ࡱ> c mbjbj rrojj !!!D!!!D"@%!l:)v-L---6|:<;$X!n<45|n<n< --&CCCn< -!-Cn<CC*38J!6-&{[(d>ZkRk<0l*6?h606!Tn<n<Cn<n<n<n<n<&Bn<n<n<ln<n<n<n<6n<n<n<n<n<n<n<n<n<j :  ASK SUBJECT " Insert subject title" Mathematics A (1MA0) ASK DATE " Insert date (month & year)" December 2010 ASK CODE " Insert Publication code" UG025517 ASK PREP "Insert QL's name" Sharon Wood and Ali Melville ASK ISSUE " Insert draft/issue" Issue 2  MACROBUTTON FieldsUpdate.Main "Double click here (or Ctrl + A then F9) to update field codes"       The Edexcel Award in Algebra Level 3 (AAL30) Scheme of work Level 3 course overview The table below shows an overview of modules in the Level 3 scheme of work. Teachers should be aware that the estimated teaching hours are approximate and should be used as a guideline only. GREEN is presumed knowledge and will not be covered for the one term course. RED is knowledge that will be covered for the one term course. Module numberTitleEstimated teaching hours1 HYPERLINK \l "_Roles_of_symbols" Roles of symbols1.52 HYPERLINK \l "_Algebraic_manipulation" Algebraic manipulation73 HYPERLINK \l "_Formulae" Formulae74 HYPERLINK \l "_Simultaneous_equations" Simultaneous equations55 HYPERLINK \l "_Quadratic_equations" Quadratic equations56 HYPERLINK \l "_Roots_of_a" Roots of a quadratic equation1.57 HYPERLINK \l "_Inequalities" Inequalities38 HYPERLINK \l "_Arithmetic_series" Arithmetic series49 HYPERLINK \l "_Coordinate_geometry" Coordinate geometry510 HYPERLINK \l "_Graphs_of_functions" Graphs of functions711 HYPERLINK \l "_Graphs_of_simple" Graphs of simple loci212 HYPERLINK \l "_Distance-time_and_speed-time" Distance-time and speed-time graphs313 HYPERLINK \l "_Direct_and_inverse" Direct and inverse proportion414 HYPERLINK \l "_Transformations_of_functions" Transformation of functions415 HYPERLINK \l "_Area_under_a" Area under a curve216 HYPERLINK \l "_Surds" Surds4Total65 Module 1 Time: 1 2 hours Awards Tier: Level 3 Roles of symbols 1.1Distinguish between the roles played by letter symbols in algebra using the correct notation, and between the words equation, formula, identity and expression GCSE SPECIFICATION REFERENCES A aDistinguish the different roles played by letter symbols in algebra, using the correct notationA bDistinguish in meaning between the words equation, formula, identity and expressionA cManipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors, multiplying two linear expressions, factorise quadratic expressions including the difference of two squares and simplify rational expressions PRIOR KNOWLEDGE Experience of using a letter to represent a number Ability to use negative numbers with the four operations Recall and use BIDMAS Writing simple rules algebraically LINKS TO LEVEL 2 CONTENT Module 1 Roles of symbols OBJECTIVES By the end of the module the student should be able to: Use notation and symbols correctly Select an expression/identity/equation/formula from a list LINKS TO GCSE SCHEME OF WORK (2-year) A/Module 4 B/Module 2-8 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Compare different expressions in mathematics and determine their form Use examples where generalisation skills are required Extend the above ideas to the equation of the straight line, y = mx + c Look at word formulae written in symbolic form, eg F = 2C + 30 to convert temperature (roughly) and compare with F =  EMBED Equation.3 C + 32 NOTES Emphasise good use of notation, eg 3n means 3 ( n Present all working clearly  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 2 Time: 6 8 hours Awards Tier: Level 3 Algebraic manipulation 2.1Multiply two linear expressions2.2Factorise expressions including quadratics and the difference of two squares, taking out all common factors2.3Use index laws to include fractional and negative indices2.4Simplify algebraic fractions2.5Complete the square in a quadratic expression GCSE SPECIFICATION REFERENCES A cManipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors, multiplying two linear expressions, factorise quadratic expressions including the difference of two squares and simplify rational expressionsA cSimplify expressions using rules of indices PRIOR KNOWLEDGE Ability to use negative numbers with the four operations Recall and use BIDMAS LINKS TO LEVEL 2 CONTENT Module 2 Algebraic manipulation OBJECTIVES By the end of the module the student should be able to: Simplify expressions using index laws Use index laws for integer, negative and fractional powers and powers of a power Factorise quadratic expressions Recognise the difference of two squares Simplify algebraic fractions Complete the square in a quadratic