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Edexcel A-level Mathematics Year 1 and AS Scheme of Work © HarperCollinsPublishers Ltd 2017 Scheme of Work Edexcel A-level Mathematics Year 1 and AS This course covers the requirements of the first year of the Edexcel AS and A-level Mathematics specification. This scheme of work is designed to accompany the use of Collins’ Edexcel A-level Mathematics Year 1 and AS Student Book, and references to sections in that book are given for each lesson. The scheme of work follows the order of the Student Book, which broadly corresponds to the Edexcel 2017 specification order: this is not a suggested teaching order but is designed for flexible planning. Each chapter from the Student Book is divided into one-hour lessons. We have assumed that approximately 120 one-hour lessons are taught during the year, however lessons can be combined as you choose depending on your timetabling and preferred teaching order, and we have left some free time for additional linking, problem-solving or investigation work and assessments. We have suggested at the start of each chapter which chapters it would be beneficial to have already covered, to get the most out of the materials. In addition, we have brought out links and cross-references between different topics in column 4 to assist you with planning your own route through the course and making synoptic connections between topics for students. Chapters 1 – 11 cover the pure mathematics content that is tested on AS paper 1 / A-level papers 1 & 2. Chapters 12 – 14 cover the statistics content that is tested on AS paper 2 / A-level paper 3. Chapters 15 – 16 cover the mechanics content that is tested on AS paper 2 / A-level paper 3. Each lesson is matched to the specification content and the corresponding Student Book section with all relevant exercises and activities listed. Learning objectives for each lesson are given, and a brief description of the lesson’s coverage to aid with your planning. A review lesson with assessment opportunities is plotted at the end of each chapter, with exam-style practice available from the Student Book and our Collins Connect digital platform. At the start of each chapter, we have suggested opportunities for you to use technology in the classroom. These ideas are developed further in the Student Book with in-context Technology boxes. In addition, you’ll find activities in the Student Book to explore the Edexcel-provided large data set using a range of technologies. These are clearly indicated in the scheme of work, and can be flexibly adapted to be the focus of class, group or individual work. The scheme suggested is of course flexible and editable to correspond with your timetabling and to enable you to plan your own route through the course. We hope that you will find this a useful aid to your planning and teaching. KEY Ex – exercise DS – Data set activity (Statistics chapters only) © HarperCollinsPublishers Ltd 2017 Scheme of Work Edexcel A-level Mathematics Year 1 and AS (120 hours) CHAPTER 1 – Algebra and functions 1: Manipulating algebraic expressions (8 Hours) Prior knowledge needed CHAPTER 2 – Algebra and functions 2: Equations and inequalities (8 Hours) GCSE Maths only Prior knowledge needed Technology Chapter 1 Use the factorial and combinations button on a calculator. Technology Onetohour lessons objectives Specification content Topic links Use a calculator check solutions to quadratic equationsLearning that have been solved by hand. Use a graphing software to plot lines when solving simultaneous equationsalgebraically so that studentsAS can see that wherePapers the lines is the solution. Paper 1/A-level 1 &intersect 2: Links back to GCSE 1 Manipulating polynomials algebraically Manipulate polynomials Use graphing software to show the solutions to quadratic inequalities. 1.1 – mathematical proof including: Students collect like terms with 2.6 – manipulate polynomials o expanding brackets and simplifying One include hour lessons Learning objectives Specification content Topic links expressions that fractional algebraically by collecting like terms are expected to write AS Paper 1/A-level Papers 1 & 2: Make connections back to 1coefficients Quadratic and functions Work with quadratic functions and expressions with terms written in 2.3 – work with quadratics factorising in 1.4 their graphs Students draw andofwork outThey key expand aspects a functions and their graphs Factorisation descending order index. of quadratic functions, theyand alsoexpand work out single term over a bracket equations of given graphs. two binomials. Theyquadratic use the same Make students aware that the techniques to go on to expand notation three f(x) maybinomials. be used. simple Consider the multiple difference between drawing 2 Expanding binomials and sketching. Students are introduced to Pascal’s 2triangle The discriminant quadratic and use thisoftoa expand multiple function binomials. One check students can use when expanding multiple binomials is to Students are reminded that the 2 the indices in each ensure the sum of discriminant is 𝑏 – 4𝑎𝑐 and that the value term the total of the index what of thetype of of theisdiscriminant indicates original expansion. Students use factorial roots and number of solutions a quadratic notation