Unit+3+Higher

=Unit 3 Higher Maths= Welcome to the Unit 3 Higher page for Edexcel GCSE Maths. You will find all sorts of useful resources to help with your revision flat

=Past Papers= Don't forget to visit the Unit 3 MOCK page to review FULL VIDEO SOLUTIONS of the Year 11 December 2011 Mock.
 * ~ Exam Paper ||~ Mark Scheme ||~ FULL VIDEO SOLUTIONS ||
 * [[file:3999_GCSE 2010_Spec B Unit 3H.pdf]] || [[file:4000_GCSE 2010_Spec B Unit 3H mark scheme.pdf]] || Follow this link for a Playlist of the Unit 3 Higher Specimen paper ||
 * [[file:6476_11b 5MB3H Unit 3 - Mock Paper.doc]] || [[file:6411_12 5MB3H Unit 3 - Mock Paper mark scheme.doc]] ||  ||
 * [[file:6187_11 Practice Paper 3H - Set A.doc]] || [[file:6188_12 Practice Paper 3H - Set A mark scheme.doc]] ||  ||
 * [[file:6229_25 Practice Paper 3H - Set B.doc]] || [[file:6230_26 Practice Paper 3H - Set B mark scheme.doc]] ||  ||
 * [[file:6233_39 Practice Paper 3H - Set C.doc]] || [[file:6234_40 Practice Paper 3H - Set C mark scheme.doc]] ||  ||
 * *New papers added* ||  ||   ||
 * [[file:8794_43 Practice Paper 3H - Set D.doc]] || [[file:8795_44 Practice Paper 3H - Set D mark scheme.doc]] ||  ||
 * [[file:7406_11a 5MB3H - June 2012.pdf]] || [[file:7537_12a 5MB3H Unit 3 mark scheme - June 2012.pdf]] ||  ||
 * [[file:7743_11a 5MB3H - November 2012.pdf]] || [[file:7810_12a 5MB3H - November 2012 mark scheme.pdf]] ||  ||
 * [[file:8674_11a 5MB3H - March 2013.pdf]] || [[file:8773_12a 5MB3H - March 2013 mark scheme.pdf]] ||  ||

=Revision Resources= Try these links for revision relating to Unit 3: Revision Booklet =Higher Revision Booklets= Find a set of exam questions collected by topic including the answers for most: (solving equations, expandind and factorising, expressions) || ||   || (plotting, using and applying) || ||  || (factorising, formula, completing the square) || ||  || (2 linear, 1 linear 1 quadratic) || ||  || (reflection, rotation, translation, enlargement) || ||  || (SOH CAH TOA, Sine rule, Cosine rule, Area of triangle) || ||  ||
 * Algebra skills
 * Bounds || [[file:bounds.pdf]] || [[file:bounds2_ans.pdf]] ||
 * Circle theorems || [[file:circletheorems.pdf]] || [[file:CircleTheorems_ans.pdf]] ||
 * Fractions || [[file:fractions.pdf]] || [[file:fractions_ans.pdf]] ||
 * Misc. Number skills || [[file:number1.pdf]] ||  ||
 * Percentages || [[file:percentages.pdf]] ||  ||
 * Probability trees || [[file:Probability Tree.pdf]] ||  ||
 * Pythagoras' theorem || [[file:pythagoras.pdf]] ||  ||
 * Quadratic graphs
 * Quadratics
 * Sequences || [[file:sequences.pdf]] ||  ||
 * Simultaneous equations
 * Surds and Indices || [[file:surds.pdf]] ||  ||
 * Transformations
 * Transformation of functions || [[file:transformation of curves.pdf]] || [[file:transformation of curves_ans.pdf]] ||
 * Trial and improvement || [[file:trial.pdf]] ||  ||
 * Trigonometry
 * Vectors || [[file:vectors.pdf]] || [[file:vectors_ans.pdf]] ||

=Topic Checklist= A list of Unit 3 topics: Percentage of an amount Percentage of an amount 2 One thing as a % of another Reverse percentages Reverese percentages 2 Compound interest ||<  || ||< Adding Fractions Adding Mixed Fractions Multiplying Mixed Fractions Dividing Fractions ||< MM Dividing Fractions || Expand & Factorise Quadratics ||< Equivalent formulas || ||< Solving linear equations Plotting Inequalities ||< Lineair equations MM Shading Inequalities || Using formulae ||<  || ||< Plotting Inequalities Solving Simultaneous Equations ||< Systems of equations MM Simultaneous 1 MM Simultaneous 2 MM Simultaneous 3 || Expand & Factorise Quadratics Notes and Videos Factorising Quadratics ||< Equivalent formulas Find the function || Reflection ||<  || Scale Drawing Constructing an Equilateral Triangle Constructing an Isosceles Triangle Constructing a Perpendicular Bisector Constructing a Perpendicular line to a point on a line Constructing a Perpendicular line to a point NOT on line ||<  || Pythagoras - Finding a short side Pythagoras' theorem and Trigonometry ||< Millionaire SohCahToa or Pythagoras? Pythagoras' Theorem Trig - Missing angles Trig - Missing sides Trig - Angles of elevation || 3D Trigonometry || Using Cosine rule to find missing side Using Sine rule to find missing angle Using Sine rule to find missing side ||<  || All Theorems reminder media type="custom" key="22457200" ||<  || Area & Circumference of circles ||< Area and Volume Exam Questions || ||< Angles in parallel lines Length Area Volume scale factors ||< MM Area scale factors MM Volume scale factors ||
 * ~ Module number ||~ Title ||~ Video Links ||~ Other ||
 * < 3-1 ||< Percentage Problems ||< Converting between F/D/P
 * < 3-2 ||< Upper & lower bounds ||<  ||<   ||
 * < 3-3 ||< Using fractions
 * < 3-4 ||< Standard form ||<  ||<   ||
 * < 3-5 ||< Factorising & algebraic fractions ||< Expand and Factorise single brackets
 * < 3-6 ||< Solving linear equations & inequalities
 * < 3-7 ||< Substitution & changing the subject ||< Substitution
 * < 3-8 ||< Straight line graphs ||< Plotting graphs from a table ||< Find the function ||
 * < 3-9 ||< Simultaneous equations and inequalities
 * < 3-10 ||< Direct and inverse proportion ||< media type="custom" key="22457108" ||<  ||
 * < 3-11 ||< Trial and improvement ||< Trial & Improvement ||<  ||
 * < 3-12 ||< Quadratic functions ||< Plotting Quadratics from a Table of Values
 * < 3-13 ||< Further simultaneous equations ||<  ||< Systems of equations ||
 * < 3-14 ||< Curved graphs ||<  ||< Find the function ||
 * < 3-15 ||< Transformations ||< Enlargement
 * < 3-16 ||< Transforming graphs ||<  ||<   ||
 * < 3-17 ||< Constructions, loci, similarity and congruency ||< Scale Drawing adn Bearings
 * < 3-18 ||< Pythagoras’ theorem & Trigonometry in 2-D ||< Pythagoras - Finding hypotenuse
 * < 3-19 ||< Applications of Pythagoras’ theorem & Trigonometry in 3-D ||< 3D Pythagoras' Theorem ||< 3D Pythagoras
 * < 3-20 ||< Trigonometry for non-right-angled triangles ||< Using Cosine rule to find an angle
 * < 3-21 ||< Circle theorems ||< Tangents
 * < 3-22 ||< Circles, cones, pyramids and spheres ||< Volume of Cuboids
 * < 3-23 ||< Similar Shapes
 * < 3-24 ||< Vectors ||<  ||<   ||
 * < 3-25 ||< Compound Units ||<  ||<   ||

