradient of a ski run Coordinate Geometry PAS.5..4 Solve co-ordinate geometry problem Coordinate Geometry Problems solving a variety of problems by applying coordinate geometry formulae and reasoning 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) 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) 7 Copyright Clive Meyers, Lloyd Dawe & Graham Barnsley 004.