expression LINKS TO GCSE SCHEME OF WORK (2-year) A/module 4 B/module 2-9, 3-5 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Consider multiplication for terms in brackets Simplification of algebra involving several variables Algebraic fractions involving multiple expressions Factorise cubic expressions Practise factorisation where the factor may involve more than one variable NOTES Avoid oversimplification Ensure cancelling is only done when possible (particularly in algebraic fractions work)  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 3 Time: 6 8 hours Awards Tier: Level 3 Formulae 3.1Substitute numbers into a formula3.2Change the subject of a formula GCSE SPECIFICATION REFERENCES A fDerive a formula, substitute numbers into a formula and change the subject of a formula PRIOR KNOWLEDGE Experience of using a letter to represent a number Ability to use negative numbers with the four operations Recall and use BIDMAS LINKS TO LEVEL 2 CONTENT Module 3 Formulae OBJECTIVES By the end of the module the student should be able to: Derive a formula Use formulae from mathematics and other subjects Substitute numbers into a formula Substitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3 Change the subject of a formula including cases where the subject is on both sides of the original formula, or where a power of the subject appears LINKS TO GCSE SCHEME OF WORK (2-year) A/module 14 B/module 2-10, 3-7 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Consider changing formulae where roots and powers are involved NOTES Break down any manipulation into simple steps all clearly shown  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 4 Time: 4 6 hours Awards Tier: Level 3 Simultaneous equations 4.1Solve simultaneous equations in two unknowns, where one may be quadratic, where one may include powers up to 2 GCSE SPECIFICATION REFERENCES A dSet up and solve simultaneous equations in two unknowns PRIOR KNOWLEDGE An introduction to algebra Substitution into expressions/formulae Solving equations LINKS TO LEVEL 2 CONTENT Module 4 Linear equations OBJECTIVES By the end of the module the student should be able to: Find the exact solutions of two simultaneous equations in two unknowns Use elimination or substitution to solve simultaneous equations Interpret a pair of simultaneous equations as a pair of straight lines and their solution as the point of intersection Set up and solve a pair of simultaneous equations in two variables Find the exact solutions of two simultaneous equations when one is linear and the other quadratic Find an estimate for the solutions of two simultaneous equations when one is linear and one is a circle LINKS TO GCSE SCHEME OF WORK (2-year) A/module 16 B/module 3-9, 3-13 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Students to solve two simultaneous equations with fractional coefficients and two simultaneous equations with second order terms, eg equations in x2 and y2 NOTES Build up the algebraic techniques slowly  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 5 Time: 4 6 hours Awards Tier: Level 3 Quadratic equations 5.1Solve quadratic equations by factorisation or by using the formula or by completing the square5.2Know and use the quadratic formula GCSE SPECIFICATION REFERENCES A eSolve quadratic equations PRIOR KNOWLEDGE An introduction to algebra Substitution into expressions/formulae Solving equations LINKS TO LEVEL 2 CONTENT Module 2 Algebraic manipulation (factorising) OBJECTIVES By the end of the module the student should be able to: Solve quadratic equations by factorisation Solve quadratic equations by completing the square Solve quadratic equations by using the quadratic formula LINKS TO GCSE SCHEME OF WORK (2-year) A/module 26 B/module 3-12 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Students to derive the quadratic equation formula by completing the square Show how the value of b2 4ac can be useful in determining if the quadratic factorises or not (i.e. square number) NOTES Some students may need additional help with factorising Students should be reminded that factorisation should be tried before the formula is used In problem-solving, one of the solutions to a quadratic equation may not be appropriate There may be a need to remove the HCF (numerical) of a trinomial before factorising to make the factorisation easier  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 6 Time: 1 2 hours Awards Tier: Level 3 Roots of a quadratic equation 6.1Understand the role of the discriminant in quadratic equations 6.2Understand the sum and the product of the roots of a quadratic