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=Unit 3 Exam Question Solutions= Try these questions then use the video solutions to check your understanding.

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=Specification VIDEO SOLUTIONS= media type="custom" key="15963630"

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=Detailed Specification= A detailed breakdown: Solve percentage problems, including increase and decrease, and reverse percentage Represent repeated proportional change using a multiplier raised to a power Use calculators for reverse percentages calculations by doing an appropriate division Use calculators to explore exponential growth and decay, using a multiplier and power key || Four operations of number Awareness that percentages are used in everyday life Percentages (Unit 2) || By the end of the module the student should be able to: • Find a percentage increase/decrease of an amount (3.1, 3.2) • Find a reverse percentage, eg find the original cost of an item given the cost after a 10% deduction (3.4) • Use a multiplier to increase by a given percent, eg 1.1 ´ 64 increases 64 by 10% (3.2, 3.3) • Calculate simple and compound interest for two, or more, periods of time (3.3) • Solve a whole host of Functional Elements involving percentages (Throughout Chapter 3) || Combine multipliers to simplify a series of percentage changes Percentages which convert to recurring decimals, eg 33 %, and situations which lead to percentages of more than 100% Problems which lead to the necessity of rounding to the nearest penny, eg real-life contexts Comparisons between simple and compound interest calculations Formulae in simple interest/compound interest methods Increase and decreases leading to a combined multiplier to use, eg 10% decrease then 5% increase || Amounts of money should always be rounded to the nearest penny where necessary, except where such rounding is premature eg in successive calculations like in compound interest In preparation for this unit, students should be reminded of basic percentages and be able to recognise their fraction and decimal equivalents from Unit 2 Use percentages in real-life situations: – VAT – Simple interest – Income tax calculations – Annual rate of inflation – Compound interest – Depreciation – Find prices after a percentage increase or decrease – Percentage profit and loss || Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction Find the upper and lower bounds of calculations involving perimeter, areas and volumes of 2-D and 3-D shapes || Some experience of rounding to a specified number of decimal places or significant figures Measurement and units Perimeter and area || By the end of the module the student should be able to: • Understand that measurements can not be precise, and write down the maximum and minimum possible values (2.1, 2.2) • Work out the maximum/minimum possible error in a calculation involving measures (2. 2) • Find when numbers are given to a specific degree of accuracy, the upper and lower bounds of perimeters and areas (2.2) • Apply upper and lower bounds to compound units, eg speed (2.2) • Give the final answer to an appropriate degree of accuracy following an analysis of the upper and lower bounds of a calculation (2.2) || Calculating areas and volumes upper and lower bounds using formulae Use examples that demonstrate finding the maximum/minimum values for a – b and a ÷ b || An effective starter is to bring in a towel and ask the class to measure it. Since it stretches, this is a good application of something not having a defined length Mention that bridges have expansion joints || Understand ‘reciprocal’ as multiplicative inverse Multiply and divide a given fraction by an integer, by a unit fraction or by a general fraction Recognise that each terminating decimal is a fraction Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals Convert any recurring decimal to a fraction (proof) Use a calculator to solve real-life problems involving Fractions || Fractions (Units 1 & 2) || By the end of the module the student should be able to: • Convert a fraction to a decimal, or a decimal to a fraction (1.1 – 1.4) • Find the reciprocal of whole numbers, fractions, and decimals (1.1) • Multiply and divide a fraction by an integer, by a unit fraction and by a general fraction (expressing the answer in its simplest form) (1.2 – 1.3) • Convert a fraction to a recurring decimal (and visa versa) (1.4) • Use fractions in contextualised problems (1.3) || Use a calculator to find fractions of given quantities Revise addition of fractions (Unit 1 and 2) Use combinations of the four operations with fractions and in real-life problems, eg to find areas using fractional values Revise algebraic fractions with very able students || Constant revision of this aspect is needed All work needs to be presented clearly with the relevant stages of working shown, even if a calculator is used Use Functional Elements problems as a source of questions involving fractions in a real-life context || Calculate with standard form Convert between ordinary and standard form representations Convert to standard form to make sensible estimates for calculations involving multiplication and/or division Use standard form display and know how to enter numbers in standard form || Rounding decimals to a given number of decimal places or significant figure Multiplying decimal numbers with, and without, a calculator Some experience with powers of 10, eg know that 102 = 100, 103 = 1000, 10–1 = 0.1 Negative indices and laws of indices Standard form (Unit 2) || By the end of the module the student should be able to: • Convert numbers to, and from, standard form (4.1) • Calculate with numbers given in standard form with, and without, a calculator (4.1) • Interpret a calculator display using standard form (4.1) • Round numbers given in standard form to a given number of significant figures (4.1) • Use standard form in real-life situations || eg stellar distances, sizes of populations and atomic sizes for small numbers || Make sure students know how to enter numbers in standard form on their particular calculator model, how to read the calculator display Ensure that students never to write down the answer as displayed by the calculator, eg 1.3 -03 This work can be enriched