when solve problems involving has. Link this to how students will sketch combinations. quadratics in the next chapter. Students need to make sure they have a quadratic © HarperCollinsPublishers Ltd 2017 Understand and use the notation 𝑛! and 𝑛𝐶𝑟 Calculate and use the discriminant of a quadratic function, including the conditions for real and repeated roots AS Paper 1/A-level Papers 1 & 2: 2.6 – manipulate polynomials algebraically AS Paper 1/A-level Papers 1 & 2: 4.1 –– work binomial 2.3 withexpansion the discriminant Links to Chapter 13 Probability and statistical distributions: 13.1 The Links to 3.1 Sketching binomial distribution curves of quadratic of a quadratic function, including the conditions for real and repeated roots functions Student Book references 1.1 Ex 1.1A Student Book references 2.1 Ex 2.1A 1.2 Ex 1.2A 2.2 Ex 2.2A Ex 1.2B AS Paper 1/A-level Papers 1 & 2: The binomial expansion is in 3 The the binomial form 𝑎𝑥 2 expansion + 𝑏𝑥 + 𝑐 = 0 before they Understand and use the binomial 2.6 – manipulate polynomials used in Chapter 13 expansion of (𝑎 + 𝑏)𝑛 calculate the discriminant. Students are the binomial CHAPTER 3 – introduced Algebra andtofunctions 3: Sketching curves (9 Hours) algebraically Probability and statistical expansion formula as a way of solving any 4.1 – binomial expansion distributions for binomial Prior knowledge needed binomial2expansion, they should state distributions Chapter formulathe they are using at the start AS Paper 1/A-level Papers 1 & 2: 5.1 Equations of circles 3which Completing square Complete the square Technology of any answer. 1.1 – mathematical proof uses completing the square Graphing software can be heavily used in this chapter; some examples are below. Students complete the to square the 2.3 – complete the square when solving problems Use graphing software checkonanswers. general expression of ato quadratic. They go of inflection and see what happens to the gradient of the curve each side of the pointinvolving circles Use graphing software zoom into a point of inflection. ontographing solving quadratic equations byhow to graphs behave. Links to 3.1 Use software to compare AS Paper 1/A-level Papers 1 & 2: Factorisationsketching is used in 4 Factorisation Manipulate polynomials algebraically completing the square.to look at approximate coordinates of turning points. curves Use graphing software 2.6 – manipulate polynomials Chapterof2quadratic Algebra and including: Students factorise different sorts of functions algebraically functionsTopic 2: Equations and o factorisation The general formula for solving a They One hour lessons Learning objectives Specification content links expressions including quadratics. inequalities when working quadratic equation can be of derived by factorise a difference twofunctions squares. AS Paper 1/A-level Papers 1 & 2: Builds on from equations Chapter 2 1also Sketching curves of quadratic Use and understand the graphs of with quadratic completing the square on the generalto Some questions are given in relation 2.3 – work with quadratic Algebra and functions 2: functions and in Chapter 3 Algebra form 𝑦= 𝑎𝑥 2 +curves 𝑏𝑥 𝑐of– quadratic this Students sketch real life problems to+ show howis question functions and their graphs Equations and3: inequalities and functions Sketching Sketch curves defined by simple 6factorisation in Ex 2.4A. used. equations andis find quadratic equations 2.7 – understand and use graphs curves quadratic equations from sketches knowing that a sketch of functions; sketch curves Algebraic division equations Link ability algebraic division to 45 Solving quadratic AS Paper 1/A-level Papers 1 & 2: The to find solutions Manipulate polynomials Solve quadratic equationsalgebraically using shows the curve along with its key defined by simplepolynomials equations 2.6 – mathematical manipulate numerical division 1.1 proof will help you solve oincluding: completing the square features. Theare keythen features being; All the skills combined write algebraically, including expanding Students use factorisation, usingtothe 2.3 – solve quadratic equations distance, speed and time o the expanding formulabrackets and simplifying maximum or minimum turning point and algebraiccompleting fractions inthe their simplest brackets, collecting like terms, formula, square andform. problems in Chapter 15 by collecting like terms o factorisation its coordinates, the quadratic roots with x Students work through the factor factorisation and simple algebraic calculators to solve equations, Kinematics and in Chapter o factorisation intercept/s if appropriate, the y intercept. theorem and the remainder theorem. division; use of the factor theorem they consider advantages and 6 Trigonometry and o simple algebraic division Students then divide disadvantages of eachpolynomials method. by an Chapter 7 Exponentials o use of the factor theorem AS Paper 1/A-level Papers 1 & 2: Builds on from Chapter 2 2expression Sketching–curves ofclear cubictofunctions make students that Use and understand the graphs of It’s important not toit be seen as a discrete and logarithms 2.7 - understand and use graphs Algebra and functions 2: functions theyofuse the same method as they do for set skills but providing different Link to graph drawing, 3.1 Students consider how the cubic