equation GCSE SPECIFICATION REFERENCES A eSolve quadratic equations PRIOR KNOWLEDGE Solve simple quadratic equations by factorisation and completing the square Solve simple quadratic equations by using the quadratic formula LINKS TO LEVEL 2 CONTENT Module 2 Algebraic manipulation (factorising) OBJECTIVES By the end of the module the student should be able to: Solve quadratic equations arising out of algebraic fractions equations Use the discriminant in making assumptions about roots of a quadratic equation Understand relationships relating to the sum and product of roots LINKS TO GCSE SCHEME OF WORK (2-year) A/module 26 B3/module 3-12 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Show how the value of b2 4ac can be useful in determining if the quadratic factorises or not (i.e. square number) Extend to general properties of the discriminant and roots NOTES Some students may need additional help with factorising Students should be reminded that factorisation should be tried before the formula is used In problem-solving, one of the solutions to a quadratic equation may not be appropriate There may be a need to remove the HCF (numerical) of a trinomial before factorising to make the factorisation easier  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 7 Time: 2 4 hours Awards Tier: Level 3 Inequalities 7.1Solve linear inequalities, and quadratic inequalities7.2Represent linear inequalities in two variables on a graph GCSE SPECIFICATION REFERENCES A gSolve linear inequalities in one variable, and represent the solution set on a number line PRIOR KNOWLEDGE Experience of finding missing numbers in calculations The idea that some operations are the reverse of each other An understanding of balancing Experience of using letters to represent quantities Understand and recall BIDMAS Substitute positive and negative numbers into algebraic expressions Rearrange to change the subject of a formula LINKS TO LEVEL 2 CONTENT Module 6 Linear inequalities OBJECTIVES By the end of the module the student should be able to: Solve linear inequalities and quadratic inequalities Change the subject of an inequality including cases where the subject is on both sides of the inequality Show the solution set of a single inequality on a graph Show the solution set of several inequalities in two variables on a graph LINKS TO GCSE SCHEME OF WORK (2-year) A/module 14, 15 B/module 3-6 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Draw several inequalities linked to regions; extend to basic linear programming Quadratic inequalities NOTES Inequalities can be shaded in or out Students can leave their answers in fractional form where appropriate  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 8 Time: 3 5 hours Awards Tier: Level 3 Arithmetic series 8.1Find and use the general term of arithmetic series8.2Find and use sum of an arithmetic series GCSE SPECIFICATION REFERENCES A iGenerate terms of a sequence using term-to-term and position-to-term definitions of the sequenceA jUse linear expressions to describe the nth term of an arithmetic sequence PRIOR KNOWLEDGE Experience of using a letter to represent a number Writing simple rules algebraically LINKS TO LEVEL 2 CONTENT Module 7 Number sequences OBJECTIVES By the end of the module the student should be able to: Generate specific terms in a sequence using the position-to-term and term-to-term rules Find and use the nth term of an arithmetic sequence Derive recurrent formulae to describe a series Investigate the terms of an arithmetic series Find and use the sum of an arithmetic series LINKS TO GCSE SCHEME OF WORK (2-year) A/module 10 B/module 2-12 LINKS TO (C1) GCE MATHEMATICS 3 Sequences and series DIFFERENTIATION & EXTENSION Sequences and nth term formula for triangle numbers, Fibonacci numbers etc Prove a sequence cannot have odd numbers for all values of n Extend to quadratic sequences whose nth term is an2 + bn + c Use of algebraic notation in generating arithmetic series NOTES When investigating linear sequences, students should be clear on the description of the pattern in words, the difference between the terms and the algebraic description of the nth term Use of sigma notation  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 9 Time: 4 6 hours Awards Tier: Level 3 Coordinate geometry 9.1Forms of the equation of a straight line graph9.2Conditions for straight lines to be parallel or perpendicular to each other GCSE SPECIFICATION REFERENCES A kUse the conventions for coordinates in the plane and plot points in all four quadrants, including using geometric informationA lRecognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding gradientsA mUnderstand that the form y = mx + c represents a straight line and that m is the gradient of the line and c is the value of the y-interceptA nUnderstand the gradients of parallel linesA sInterpret graphs of linear functions PRIOR KNOWLEDGE Substitute positive and negative numbers into algebraic expressions Rearrange to change the subject of a formula LINKS TO LEVEL 2 CONTENT Module 5 Graph sketching Module 8 Gradients of straight lines Module 9 Straight line graphs OBJECTIVES By the end of the module the student should be able to: Recognise that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane Plot and draw graphs of straight lines with equations of the form y = mx + c Find the equation of a straight line from two given points Find the equation of a straight line from the gradient and a given point Explore the gradients of parallel lines and lines perpendicular to each other Write down the equation of a line parallel or perpendicular to a given line Use the fact that when y = mx + c is the equation of a straight line then the gradient of a line parallel to it will have a gradient of m and a line perpendicular to this line will have a gradient of  EMBED Equation.DSMT4  Interpret and analyse a straight-line graph and generate equations of lines parallel and perpendicular to the given line LINKS TO GCSE SCHEME OF WORK (2-year) A/module 15 B/module 2-13, 3-8 LINKS TO (C1) GCE MATHEMATICS 2 Coordinate geometry in the (x, y) plane DIFFERENTIATION & EXTENSION Students should find the equation of the perpendicular bisector of the line segment joining two given points 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 to scatter graphs and correlation Cover lines parallel to the axes (x = c and y = c), as students often forget these NOTES Careful annotation should be encouraged; students should label the coordinate axes and origin and write the equation of the line Students need to recognise linear graphs and hence when data may be incorrect Link to graphs and relationships in other subject areas, eg science, geography  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 10 Time: 6 8 hours Awards Tier: Level 3 Graphs of functions 10.1Recognise, draw and sketch graphs of linear, quadratic, cubic, reciprocal, exponential and circular functions, and understand tangents and normals10.2Sketch graphs of quadratic, cubic, and reciprocal functions, considering asymptotes, orientation and labelling points of intersection with axes and turning points10.3Use graphs to solve equations GCSE SPECIFICATION REFERENCES A oFind the intersection points of the graphs of a linear and quadratic function A pDraw, sketch, recognise graphs of simple cubic functions, the reciprocal function y =  EMBED Equation.DSMT4 with x `" 0, the function y = kx for integer values of x and simple positive values of k, the trigonometric functions y = sin x and y = cos x PRIOR KNOWLEDGE Linear functions Quadratic functions LINKS TO LEVEL 2 CONTENT Module 11 Simple quadratic functions OBJECTIVES By the end of the module the student should be able to: Plot and recognise cubic, reciprocal, exponential and circular functions Understand tangents and normal Understand asymptotes and turning points Find the values of p and q in the function y = pqx given the graph of y = pqx Match equations with their graphs and sketch graphs Recognise the characteristic shapes of all these functions LINKS TO GCSE SCHEME OF WORK (2-year) A/module 15, 31 B/module 3-14 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION 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) This work should be enhanced by drawing graphs on graphical calculators and appropriate software Complete the square for quadratic functions and relate this to transformations of the curve y = x2 NOTES Graphical calculators and/or graph drawing software will help to underpin the main ideas in this unit Link with trigonometry and curved graphs  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 11 Time: 1 3 hours Awards Tier: Level 3 Graphs of simple loci 11.1Construct the graphs of simple loci eg circles and parabolas GCSE SPECIFICATION REFERENCES A qConstruct the graphs of simple loci PRIOR KNOWLEDGE: Substitution into expressions/formulae Linear functions and graphs LINKS TO LEVEL 2 CONTENT Module 11 Simple quadratic functions OBJECTIVES By the end of the module the student should be able to: Construct the graphs of simple loci including the circle x + y= r for a circle of radius r centred at the origin of the coordinate plane Find the intersection points of a given straight line