by using examples drawn from the sciences, eg Avogadro’s Constant 6.02 × 10^ -23 || Multiply a single term over a bracket Take out common factors Expand the product of two linear expressions Factorise quadratic expressions Manipulate Algebraic Fractions || By the end of this Unit the student should be able to: • Simplify expressions with like terms, eg ; (7.2 Unit 2) • Expand and factorise expressions with one pair of brackets, eg expand x(2x +3y); factorise 3xy2 - 6x2y (8.1, 8.2, Unit 2) • Expand and simplify expressions involving more than one pair of brackets, eg 3(x + 4) – 2(x – 3); (2x + 3)(3x – 4) (8.3 Unit 2) • Factorise quadratic expressions (including the difference of two squares) (8.4 Unit 2) • Further examples in factorising quadratic expression with non-unitary values of a (including fractional values) (8.4 Unit 2) • Simplify algebraic fractions, eg (11.1 Unit 2) • Simplify algebraic fractions, including the addition of fractions and to solve problems (11.2 Unit 2) || Expand algebraic expressions involving three pairs of brackets Link difference of two squares with surds and the rationalisation of denominators || Emphasise correct use of symbolic notation, eg 3x2 rather than 3 ´ x2 Present all work neatly, writing out the questions with the answers to aid revision at a later stage Practise some basic fraction additions, with numerical values, then compare this with the algebraic version Encourage students to look out for difference of two squares when working with algebraic fractions Remind student to always look to take out a common factor first ||  || Solve linear equations with integer or fractional coefficients, in which the unknown appears on either side or on both sides of the equation Solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution Solve simple linear inequalities in one variable, and represent the solution set on a number line || Experience of finding missing numbers in calculations The idea that some operations are ‘opposite’ to each other Experience of using letters to represent quantities Some experience of ‘balancing’ equations in order to solve them Be able to draw a number line || By the end of the module the student should be able to: • Solve linear equations with one, or more, operations (including fractional coefficients) (Throughout Chapter 6) • Solve linear equations involving brackets and/or variables on both sides (6.2, 6.3) • Solve linear inequalities in one variable and present the solution set on a number line (7.1) • Form linear equations from worded problems in a variety of contexts and relating the answer back to the original problem (Throughout Chapter 6) || Use of inverse operations and rounding to 1 significant figure could be applied to more complex calculations Derive equations from practical situations (such as finding unknown angles in polygons or perimeter problems) Solve equations where manipulation of fractions (including the negative fractions) is required Solve linear inequalities where manipulation of fractions is required || Students need to realise that not all linear equations can easily be solved by either observation or trial and improvement, and hence the use of a formal method is vital Students can leave their answers in fractional form where appropriate Interpreting the direction of an inequality is a problem for many Students should use the correct notation when showing inequalities on a number line, eg a filled in circle to show inclusion of a point, an empty circle to show exclusion of a point Inequalities in two variables will be covered again after simultaneous equations || Use formulae from mathematics and other subjects that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution Change the subject of a formula including where the subject occurs once or more than once Generate a formula || Understanding of the mathematical meaning of the words expression, simplifying, formulae and equation Experience of using letters to represent quantities Substituting into simple expressions using words Using brackets in numerical calculations and removing brackets in simple algebraic expressions Solving linear equations || By the end of the module the student should be able to: • Use letters or words to state the relationship between different quantities (Revision of prior knowledge) • Substitute positive and negative numbers into simple algebraic formulae (Revision of prior knowledge) • Substitute positive and negative numbers into algebraic formulae involving powers (Revision of prior knowledge) • Find the solution to a problem by writing an equation and solving it (Revision of prior knowledge) • Simple change of subject of a formula, eg convert the formula for converting Centigrade into Fahrenheit into a formula that converts Fahrenheit into Centigrade (7.5) • Change the subject of the formula when the variable appears more than once (questions could involve powers, roots, fractions or reciprocals) (7.6) • Generate a formula from given information, eg find the formula for the perimeter of a rectangle given its area A and the length of one side (Revision of prior knowledge) || Use negative numbers in formulae involving indices Various investigations leading to generalisations Further problems in generating formulae form given information Use equation of a straight line, eg what is the gradient of the line 4x + 2y = 12? Apply changing the subject to physics formulae, ie pendulum, equations of motion, lens formula || Emphasise good use of notation, eg 3ab means 3 ´ a ´ b Students need to be clear on the meanings of the words expression, equation, formula and identity Simple changing the subject is covered in Unit 2 Show a linear equation first and follow the same steps for the similarly structured formula to be rearranged Functional Elements material is a rich source of formulae given in everyday contexts Link with formulae for area, volume, surface area, etc || Find the gradient of lines given equations of the form y = mx + c values are given for m and c) Analyse problems and use gradients to see how one variable changes in relation to another Calculate the length of a line segment between two coordinates || Experience at plotting points in all quadrants Linear sequences and