functions of functions; sketch curves Equations and inequalities Sketch curves defined by simple cubic numerical long division. information about the same equation. Sketching curves of behaves as it tends to positive and defined by simple equations equations quadratic functions 6 Laws of indices, manipulating surds AS Paper 1/A-level Papers 1 & 2: Builds on GCSE skills Understand and use the laws of indices negative infinity, this determines the 2.7 – interpret algebraic solutions 5 Solving simultaneous equations AS Paper 1/A-level Papers 1 & 2: This is used when finding 1.1 – mathematical proof Laws of indices is used in Solve simultaneous equations in two for all rational exponents shape of the curve. The coordinates of the of equations graphically 2.4 simultaneous of7intersection with Students solve questions using the laws of variables 2.1 – solve understand and use the laws points Chapter Exponentials by Use and manipulate surds axes intercepts and point of inflection are Students go through solvingand twofractional linear equations in two variables by circles and straight lines 7.2 in indices, including negative of indices for all rational and logarithms, section o elimination or included on the graph. Students use this simultaneous by elimination elimination Chapter 5 Coordinate indexes. They equations are reminded that surds are exponents and by substitution, Manipulation of indices o by substitution, information sketch curves cubicto and by to substitution. They need including onemanipulate linear and one geometry 2:used Circles rootsthen of rational numbers andofsimplify 2.2 - use and surds, and surds is in including one linear and one quadratic functions. think about which is the best method and quadratic equation the expressions involving surds. including rationalising Chapter 8 Differentiation equation why. denominator and Chapter 9 Integration AS Paper 1/A-level Papers 1 & 2: Builds on from Chapter 2 3 Sketching curves of quartic functions Use and understand the graphs of 7 Rationalising the denominator AS Paper 1/A-leveland Papers 1 & 2: functions Use and manipulate surds, including 2.7 - understand use graphs Algebra and function 2: Students turning points and points of Solve 6 Solving find linear and quadratic AS Paper 1/A-level 1.1functions; – mathematical proof simultaneous equations in quartic two rationalising denominator of sketchPapers curves1 & 2: Equations and inequalities Sketch curvesthe defined by simple simultaneous variables by intersection toequations sketch curves of quartic defined by simple equations equations © HarperCollinsPublishers Ltd 2017 1.3 Ex 1.3A 2.3 Ex 2.3A 1.4 Ex 1.4A Student Book references 3.1 Ex 3.1A 1.5 2.4 Ex 1.5A 2.4A 3.2 Ex 3.2A Ex 3.2B 1.6 2.5 1.7 Ex 2.5A 1.6A Ex 2.5B 1.7A 3.3 1.8 Ex 3.3A 2.6 Ex Ex 1.8A 3.3B Ex 2.6A Recap manipulating and complete o elimination or functions. They look surds at maximum and Interpret the algebraic solution of Student Ex 1.7A ifsolve required linear and andthen quadratic go o by substitution, equations graphically minimum turning points andstudents then at the simultaneous on to of rationalise equations the denominator considering ofwhich including one linear and one quadratic roots the equation. A typical example method algebraic is fractions. the most appropriate to use. equation of a quartic to be sketched is 𝑦 = 2Chapter review 2 8 lesson Identify areas of strength and 𝑥 (2𝑥 − 1) . 7 Solving linear and quadratic development within the content of the Solve linear and quadratic inequalities inequalities Recap key points ofof chapter. Students to 4 Sketching curves reciprocal functions chapter a single variable and interpret such inUse and understand the graphs of complete the exam-style questions, inequalities functions graphically, including extensionsketch solve questions, inequalities, or of online useend-ofset of Students graphs reciprocals with brackets fractions inequalities Sketch curves defined by and simple 𝑎 𝑎 notation chapter test to list integer members of two or Express solutions through correct use reciprocals linear 𝑦 = and quadratic 𝑦 = 2 𝑥 𝑥 more combined inequalities andto the Assessment ‘and’ and or through set of of Interpret the‘or’, algebraic solution functions. They are introduced represent inequalities Strict Exam-style questions 1graphically. in Student Book notation equations graphically word discontinuity and know that this is a inequalities are aindotted Exam-style Student Book point whereextension therepresented valuequestions of x orwith y is1not line, inclusive inequalities are represented End-of-chapter testare 1 on Collins Connect defined. They also told that an with a solidisline. asymptote a straight line that a curve Link to solving drawing approaches butequations, never meets, thesegraphs should and looking at inequalities from the They be clearly highlighted on the graph. sketch of the a graph. also give points of interception on the 8 Chapter review lesson Identify areas of strength and axis. development within theofcontent of the 5 Intersection points Use intersection points graphs to Recap key points of chapter. Students to chapter solve equations complete the exam-style questions, Students sketch more than one function extension or the online end-ofon a graphquestions, to work out intersection chapter test points of the functions and link this to Assessment solve an equation which includes these Exam-styleThey questions 2 in Student Book functions. use the information they Exam-style questions 2 in in Student have learnt extension for the previous section this Book End-of-chapter 2 on Collins Connect exercise to solvetest these problems. 