with a circle graphically Select and apply construction techniques and understanding of loci to draw graphs based on circles and perpendiculars of lines LINKS TO GCSE SCHEME OF WORK (2-year) A/module 31 B/module 3-13, 3-14 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Find solutions to equations of the circular functions y = sin x and y = cos x over more than one cycle (and generalise) This work should be enhanced by drawing graphs on graphical calculators and appropriate software NOTES Emphasise that inaccurate graphs could lead to inaccurate solutions; encourage substitution of answers to check they are correct Graphical calculators and/or graph drawing software will help to underpin the main ideas in this unit Link with trigonometry and curved graphs  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 12 Time: 2 4 hours Awards Tier: Level 3 Distance-time and speed-time graphs 12.1Draw and interpret distance-time graphs and speed-time graphs12.2Understand that the gradient of a distance-time graph represents speed and that the gradient of a speed-time graph represents acceleration12.3Understand that the area under the graph of a speed-time graph represents distance travelled GCSE SPECIFICATION REFERENCES A pDraw, sketch, recognise graphs of simple cubic functions, the reciprocal function y =  EMBED Equation.DSMT4 with x `" 0, the function y = kx for integer values of x and simple positive values of k, the trigonometric functions y = sin x and y = cos xA rConstruct linear functions from real-life problems and plot their corresponding graphsA sInterpret graphs of linear functions PRIOR KNOWLEDGE A basic understanding of speed Interpret the slope of a graph as its gradient Interpret the gradient within a real life context LINKS TO LEVEL 2 CONTENT Module 12 Distance-time and speed-time graphs OBJECTIVES By the end of the module the student should be able to: Draw distance-time and speed-time graphs Interpret distance-time and speed-time graphs Understand that the gradient of a distance-time graph represents speed Understand that the gradient of a speed-time graph represent acceleration Calculate speed using distance-time graphs and acceleration using speed-time graphs Understand that the area under the graph of a speed-time graph represents distance travelled LINKS TO GCSE SCHEME OF WORK (2-year) A/module 15 B/module 2-13, 3-8 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Draw distance-time graphs of journeys of several stages Consider the link between distance-time and speed-time graphs NOTES Consider the importance of understanding zero gradient in both types of graph Use of different scales in accurate reading and drawing of graphs Accuracy of plotting is important Calculating the gradient from given information regarding speed and/or acceleration is worth practising  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 13 Time: 3 5 hours Awards Tier: Level 3 Direct and inverse proportion 13.1Set up and use equations to solve word and other problems using direct and inverse proportion and relate algebraic solutions to graphical representations of the equations GCSE SPECIFICATION REFERENCES N nUnderstand and use direct and indirect proportionN qUnderstand and use number operations and the relationships between them, including inverse operations and hierarchy of operationsA uDirect and indirect proportion PRIOR KNOWLEDGE Fractions Deriving algebraic formulae LINKS TO LEVEL 2 CONTENT Module 3 Formulae (Derivation of) OBJECTIVES By the end of the module the student should be able to: Solve word problems about ratio and proportion Calculate an unknown quantity from quantities that vary in direct or inverse proportion Set up and use equations to solve word and other problems involving direct proportion or inverse proportion and relate algebraic solutions to graphical representation of the equations LINKS TO GCSE SCHEME OF WORK (2-year) A/module 18 B/module 3-10 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Harder problems involving multi-stage calculations Relate ratios to Functional Elements situations, eg investigate the proportions of the different metals in alloys or the new amounts of ingredients for a recipe for different numbers of guests Harder problems involving multi-stage calculations NOTES A statement of variance is a pre-cursor to writing a formula A formulae describing the proportionality is required in all cases, even if not asked for  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 14 Time: 3 5 hours Awards Tier: Level 3 Transformations of functions 14.1Apply to the graph of y = f(x) transformations