straight line graphs || By the end of the module the student should be able to: • Draw linear graphs from tabulated data, including real-world examples (Revision of prior knowledge) • Interpret linear graphs, including conversion graphs and distance-time graphs (Revision of prior knowledge) • Draw and interpret graphs in the form y = mx + c (when values for m and c are given) (Revision of prior knowledge) • Understand that lines are parallel when they have the same value of m (Revision of prior knowledge) • Find the gradient and intercept of a straight line graph (Revision of prior knowledge) • Find the distance between any two coordinates (link with Pythagoras’ theorem) (18.2) || Plot graphs of the form y = mx + c where pupil has to generate their own table and set out their own axes Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines Use a graphical calculator or graphical ICT package to draw straight-line graphs Link with length of line to Pythagoras’ theorem and contrast with midpoint formula (Unit 2) || Clear presentation with axes labelled correctly is vital Recognise linear graphs and hence when data may be incorrect Link to graphs and relationships in other subject areas, ie science, geography etc Interpret straight line graphs for Functional situations and link gradient to cost per unit or rate of change and intercept to initial cost etc – Ready reckoner graphs – Conversion graphs – Fuel bills – Fixed charge (standing charge) and cost per unit || Solve several linear inequalities in two variables and finding the solution set || Algebraic manipulation Ability to solve simple linear equations Some experience with solving inequalities Straight line graphs || By the end of the module the student should be able to: • Solve algebraically two simultaneous equations (9.2, 9.3) • Interpret the solution of two simultaneous equations as the point of intersection the corresponding lines (9.4) • Model worded problems as a pair of linear simultaneous equations and interpret the answer (9.3) • Draw the graphs of linear inequalities in two variables and interpret the solution sets given by regions in the coordinate plane, or to identify all the integer coordinates with crosses (7.4) || Solve two simultaneous equations with fractional coefficients Solve two simultaneous equations with second order terms, eg equations in x and y2 (link to further simultaneous equations) || Build up the algebraic techniques slowly Link the graphical solutions with straight line graphs (Units 1 and 2) and changing the subject (above) Inaccurate graphs could lead to incorrect solutions, encourage substitution of answers to check they are correct Clear presentation of working out is essential Students should use the correct notation when giving graphical solutions to inequalities, eg a dotted boundary line for || Set up and use equations to solve word and other problems involving direct proportion or inverse proportion and relating algebraic solutions to graphical representations of the equations Calculate an unknown quantity from quantities that vary in direct or inverse proportion || Substitute numbers into algebraic formulae Rearrange the subject of a formula || By the end of the module the student should be able to: • Solve simple direct & inverse proportion problems using the Unitary method or by proportional change (from a table of values (Chapter 5) • Interpret direct and inverse proportions as algebraic functions, eg y µ x2 as y = kx2 (Chapter 10) • Use given information to find the value of the constant of proportionality (Chapter 10) • Use algebraic functions for direct and inverse proportionality, with their value of k, to find unknown values (Chapter 10) • Recognise and sketch the graphs for direct and inverse proportions (y µ x, y µ x2, y µ x3, y µ, y µ ) (Chapter 10) || Link Unitary method with Ratio topic (Unit 1 and 2) Problems involving other types of proportionality, eg surface area to volume of a sphere Link to graphs to show direct and inverse proportion, eg etc || Students should be encouraged to show all steps in their working Students often forget the “square” in inverse square proportionality || Dealing with decimals on a calculator Ordering decimals and decimal place approximations. || By the end of the module the student should be able to: • Solve quadratic and cubic functions by successive substitution of values of x || Solve functions of the form (link with ‘changing the subject’) || Students should be encouraged to use their calculators efficiently, by using the "replay" or ANS/EXE functions The square/cube function on a calculator may not be the same for different makes Take care when entering negative values to be squared (always use brackets) Students should write down all the digits on their calculator display and only round the final answer declared to the degree of accuracy || Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function Factorise quadratic expressions Solve simple quadratic equations by factorising, completing the square and using the quadratic formula || Graphs Factorising Quadratics || By the end of the module the student should be able to: • Plot the graphs of quadratic functions for positive and negative values of x (8.1) • Find graphically the solutions of quadratic equations by considering the intercept on the x-axis (9.10) • Solve quadratic equations by factorising (including values of a not equal to 1) (9.5) • Use the quadratic formula to solve quadratic equations giving the answers to a specified degree of accuracy (9.7) • Use the quadratic formula to solve quadratic equations leaving the answer in surd form or decimal form (9.7) • Complete the square of a quadratic function (using this to write down the maximum/minimum of the function) (9.1, 9.6) || Solve equations involving algebraic fractions which lead to quadratic equations Solve quadratic equations by completing the square Derive the quadratic equation by completing the square Use graphical calculators or ICT graph package where appropriate to enable students to get through examples more rapidly Show how the value of ‘b2 – 4ac’ can be useful in determining if the quadratic factorises or not (ie square number) Extend to discriminant’s properties and roots (for