6 Proportional relationships Students express the relationship between two variables using the proportion symbol and then find the equation between the two variables. They also recognise and use graphs that show proportion. © HarperCollinsPublishers Ltd 2017 Understand and use proportional relationships and their graphs 2.4 2.2 –- use solve and simultaneous manipulate surds, 2.7 interpret algebraic solutions equations including rationalising in two variables the by of equations graphically elimination denominator and by substitution, including one linear and one quadratic equation AS Paper 1/A-level Papers 1 & 2: 2.5Paper - solve1/A-level linear and quadratic AS Papers 1 & 2: inequalities in a single variable 2.7 - understand and use graphs andfunctions; interpretsketch such inequalities of curves graphically, including inequalities defined by simple equations 𝑎 with brackets and fractions including polynomials 𝑦 = and 𝑥 𝑎 2.5 - express solutions through 𝑦 = 2 (including their vertical 𝑥 use of ‘and’ and ‘or’, or correct and horizontal asymptotes) through set notation This is used in Chapter 5 Coordinate geometry 2: Graph sketching skills are Circles when solving further developed in problems7involving Chapter Exponentials equations of circles and logarithms See below 2.7 Ex 2.7A 3.4 Ex 3.4A 2.83.4B Ex Ex 2.8A See below AS Paper 1/A-level Papers 1 & 2: 2.7 interpret algebraic solution of equations graphically; use intersection points of graphs to solve problems Links back to 2.5 Solving simultaneous equations 3.5 Ex 3.5A AS Paper 1/A-level Papers 1 & 2: 2.7 – understand and use proportional relationships and their graphs Links back to GCSE 3.6 Ex 3.6A AS Paper 1: Translations and stretches 3.7 Understand the effects of simple 2.8 – effects of simple arise in 6.5 solving Ex 3.7A transformations on the graphs of CHAPTER 4 – Coordinate 5 geometry 1: Circles 2: Equations (6 Hours) lines (5 𝑦 Hours) Students transform functions and sketch transformations including trigonometric equations 𝑦of=straight f(𝑥), including = f(𝑥) + 𝑎 and the graphs using the two sketching associated graphs 𝑦 = f(𝑥 + 𝑎) Priorresulting knowledge needed transformations identified in the learning Chapter 1 2 objective. A-level Papers 1 & 2: Chapter 4 Technology 2.9 – effects of simple Use graphing software to check answers. Technology transformations to investigate location as the varies both positive and negative and integer and fractional values. Use graphing software draw 𝑥 2 + 𝑦 2 the = 25 and 𝑥 2of+the 𝑦 2 x-intercept = 36 to show howgradient the centre of awith circle is always (0,including 0) and the radius is the square root of the number on the rightsketching associated graphs Use graphing software to find equations of lines that are parallel. hand side of the equation. Get students to predict what other circles will look like given an equation of a circle in this form. 8 Stretches AS Paper 1:not about Translations and stretches circles Understand the effects of simple Use graphing software thatatintersect but are not perpendicular. What do the students notice the products of their gradients? Graphing software can to beplot usedlines to look other (still with the equation written in this form but with centre (0, 0) ) and link the equation of the circle 3.8 to the 2.8 – effects of simple arise in 6.5 solving Ex 3.8A Book transformations on the graphs of coordinates of the centre and length of radius. Student One hour lessonsand sketch Learning objectives Specification content Topic links Students transform functions transformations including trigonometric equations 𝑦 = f(𝑥), including 𝑦 = 𝑎f(𝑥) and Use graphing software to find midpoints of line segments. references the resulting the1 two sketching associated graphs# 𝑦 = f(𝑎𝑥). Use graphing software plot a circle and tangent. AS Paper 1/A-level Papers 1 & 2: 4.2 and 4.3 link to Chapter 4.1 1 Straight linegraphs graphsusing –topart Write the equation of a straight line in transformations identified in the learning Student 3.1 – understand and use the 15 Kinematics Ex 4.1A Book the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 One hour lessons of straight Learning objectives Specification Topic links Students equations objective.rearrange A-level Papers 1 & 2:content references equation of a straight line, Find the equation of a straight line lines to the form 𝑎𝑥 +–𝑏𝑦 + 𝑐1 = 0 2.9Paper – effects simple including theof forms 4.2 AS 1/A-level Papers 1 & 2: Links back to GCSE 5.1 1 Equations of circles part through point withcircle a given Find the one centre of the andgradient the transformations including 𝑦– 𝑦 = 𝑚(𝑥– 𝑥 ) and 4.2A 3.2 – understand and use the Ex 5.1A using the radius from formula the equation of a circle 1 1 Students find equations the equation of a line Students link of circles to given sketching associated graphs 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 coordinate geometry of the circle 𝑦– 𝑦1 the = 𝑚(𝑥– 𝑥1 ) of a circle