of y = f(x) a, y = f(ax), y = f(x a), y = af(x) for any function in x GCSE SPECIFICATION REFERENCES A vTransformation of functions PRIOR KNOWLEDGE Transformations Using f(x) notation Graphs of simple functions LINKS TO LEVEL 2 CONTENT None OBJECTIVES By the end of the module the student should be able to: Apply to the graph of y = f(x) the transformations of y = f(x) a, y = f(ax), y = f(x a), y = af(x) for linear, quadratic, sine and cosine functions Select and apply the transformations of reflection, rotation, stretch, enlargement and translation of functions expressed algebraically Interpret and analyse transformations of functions and write the functions algebraically LINKS TO GCSE SCHEME OF WORK (2-year) A/module 32 B/module 3-16 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION 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 Investigate simple relationships such as sin(180 x) = sin x and sin(90 x) = cos x NOTES Make sure students understand the notation y = f(x), start by comparing y = x2 with y = x2 + 2 before mentioning y = f(x) + 2 etc Graphical calculators and/or graph drawing software will help to underpin the main ideas in this unit Link with trigonometry and curved graphs  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 15 Time: 1 3 hours Awards Tier: Level 3 Area under a curve 15.1Find the area under a curve using the trapezium rule GCSE SPECIFICATION REFERENCES A pDraw, sketch, recognise graphs of simple cubic functions, the reciprocal function y =  EMBED Equation.DSMT4 with x `" 0, the function y = kx for integer values of x and simple positive values of k, the trigonometric functions y = sin x and y = cos x PRIOR KNOWLEDGE Sketching graphs Plotting graphs LINKS TO LEVEL 2 CONTENT None OBJECTIVES By the end of the module the student should be able to: Interpret the area under a curve Use the trapezium rule to find the area under a curve LINKS TO GCSE SCHEME OF WORK (2-year) A/module 26, 31 B/module 3-12, 3-14 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION This work should be enhanced by drawing graphs on graphical calculators and appropriate software NOTES Careful annotation should be encouraged; students should label the coordinate axes and origin and write the equation of the line Students need to recognise linear graphs and hence when data may be incorrect Link to graphs and relationships in other subject areas,  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Module 16 Time: 3 5 hours Awards Tier: Level 3 Surds 16.1Use and manipulate surds, including rationalising the denominator of a fraction written in the form  EMBED Equation.DSMT4  GCSE SPECIFICATION REFERENCES N eUse index notation for squares, cubes and powers of 10N fUse index laws for multiplication and division of integer, fractional and negative powersN qUnderstand and use number operations and the relationships between them, including inverse operations and hierarchy of operationsN vUse calculators effectively and efficientlyN rCalculate with surdsA cSimplify expressions using rules of indices PRIOR KNOWLEDGE Knowledge of squares, square roots, cubes and cube roots Fractions and algebra Rules of indices LINKS TO LEVEL 2 CONTENT Module 2 Algebraic manipulation (Indices) OBJECTIVES By the end of the module the student should be able to: Find the value of calculations using indices Rationalise the denominator, eg  EMBED Equation.3  =  EMBED Equation.3 , and eg write ((18 + 10)((2 in the form p+q(2 Write (8 in the form 2(2 LINKS TO GCSE SCHEME OF WORK (2-year) A/module 27 B/module 2-11 LINKS TO (C1) GCE MATHEMATICS 1 Algebra and functions DIFFERENTIATION & EXTENSION Use index laws to simplify algebraic expressions Treat index laws as formulae (state which rule is being used at each stage in a calculation) Explain the difference between rational and irrational numbers as an introduction to surds Prove that (2 is irrational Revise the difference of two squares to show why we use it, for example ((3 2) as the multiplier to rationalise ((3 + 2) Link to work on circle measures (involving ) and Pythagoras calculations in exact form NOTES Link simplifying surds to collecting like terms together, eg 3x + 2x = 5x, so therefore 3(5 + 2(5 = 5(5 Stress it is better to write answers in exact form, eg  EMBED Equation.3  is better than 0.333333 Useful generalisation to learn (x  EMBED Equation.3  (x = x  HYPERLINK \l "_Level_3_course" Back to OVERVIEW Level 3 concepts and skills What students need to learn: The content of the Level 2 Award in Algebra is assumed knowledge and this content may be assessed in the Level 3 award. Topic Concepts and