those going on to C1) || There may be a need to remove the HCF (numerical) of a trinomial before factorising it to make the factorisation easier to do Students should be reminded that factorisation should be tried before the formula is used In problem-solving, one of the solutions to a quadratic may not be appropriate, eg negative length || Construct the graph of x2 + y2 = r2 for a circle of radius r centred at the origin of coordinates Find graphically the intersection points of a given straight line and a circle and knowing that this corresponds to solving the simultaneous equations representing the line and the circle Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other, of where the second is of the form x2 + y2 = r2 || Quadratic functions Straight line graphs Algebraic manipulation and solving linear and quadratic equations || By the end of the module the student should be able to: • Find graphically the approximate solutions of linear and quadratic simultaneous equations (9.10) • Find the exact solutions of linear and quadratic simultaneous equations (9.11) • Draw a circle of radius r centred at the origin (9.10) • Find the approximate solutions of linear and circular simultaneous equations graphically (9.11) • Find the exact solutions of linear and circular simultaneous equations (9.10) || Find the approximate solutions of quadratic and circular simultaneous equations graphically Find the exact solutions of quadratic and circular simultaneous equations using algebraic methods Look at circles whose centre is not the origin (x – 2)2 + (y – 3)2 = 4 (link with transforming graphs) || Clear presentation of workings is essential Stress which variable it is easiest to work with when assessing the linear equation ICT graph drawing packages make this topic more dynamic and easier to picture Further examples and questions can be obtained from A-Level (C1) texts || Plot graphs of simple cubic functions, the reciprocal function y = with x 0, the exponential function, y = kx for integer values of x and simple positive values of k, the circular functions y = sin x and y = cos x Recognise the characteristic shapes of all these functions || Straight line graphs BIDMAS || By the end of the module the student should be able to: • Plot and recognise quadratic, cubic, reciprocal, exponential and circular (trig) functions (see above) within the range –360º to +360º (8.1 – 8.4) • Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa) (8.1 – 8.4) • Match equations with their graphs (8.1 – 8.4) • Sketch graphs of given functions (8.1 – 8.4) || Explore the function y = ex (perhaps relate this to y = ln x) Explore the function y = tan x Find solutions to equations of the circular functions y = sin x and y = cos x over more than one cycle (and generalise) Start to investigate transformations, eg y = sin(2x) or y = x2 + 5 || This work should be enhanced by drawing graphs on graphical calculators and appropriate software Group work with each group assigned a different type of graph is an effective way to share the graphs’ properties. Each group reports to the whole class and creates a display There are plenty of old exam papers with matching tables testing knowledge of the ‘Shapes of Graphs’ || Understand that translations are specified by a distance and direction (or a column vector), and enlarging by a centre and a scale factor Rotate a shape about the origin, or any other point Measure the angle of rotation using right angles, simple fractions of a turn or degrees Understand that rotations are specified by a centre and an (anticlockwise) angle Understand that reflections are specified by a mirror line, at first using a line parallel to any axis, then a mirror line such as y = x or y = –x Recognise, visualise and construct enlargements of objects using positive and negative scale factors greater and less than one Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations || Recognition of basic shapes Line and rotational symmetry (Unit 2) An understanding of the concept of rotation and enlargement Coordinates in four quadrants Linear equations parallel to the coordinate axes || By the end of the module the student should be able to: • Understand translation as a combination of a horizontal and vertical shift including signs for directions (17.2) • Understand rotation as a (clockwise) turn about a given origin (17.4) • Reflect shapes in a given mirror line; parallel to the coordinate axes and then y = x or y = –x (17.3) • Enlarge shapes by a given scale factor from a given point; using positive and negative scale factors greater and less than one (and understand the effects that negative and fractional scale factors have on the image) (17.5, 17.6) • Understand that shapes produced by translation, rotation and reflection are congruent to its image (17.1 – 17.4) || The tasks set should be extended to include combinations of transformations || Emphasise that students describe the given transformation fully Diagrams should be drawn carefully The use of tracing paper is allowed in the examination (although students should not have to rely on the use of tracing paper to solve problems) || y = af(x) for linear, quadratic, sine and cosine functions f(x) Draw, sketch and describe the graphs of trigonometric functions for angles of any size, including transformations involving scalings in either or both the x and y directions || Transformations Curved graphs (Unit 1 and 3) || By the end of the module the student should be able to: • Understanding of the notation y = f(x) (11.1) • Represent translations in the x and y direction, reflections in the x-axis and the y axis, and stretches parallel to the x-axis and the y-axis (11.2 – 11.4) • Apply to general graphs or specific curves such as trigonometric functions (ie curved graphs) (11.2 – 11.4) • Sketch the graph of y = 3sin(2x), given the graph of y = sinx (11.2 – 11.4) • Sketch the graph of y = f(x + 2), y = f(x) + 2, y=2f(x), y = f(2x) given the shape of the graph y = f(x) (11.2 – 11.4) • Find the coordinates of the minimum of y = f(x + 3), y = f(x) + 3 given the coordinates of the minimum of y = x2 – 2x (11.2 – 11.4) || Complete the square of quadratic functions and relate this to transformations of the curve y = x2 Use a