Write equation the gradient and alesson point on learn the line Pythagoras’ theorem. They theand 9 Chapter review See below Identify areas of strength and including using the equation of a solve problems using general equation of a this. circle and solve development within the content of the circle in the form Recap key points of chapter. Students chapter problems. They find the equation of a to (𝑥– 𝑎)2 + (𝑦– 𝑏)2 = 𝑟 2 2 Straight line graphs – part 2 AS 1/A-level 2: Graph sketching skills are 4.3 Find the gradient of the straight line complete the exam-style questions, circle from its centre and a point on the 3.2Paper – complete the Papers square 1to&find 3.1 – understand and use the further developed in Ex 4.3A between two points extension questions, end-ofthe centre and radius of a circle circumference. Givenor anonline equation of a Students find the gradient given two equation of a straight line, Chapter 7 Exponentials Find the equation of a straight line chapter test find the centre and radius circle students 𝑦–𝑦1 𝑥–𝑥1 points on the line and solve problems including the forms and logarithms 4.4 using the formula = Assessment by completing the square on 𝑥 and 𝑦. 𝑦2–𝑦1 𝑥2–𝑥1 using this. 𝑦– 𝑦 = 𝑚(𝑥– 𝑥 ) and Ex 4.4A 1 1 Exam-style questions 3 in Student Book 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 Exam-style extension questions 3 in Student Book Given two points on3aon students use AS Paper 1/A-level Papers 1 & 2: 5.1 2 Equations of circles –line part 2 Connect Find the centre of the circle and the End-of-chapter test Collins the formula to find the equation of the 3.2 – understand and use the Ex 5.1B radius from the equation of a circle Recap straightlesson line. one and then students find coordinate geometry of the circle Write the equation of a circle the equation a circle from the including using the Papers equation 3 Parallel andofperpendicular linesends of a Understand and use the gradient AS Paper 1/A-level 1 &of2:a Links back to GCSE 4.5 diameter, this involves them using the circle in the form 3.1 – gradient conditions for two Ex 4.5A conditions for parallel lines formula of agradients line (𝑥– 𝑎)2 lines + (𝑦– = 𝑟 2 or Studentsfor usethe themidpoint facts about of straight to𝑏) be2 parallel Ex 4.5B Understand and use the gradient 3.2 – complete the square to find segment. parallel and perpendicular straight lines to perpendicular conditions for perpendicular lines the centre and radius of a circle solve problems. 4 Straight line models AS Paper 1/A-level Papers 1 & 2: This is used in Chapter 5 4.6 Be able to use straight line models in a 3.1 – be able to use straight line Coordinate geometry 2: Ex 4.6A variety of contexts Students apply what they have learnt in models in a variety of contexts Circles when finding a this chapter to solve problems in a variety gradient to solve problems 7 Translations © HarperCollinsPublishers Ltd 2017 ofAngles different contexts including real-world 3 in semicircles Find the equation of a circle given scenarios. different information such as the end A geometric proof is used to show that for points of a diameter three points on the circumference of a Use circle theorems to solve problems circle, if two pointslesson are on a diameter, 5 Chapter review Identify areas of strength and then the triangle is right-angled. Students development within the content of the are alsokey reminded Pythagoras’ Recap points ofthat chapter. Students to chapter theorem that a questions, triangle has a completewill theshow exam-style extension or online end-ofright anglequestions, and gradients can be used to chapter test show that two lines are perpendicular. Assessment 4 Perpendicular from 4the centre toBook a Find the equation of a perpendicular Exam-style questions in Student bisector chord Exam-style extension questions 4 in Student Book Use circle theorems to solve problems End-of-chapter test 4 on Collins Connect A geometric proof is used to show that congruent triangles are identical and then links to show that the perpendicular from the centre of the circle to a chord bisects the chord. 5 Radius perpendicular to the tangent A geometric proof by contradiction is used to show that a radius and tangent are perpendicular at the point of contact. Pythagoras’ theorem is used to find the length of a tangent. Students then go on to consider that for any circle there are two tangents with the same gradient. Given the gradient of the tangent and the equation of the circle they can find both points where the two tangents touch the circle. The second method used it to show that a line is tangent to a circle is that the discriminant will be zero i.e. 𝑏 2 – 4𝑎𝑐 = 0. 