skills1. Roles of symbolsDistinguish between the roles played by letter symbols in algebra using the correct notation, and between the words equation, formula, identity and expression2. Algebraic manipulationMultiply two linear expressions Factorise expressions including quadratics and the difference of two squares, taking out all common factors Use index laws to include fractional and negative indices Simplify algebraic fractions Complete the square in a quadratic expression3. FormulaeSubstitute numbers into formulae Change the subject of a formula4. Simultaneous equationsSolve simultaneous equations in two unknowns, where one may be quadratic, where one may include powers up to 25. Quadratic equationsSolve quadratic equations by factorisation or by using the formula or by completing the square Know and use the quadratic formula6.Roots of a quadratic equationUnderstand the role of the discriminant in quadratic equations Understand the sum and the product of the roots of a quadratic equation7. InequalitiesSolve linear inequalities, and quadratic inequalities Represent linear inequalities in two variables on a graph8.Arithmetic seriesFind and use the general term of arithmetic series Find and use sum of an arithmetic series9.Coordinate geometryForms of the equation of a straight line graph Conditions for straight lines to be parallel or perpendicular to each other10.Graphs of functionsRecognise, draw and sketch graphs of linear, quadratic, cubic, reciprocal, exponential and circular functions, and understand tangents and normals Sketch graphs of quadratic, cubic, and reciprocal functions, considering asymptotes, orientation and labelling points of intersection with axes and turning points Use graphs to solve equations11.Graphs of simple lociConstruct the graphs of simple loci eg circles and parabolas12.Distance-time and speed-time graphsDraw and interpret distance-time graphs and speed-time graphs Understand that the gradient of a distance-time graph represents speed and that the gradient of a speed-time graph represents acceleration Understand that the area under the graph of a speed-time graph represents distance travelled13.Direct and inverse proportionSet up and use equations to solve word and other problems using direct and inverse proportion and relate algebraic solutions to graphical representations of the equations14.Transformations of functionsApply to the graph of y = f(x) transformations of y = f(x) a, y = f(ax), y = f(x a), y = af(x) for any function in x15.Area under a curveFind the area under a curve using the trapezium rule16.SurdsUse and manipulate surds, including rationalising the denominator of a fraction written in the form  EMBED Equation.DSMT4  Resources Table Level 2 Module numberTitle ResourcesWeb resources1Roles of symbols2Algebraic manipulation3Formulae4Linear equations5Graph sketching6Linear inequalities7Number sequences8Gradients of straight line graphs9Straight line graphs10Graphs for real-life situations11Simple quadratic functions12Distance-time and speed-time graphs Level 3 Module numberTitle ResourcesWeb resources1Roles of symbols2Algebraic manipulation3Formulae4Simultaneous equations5Quadratic equations6Roots of a quadratic equation7Inequalities8Arithmetic series9Coordinate geometry10Graphs of functions11Graphs of simple loci12Distance-time and speed-time graphs13Direct and inverse proportion14Transformation of functions15Area under a curve16Surds Edexcel, a Pearson company, is the UKs largest awarding body, offering academic and vocational qualifications and testing to more than 25,000 schools, colleges, employers and other places of learning in the UK and in over 100 countries worldwide. Qualifications include GCSE, AS and A Level, NVQ and our BTEC suite of vocational qualifications from entry level to BTEC Level 2 National Diplomas, recognised by employers and Level 2 education institutions worldwide. We deliver 9.4 million exam scripts each year, with more than 90% of exam papers marked onscreen annually. As part of Pearson, Edexcel continues to invest in cutting-edge technology that has revolutionised the examinations and assessment system. This includes the ability to provide detailed performance data to teachers and students which help to raise attainment. Acknowledgements This document has been produced by Edexcel on the basis of consultation with teachers, examiners, consultants and other interested parties. Edexcel would like to thank all those who contributed their time and expertise to its development. &';=efsu  - 9 l m n o p q r }  qgc[cjhUhjhCJUjh? 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