graphical calculator/software to investigate transformations Investigate curves which are unaffected by particular transformations Investigations of the simple relationships such as sin(180 – x) = sin x, and sin(90 – x) = cos x || Make sure the students understand the notation y = f(x). Perhaps start with comparing y = x2 with y = x2 + 2 before mentioning y = f(x) + 2 Graphical calculators and/or graph drawing software will help to underpin the main ideas in this unit Link with trigonometry and curved graphs || Understand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA are not Use straight edge and compasses to do standard constructions Construct loci Understand similarity of triangles and of other plane figures and use this to make geometric inferences Recognise that enlargements preserve angle but not length Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles by using formal arguments, and also to verify standard ruler and compass constructions || An ability to use a pair of compasses The special names of triangles (and angles) Understanding of the terms perpendicular, parallel and arc Transformations (particularly enlargements) || By the end of the module the student should be able to do a range of standard constructions including: • An equilateral triangle with a given side (16.2 – 16.5) • The mid point and perpendicular bisector of a line segment (16.1) • The perpendicular from a point on a line (16.1) • The bisector of an angle (16.2) • The angles 60°, 30° and 45° (16.2) • A regular hexagon inside a circle, etc (16.2) • A region bounded by a circle and an intersecting line (16.4) • A path equidistant from two points or two line segments (16.3) • Prove geometric properties of triangles formally, eg that the base angles of an isosceles triangle are equal (16.5) • Prove that two triangles are congruent formally (14.1 – 16.5) • Use integer and non-integer scale factors to find the length of a missing side in each of two similar shapes, given the lengths of a pair of corresponding sides (14.2 – 14.3) || Solve loci problems that require a combination of loci Construct combinations of 2-D shapes to make nets Link with tessellations (Unit 2) and enlargements (Unit 3) Link with similar areas and volumes Use harder problems in congruence Relate this unit to circle theorems || All working should be presented accurately and clearly A sturdy pair of compasses are essential Construction lines should not be erased as they carry method marks +H54 || Understand, recall and use trigonometrical relationships in right-angled triangles, and use these to solve problems, including those involving bearings Find angles of elevation and angles of depression || Names of triangles & quadrilaterals Knowledge of the properties of rectangles, parallelograms and triangles Indices, equations and changing the subject Similarity of triangles and other plane figures || By the end of the module the student should be able to: • Find missing sides of right-angle triangles by using Pythagoras (18.1 – 18.2) • Find the distance between two coordinates using Pythagoras (18.2) • Giving answers as decimals or surds for Pythagoras problems (18.2) • Use trigonometric ratios (sin, cos and tan) to calculate angles in right-angled triangles (18.3) • Use the trigonometric ratios to calculate unknown lengths in right-angled triangles (2-D) (18.4) • Understand how bearings work and solve problems involving bearings using Pythagoras/trigonometry (16.6, 18.4) • Solve problems involving geometric figures (including triangles within circles) in which a right-angle triangle has to be extracted in order to solve it by Pythagoras and/or trigonometry (18.2, 18.4) || Introduce 3-D trigonometry and show that the trigonometric ratios will only work for right-angled triangles || Students should be encouraged to become familiar with one make of calculator Calculators should be set to “deg” mode Emphasise that scale drawings will score no marks for this type of question A useful mnemonic for remember trig ratios is “Sir Oliver’s Horse, Came Ambling Home, To Oliver’s Aunt” or ‘SOH/CAH/TOA’ Calculated angles should be given to at least 1 dp and sides are determined by the units used or accuracy asked for in the question Students should not forget to state the units in their answers Organise a practical surveying lesson to find the heights of buildings/trees around your school grounds. All that is needed is a set of tape measures (or trundle wheels) and clinometers || Use the trigonometric ratios and the sine and cosine rules to solve 3-D problems Find the angles between a line and a plane || Knowledge of Pythagoras’ theorem and trigonometry 3-D solids (Unit 2) || By the end of the module the student should be able to: • Calculate the length of a diagonal of a rectangle given the lengths of the sides of the rectangle (18.2) • Calculate the diagonal through a cuboid, or across the face of a cuboid (19.1, 19.2) • Find the angle between the diagonal through a cuboid and the base of the cuboid (19.1, 19.2) • Find the angle between a sloping edge of a pyramid and the base of the pyramid (19.1, 19.2) || Use harder problems involving multi-stage calculations || The angle between two planes or two skew lines is not required || Calculate the area of a triangle using absinC
 * ~ MODULE ||~ CONTENTS ||~ PRIOR KNOWLEDGE ||~ OBJECTIVES ||~ DIFFERENTIATION & EXTENSION ||~ NOTES ||
 * 3-1 || Percentage problems || Use percentages in real-life situations
 * 3-2 || Upper & lower bounds || By using calculators, or written methods, calculate the upper and lower bounds, particularly when working with measurements
 * 3-3 || Using fractions || Calculate a given fraction of a quantity, expressing the answer as a fraction
 * 3-4 || Standard form || Use standard form, expressed in standard notation and on a calculator display
 * 3-5 || Factorising & algebraic fractions || Simplify terms, products and sums
 * 3-6 || Solving linear equations & inequalities || Solve equations by using inverse operations or by transforming both sides in the same way
 * 3-7 || Substitution & changing the subject || Substitute numbers into formulae
 * 3-8 || Straight line graphs || Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding gradients.