6 Chapter review lesson © HarperCollinsPublishers Ltd 2017 Find the equation and length of a tangent Use circle theorems to solve problems Identify areas of strength and development within the content of the chapter AS Paper 1/A-level Papers 1 & 2: 1.1 - mathematical proof 3.2 - use of the following properties: the angle in a semicircle is a right angle Links backtangents involving to GCSE and radii of circles and points of intersections of circles with straight lines AS Paper 1/A-level Papers 1 & 2: 3.2 - use of the following properties: the perpendicular from the centre to a chord bisects to chord Links back to GCSE AS Paper 1/A-level Papers 1 & 2: 3.2 - use of the following properties: the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point 5.2 Ex 5.2A See below 5.3 Ex 5.3A 5.4 Ex 5.4A Ex 5.4B See below Recap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-ofchapter test Assessment Exam-style questions 5 in Student Book Exam-style extension questions 5 in Student Book End-of-chapter test 5 on Collins Connect CHAPTER 6 – Trigonometry (7 Hours) Prior knowledge needed Chapter 2 Chapter 3 Technology Use calculators to check values. Use graphic software to confirm the shapes of graphs. One hour lessons 1 Sine and cosine Students extend their knowledge of sine and cosine beyond 90° to between 0° and 180° by imagining a rod of length one unit on a coordinate grid. 2 The sine rule and the cosine rule Students work through proofs for the sine rule, cosine rule and formula for the area of a triangle and then use these to solve problems. Learning objectives Define sine for any angle Define cosine for any angle Use the sine rule Use the cosine rule Use the formula for the area of a triangle CHAPTER 7 – Exponentials and logarithms (8 Hours) © HarperCollinsPublishers Ltd 2017 Specification content Topic links AS Paper 1/A-level Papers 1 & 2: 5.1 – understand and use the definitions of sine and cosine for all arguments Links back to GCSE AS Paper 1/A-level Papers 1 & 2: 5.1 - the sine and cosine rule 5.1 – the area of a triangle in the 1 form 𝑎𝑏sin𝐶 Links back to GCSE 2 Student Book references 6.1 Ex 6.1A Bearings will be extended further in Chapter 10 Vectors 6.2 Ex 6.2A AS Paper 1: 3 Trigonometric Understand the graph of the sine Prior knowledgegraphs needed 5.2 – Understand and use the sine function and interpret its symmetry Chapter 1 Students start by considering the sine and CHAPTER and cosine functions; their graphs, and periodicity Chapter 2 8 – Differentiation (8 Hours) cosine of3any angle using x and y symmetries and periodicity Chapter Understand the graph of the cosine Prior knowledge needed coordinates 4 of points on the unit circle. function and interpret its symmetry Chapter 1 This is then A-level Papers 1 & 2: and periodicity Chapter 3 linked to the trigonometric Technology graphsplotting and symmetries andthe graphs 𝑦 = 1.2𝑥 . 5.3 – Understand and use the sine Chapter 4 their Graph software to draw and cosine functions; their graphs, periodicity. Use the log, power of e and ln button on a calculator. Technology Use spreadsheet software to calculate values of logarithms. graphical software to find the gradients of a quadratic curve in example 2 in section 8.2.symmetries and periodicity 4 The tangent function AS Paper 1: Define the tangent foralso any need angleto find them Graphical software can be used to look for stationary points – students using differentiation. One hour lessons Learning objectives Specification 5.1 – understand andcontent use the Understand the graph of the tangent One hour lessons Learning objectives Specification content 𝒙 when the tangent Students consider definition of tangent for all1 & 2: function and interpret its symmetry AS Paper 1/A-level Papers 1 The function 𝒂 Use and manipulate expressions in the 𝑥 cannot be calculated and look at the arguments and periodicity 6.1Paper – Know and usePapers the function 1/A-level 1 & 2: 1 The gradient of a curve form 𝑦 = 𝑎 the where the variable x is an AS To consider relationship between Students learn the terms exponential similarities between the tangent graph -– understand Understand and use use the the Usetangent an important formula connecting 𝑎 𝑥 and its graph, where a the is 7.1 and index to a curve and the gradient 5.2 Students areexponential introduced to the concept growth decay. The and theand sine and cosine graphs. Theydraw use tangent graph, sin 𝑥,and cosinterpret 𝑥 at and tangraphs 𝑥 positive functions; derivative of f(𝑥) asitsthe gradient the curve a point of Draw of the form sinƟ 𝑥 that of𝑦a= curve at any symmetry and periodicity 𝑥 graphs theto form 𝑎 and arepoint madeis Introduction tanƟthe =ofgradient solve problems. of the tangent to a graph of 𝑦 = 𝑦=𝑎 of the term derivative cosƟ 5.3 – understand and use equal thethere gradient of the tangent at awareto that is a difference in the f(𝑥) at a general point (𝑥, 𝑦); the Use the graph to find approximate Recognise the derivative function of sinƟas a limit; 2 that point. shape of the graph when a < 1 and a > 1, gradient of the tangent solutions 𝑦 =𝑥 tanƟ = interpretation as a cosƟ rate of change they should understand this.2 Students then look at 𝑦 = 𝑥 and see that A-level Papers 1 & 2: the gradient at any point on the curve is 5.1 – understand and use the 2𝑥. The derivative is then introduced as AS Paper 1/A-level Papers 2 Logarithms Understand and use logarithms and definition of tangent for all1 & 2: the connection between the x coordinate 6.3 – know and use the definition their connection with indices arguments of a pointstart and the curve at Students withgradient the lawsofofthe logarithms 𝑥 of as the inverse of 𝑎 Understand the proof of the laws of 5.3log𝑎𝑥 - Understand and use the, that point. in base 10 and the laws of logarithms are where is positiveits and 𝑥≥0 logarithms tangent𝑎functions; graph, linked to laws of indices with a proof for 6.4 – understand and use symmetry and periodicity AS Paper 1/A-level Papersthe 1 &laws 2: 2 The gradient of a quadratic curve Extend the result for the derivative of each law. They work out basic logarithms of logarithms 2 2 5.5 – sketching understand and use 7.1 the gradient 𝑦 = 𝑥 to the general case 𝑦 = 𝑎𝑥 From exercise 8.2Aastudents areThey lead then to in base 10 without calculator. sinƟ function for a given Be able to differentiate any quadratic 2 tanƟ = curve the general resultthe thatlaws if 𝑦for = 𝑎𝑥 then 𝑛 move onto using logarithms 7.2 – differentiate 𝑥cosƟ , for rational equation 𝑑𝑦 5 Solving trigonometric equations AS Paper 1: = 2𝑎𝑥. After some the examples students of any base and using log𝑎𝑏 button on Solve simple trigonometric equations values of 𝑛, and related constant 𝑑𝑥 5.4 – solvesums simple trigonometric a calculator. multiples, and differences are given the general result when Students thing aabout the multiple equations in a given interval, differentiating quadratic. It is also made 𝒙 AS Paper 1/A-level 1 & in 2: 3 Thethat equation 𝒂should = 𝒃 be equations Solve equations in the form 𝑎 𝑥 = 𝑏 solutions of trigonometric and including quadraticPapers equations clear terms differentiated 6.5 – solve equations in the form Understand a proof of the formula to link this to in the graphs of the function. sin, cos and tan and equations separately quadratic equation. They Students solvea equations of the form 𝑥 𝑎 = 𝑏 multiples of the change the base of a logarithm involving start to sketch curves of the derivative. 𝒙 𝒂 = 𝒃 by taking the logarithms of both unknown angle sides. The work through a proof of the A-level Papers 1 & 2: © HarperCollinsPublishers Ltd 2017 Link back to GCSE 6.3 Ex 6.3A Link to Chapter 4 Topic links 1: Coordinate geometry Topic links Equations Links back of to straight GCSE lines 6.4 Student Book Ex 6.4A Book references Student 7.1references Ex 7.1A 8.1 Ex 8.1A The skills learnt in this chapter are essential when students learn Chapter 9 Integration 8.2 Ex 8.2A The skills learnt in 1.6 Laws of indices are used here 7.2 Ex 7.2A Ex 7.2B 8.2 Ex 8.2B Link this back to transformation of graphs in Chapter 3 Algebra and functions 3 6.5 Ex 6.5A 7.3 Ex 7.3A formula to changeofthe 3 Differentiation 𝒙𝟐 base and 𝒙of𝟑 a Understand a proof that if 𝑓(𝑥) = 𝑥 2 then 𝑓’(𝑥) = 2𝑥 logarithm. Students work through a proof that the derivative of 𝑥 2 is 2𝑥, they are told that using the formula is called differentiation from first principles. 4 Logarithmic graphs Use and manipulate expressions in the 6 A useful formula forman𝑦 important = 𝑘𝑎 𝑥 where the variable x is Use formula connecting Students look at two types of 4 Differentiation of adifferent polynomial an sinindex 𝑥, cos 𝑥 and tan 𝑥 containing Differentiate expressions 𝑥 curve 𝑦 =diagrammatically 𝑎𝑥 𝑛 and 𝑦 = 𝑘𝑏 see . To find the Students that powers graphs to estimate integer Use logarithmic 2 2 parameters the logarithms of sin Ɵ The general + costhey result Ɵ =take 1for and differentiation then use thisis parameters in relationships of the form both sides then rearrange this are identity introduced to and go to on students to solve and trigonometric examples 𝑦 = 𝑎𝑥 𝑛 and 𝑦 = 𝑘𝑏 𝑥 , given data for 𝑥 equation into theThey formthen 𝑦 = go 𝑚𝑥on + to 𝑐. In the equations. worked through. and 𝑦 first caseatthey plot log 𝑦 against looking differentiating the sumlog or𝑥 and obtain straight linetheorem. where the intercept difference of terms. Link to aPythagoras’ is log 𝑎 and the gradient 5 Differentiation 𝒙𝒏 is 𝑛 and in the 7 Chapter reviewof lesson Differentiate Identify areasexpressions of strengthcontaining and second case they plot log 𝑦 against 𝑥 and powers development of variables within the content of the obtain a straight where the intercept Recap key Students move points online oftochapter. work with Students powers to chapter is logare 𝑘 and the gradient isquestions, log complete that not the integer exam-style values and𝑏.variables 5 The than number e y. Differentiation other x and is again Know that the gradient of 𝑒 𝑘𝑥 is equal extension questions, or online end-ofchaptertotest linked being the gradient of the graph to 𝑘𝑒 𝑘𝑥 and hence understand why the Students lookofatchange. graphs of the form and the rate exponential model is suitable in many Assessment 𝑦Stationary = 𝑎 𝑥 . and see the is always Exam-style 6 questions points and 6gradient inthe Student second Book a applications Determine geometrical properties of a multiplier 𝑦. When the multiplier one Book Exam-stylex extension derivative questions 6 inisStudent curve then the function is 6ofon theCollins form 𝑦 = 𝑒𝑥 End-of-chapter test Connect and go on to generalise 𝑦 = 𝑒 𝑎𝑥 Differentiation is used tothat findfor stationary 𝑎𝑥 𝑑𝑦 for any then thei.e. gradient = 𝑎𝑒 = 𝑎𝑦 points, the point where = 0. value of 𝑎. Students should 𝑑𝑥 realise that Students are shown that the second when the rate of change is proportional to derivative will show if a turning point is a the 𝑦 value, an exponential model should maximum or minimum point so there is be used. Insure that students include the no need to draw𝑎𝑥the graph to ascertain graphs of 𝑦 = 𝑒 + 𝑏 + 𝑐. this information. 