 * 3-9 || Simultaneous equations and inequalities || Find the exact solution of two (linear) simultaneous equations in two unknowns by eliminating a variable and interpreting the equation as lines and their common solution as the point of intersection
 * 3-10 || Direct and inverse proportion || Use repeated proportional change
 * 3-11 || Trial and improvement || Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them || Substituting numbers into algebraic expressions
 * 3-12 || Quadratic functions || Generate points and plot graphs of quadratic functions
 * 3-13 || Further simultaneous equations || Find the intersection points of the graphs of linear and quadratic functions, knowing that these are the approximate solutions of the corresponding simultaneous equation representing the linear and quadratic functions
 * 3-14 || Curved graphs || Generate points and plot graphs of simple quadratic functions
 * 3-15 || Transformations || Transform triangles and other 2-D shapes by translation, rotation and reflection and combinations of these transformations
 * 3-16 || Transforming graphs || Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a),
 * 3-17 || Constructions, loci, similarity and congruency || Draw triangles and other 2-D shapes, using a ruler and protractor, given information about side lengths and angles
 * 3-18 || Pythagoras’ theorem & Trigonometry in 2-D || Understand, recall and use Pythagoras’ theorem in 2-D
 * 3-19 || Applications of Pythagoras’ theorem & Trigonometry in 3-D || Understand and recall and use trigonometric relationships in right angled triangles and use these to solve problems in 3-D (include the use of Pythagoras’ Theorem)
 * 3-20 || Trigonometry for non-right-angled triangles || Use the sine and cosine rules to solve 2-D problems

Use the sine and cosine rules to solve 3-D problems || Trigonometry Formulae || By the end of the module the student should be able to: • Find the unknown lengths, or angles, in non right-angle triangles (in 2-D and 3-D) using the sine and cosine rules (19.3, 19.5 – 19.9) • Find the area of triangles given two lengths and an included angle (19.4) || • Use trigonometrical ratios to solve problems in 3-D and decide if it is easier to extract right-angle triangles to use simple trigonometry • Establish strategies for solving problems in 3-D. Decide when it is more appropriate to use cosine rule rather than sine rule || Students find the cosine rule more difficult for obtuse angles. Demonstrate an example of the ambiguous case Reminders of simple geometrical facts will always be helpful, eg angle sum of a triangle, the shortest side is opposite the smallest angle Show the form of the cosine rule on the formula page and re-arrange it to show the form which finds missing angles || Prove and use the fact that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference Prove and use the fact that the angle subtended at the circumference by a semicircle is a right angle Prove and use the fact that angles in the same segments are equal Prove and use the fact that opposite angles of a cyclic quadrilateral sum to 180° Prove and use the alternate segment theorem || Recall the words centre, radius, diameter and circumference Have practical experience of drawing circles with compasses Circle theorem covered in Unit 2 (tangent/radius property) Recall that the tangent at any point on a circle is perpendicular to the radius at that point Recall and use the fact that tangents from an external point are equal in length || By the end of the module the student should be able to: • Understand, prove and use circle theorems (see above) (Chapter 15) • Use circle theorems to find unknown angles and explain their method - quoting the appropriate theorem(s) (Chapter 15) || Harder problems involving multi-stage calculations || Any proof required will be in relation to a diagram, not purely by reference to a named theorem Reasoning needs to be carefully constructed as ‘Quality of Written Communication’ marks are allotted || Solve problems involving surface areas and volumes of prisms and cylinders Solve problems involving volumes of prisms and cylinders Solve problems involving surface areas and volumes of cones and pyramids Solve problems involving more complex shapes, including segments of circles, length of a chord and frustums of cones Use surds and π in exact calculations, without a calculator Find the surface areas and volumes of compound solids constructed from; spheres, hemispheres and cylinders (include examples of solids in everyday use) || Perimeter and area (Unit 2) Surface area and volume (Unit 2) Formulae, substitution and changing the subject Surds || By the end of the module the student should be able to: • Learn that the ratio between the circumference and diameter is always constant for a circle (ie π) (12.1) • Learn the formula for circumference and area of a circle (12.1) • Solve problems involving the circumference and area of a circle (and simple fractional parts of a circle) (12.1 – 12.4) • Solve problems involving the volume of a cylinder (13.1) • Find the surface area and the volume of more complex shapes, eg find the volume of an equilateral triangular prism (13.2 – 13.6) • Solve more complex problems, eg given the surface area of a sphere find the volume (13.2 – 13.6) || Find the volume of a cylinder given its surface area, leaving the answer in terms of l Find the volume of a right hexagonal cone of side x and height h (researching the method for finding the volume of any cone) || ‘Now! I Know Pi’ is a good way to learn the approx value (the number of letters of each word and the ! is the decimal point) Also ‘Cherry Pie Delicious’ is C = πD and ‘Apple Pies are too’ is A = πr2 Answers in terms of π may be required or final answers rounded to the required degree of accuracy Need to constantly revise the expressions for area/volume of shapes Students should be aware of which formulae are on the relevant page on the exam paper and which they need to learn || Understand the implications of enlargement for perimeter Understand and use the effect of enlargement on areas and volumes of shapes and solids || Use ruler and compasses to construct triangles with given dimensions Some concept of enlargement (magnification) Similar triangles || By the end of the module the student should be able to: • Use integer and non-integer scale factors to find the length of a missing side in each of two similar shapes, given the lengths of a pair of corresponding sides (14.2 – 14.3) • Know the relationship between linear, area and volume scale factors of similar shapes (14.4 – 14.6) • Prove formally geometric properties of triangles, eg that the base angles of an isosceles triangle are equal (14.5) || Find algebraic formulae for the areas and volumes of similar shapes Extend to questions which give the ratio for area and the need to square root the ratio to get back to length first etc || Students will need to be reminded of this work on a regular basis, link it to ratios 1 : L (Length) 1: L2 (Area) 1: L3 (Volume) A good starter is to bring in a small bottle of water and a larger bottle (preferably for the larger bottle to be twice the length). Show by pouring that eight small bottles will fill the larger bottle. [This can also be done with one small cube and a larger box with lengths twice as long] Initially the class will say two bottles will fill the larger double size bottle || Calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector Calculate the resultant of two vectors Understand and use the commutative and associative properties of vector addition Solve simple geometrical problems in 2-D using vector methods || Vectors to describe translations. Algebraic manipulation || By the end of the module the student should be able to: • Understand that 2a is parallel to a and twice its length (20.1 – 20.3, 20.5) • Understand that a is parallel to -a and in the opposite direction (20.1 – 20.3, 20.5) • Use and interpret vectors as displacements in the plane (with an associated direction) (20.1) • Use standard vector notation to combine vectors by addition (20.4) • Represent vectors, and combinations of vectors, in the plane (20.1 – 20.6) • Solve geometrical problems in 2-D, eg show that joining the mid points of the sides of any quadrilateral forms a parallelogram (20.6) || Harder geometric proof, eg show that the medians of a triangle intersect at a single point Illustrate use of vectors by showing ‘Crossing the flowing River’ example or navigation examples Vector problems in 3-D (for the most able) Use i and j (and k) notation (preparation for A-Level but not in GCSE specification) Lengths are sometimes given in the ratio of 1:3. It is important to explain this concept carefully || Students often find the pictorial representation of vectors more difficult than the manipulation of column vectors The geometry of a hexagon provides a rich source of parallel, reverse and vectors with double the magnitude Stress that parallel vectors are equal to one another Link with ‘like terms’ and ‘brackets’ when simplifying Show there is more than one route round a geometric shape, but that the answer simplifies to the same vector Remind students to underline vectors or they will be regarded as just lengths with no direction Extension material can be found in M1 texts || Convert measurements from one unit to another (area and volume) || Knowledge of metric units of length, area, volume, and weight, eg 1 m = 100 cm, etc Speed calculations Experience of multiply by powers of 10, eg 100 ´ 100 = 10 000 Substitution and changing the subject || By the end of the module the student should be able to: • Use the relationship between distance, speed and time to solve problems (revision of Unit 2) (Revision of prior knowledge) • Know that density is found by mass ÷ volume (13.6) • Use the relationship between density, mass and volume to solve problems (13.6) • Convert between metric units of density, eg kg/m to g/cm (13.7) • Convert between area measures, eg 1 m2 = 100 00 cm2 (12.4) • Convert between volume measures (using metric units) (13.5) • Convert between volume and capacity measures, eg 1 ml = 1 cm³ (1cc) (13.5) || Perform calculations on a calculator by using standard form Convert imperial units to metric units, eg mph into km/h Mention other units (not on course) like hectares || Use a formula triangle to help students see the relationship between the variables for density Borrow a set of electronic scales and a Eureka Can from Physics for a practical density lesson Look up densities of different elements from the internet Link converting area and volume units to similar shapes Draw a large grid made up of 100 by 100 cm squares to show what 1 square metre looks like Use of Calculators (Spec Ref:- Nv) Students are well advised to regularly bring and use their particular calculator to lessons and get used to its functions throughout this Unit 3 course. They should be confident in entering a range of calculations including those involving time and money. They should realise that 2.33333333…. hrs is 2 hours 20 minutes etc ||
 * 3-21 || Circle theorems || Explain why the perpendicular from the centre to a chord bisect the chord
 * 3-22 || Circles, cones, pyramids and spheres || Find circumferences of circles and areas enclosed by circles, recalling relevant formulae
 * 3-23 || Similar Shapes || Identify the scale factor of an enlargement as a ratio of the lengths of any two corresponding line segments
 * 3-24 || Vectors || Understand and use vector notation (revise column vectors)
 * 3-25 || Compound Units || Understand and use compound measures, including density