6 Natural logarithms Know and use the function ln𝑥 and its graphs 7 Tangents and normal Find equations of tangents and normal Students are introduced to the fact that using Know differentiation and use ln𝑥 as the inverse ln𝑦 means to the 𝑒 and function of 𝑒 𝑥 The normallogarithm at any point on abase curve is that if 𝑒 𝑥 = 𝑦 then = 𝑥. The graphs of perpendicular to theln𝑦 tangent at that point. 𝑥 𝑦 =the ln𝑥product and 𝑦 = show students As of 𝑒their gradients is that they are a reflection of each other in the line 𝑦 = 𝑥 due to the fact they are inverse © HarperCollinsPublishers Ltd 2017 AS Paper 5.7 – solve1/A-level simple trigonometric Papers 1 & 2: equations 7.1 – differentiation in a given from interval, first including quadratic principals for small positive equations in sin, cos powers integer and tan of and 𝑥 equations 𝑛 involving 7.2 - differentiate multiples𝑥of , the for rational unknown values of 1/A-level 𝑛, angle and related constant AS Paper Papers 1 & 2: multiples, differences 6.3 – know and and use the definition AS Paper 1:sums of as inverse of sin 𝑎1𝑥& ,2 Ɵ2:+ 5.3log𝑎𝑥 AS Paper – understand 1: the A-level and Papers use 2 𝑛 where 𝑎 ≥0 cos - Ɵ 7.2 differentiate = is1 positive𝑥and , for𝑥 rational 6.6 – use graphs to values of logarithmic 𝑛 , and related constant A-level Papers & 2:differences estimate parameters in multiples, sums1 and 5.5 – understand and use sin relationships of the form 𝑦 =2 Ɵ𝑎𝑥+𝑛 2 𝑥 cos 𝑦Ɵ = and = 𝑘𝑏 1 , given data for 𝑥 and 𝑦 AS Paper 1/ A-level Papers 1 & 2: 7.2 - differentiate 𝑥 𝑛 , for rational values of 𝑛, and related constant multiples, sums and differences AS Paper 1/A-level Papers 1 & 2: 6.1 – Know and use the function 𝑒 𝑥 and its graph 6.2Paper – Know thePapers gradient AS 1: that A-level 1 &of2: 𝑒 𝑘𝑥 –issecond equal to 𝑘𝑒 𝑘𝑥 and hence 7.1 derivatives understand why the 7.1 – understand andexponential use the model is suitable in many second derivative as the rate of applications change of gradient 7.3 – apply differentiation to find gradients, tangents and normal, maxima and minima and stationary points 7.3 – identify where functions are AS Paper 1/A-level Papers 1 & 2: increasing and decreasing 6.3Paper – know use Papers the function AS 1:and A-level 1 & 2: ln𝑥 –and its graph 7.3 apply differentiation to find 6.3 – knowtangents and use and ln𝑥 as the gradients, normal, inverse function of 𝑒 𝑥and maxima and minima stationary points 8.3 Ex 8.3A Link to Pythagoras’ theorem. 7.4 Ex 6.67.4A Ex 6.6A 8.4 Ex 8.4A 8.5 See below Ex 8.5A Ex 8.5B 7.5 Ex 7.5A 8.6 Ex 8.6A 7.6 Ex 7.6A 8.7 Ex 8.7A -1 differentiation function. Solutionscan of be equations used to of find thethe equations tangent and the=normal. form 𝑒 𝑎𝑥+𝑏of=the 𝑝 and ln(𝑎𝑥 + 𝑏) 𝑞 is expected. 7 Chapter 8 Exponential review growth lesson and decay 7.3 – identify where functions are increasing and decreasing Recognise Identify areas and of interpret strength exponential and growth development and decay within in modelling the content change of the chapter Recap key of Equations points exponential of chapter. growth Students and to decay are the complete often exam-style written inquestions, terms of powers of questions, extension e as it makes or online it easy end-ofto find the rate of change chapter test at any time. Exponential growth and decay are often described by Assessment a relationship of the form 𝑦 = 𝑎𝑒 𝑘𝑥Book Exam-style questions 8 in Student where 𝑎 and 𝑘 are parameters. Exam-style extension questions For 8 in Student Book growth 𝑎 > 0 and decay 𝑎 <Connect 0. End-of-chapter testfor 8 on Collins 8 Chapter review lesson Identify areas of strength and development within the content of the chapter AS Paper 1/A-level Papers 1 & 2: 6.7 – understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models See below 7.7 Ex 7.7A See below Recap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-ofchapter test Assessment Exam-style questions 7 in Student Book Exam-style extension questions 7 in Student Book End-of-chapter test 7 on Collins Connect CHAPTER 9 – Integration (5 Hours) Prior knowledge needed CHAPTER 10 – Vectors (5 Hours) Chapter 1 Prior knowledge needed Chapter 3 Chapter Chapter 6 8 Technology Technology Use a graphics package to plot can vectors and to investigate how under the angle changes depending on the orientation. Graphics calculators/software be used look at areas a curve and model changes. Plotting diagrams using graphics software enables you to visualise the situation and see how the vectors will still satisfy the rules even if the orientation of the vectors not Studentis Book Learning objectives Specification content Topic links identical. One hour lessons references © HarperCollinsPublishers Ltd 2017 1 Indefinite integrals One hour lessons Students look at integration being the 1 Definition of a vector reverse process of differentiation. Because the the finaland use Students lookderivative at vectorof notation constant is zero the result of integration Pythagoras’ theorem to calculate the is sometimes of magnitude called vectors. the indefinite They use integral because the value trigonometry to find of the 𝑐 is direct not fixed. of a vector Students integrate thecalculate sum or difference or use bearings. They the unit of terms vector and byuse integrating this to find them a velocity separately. vector. 2 Indefinite integrals continued Students move the learning from the first lesson forward by finding the equations of the curve given the derivative and a point on the curve 2 3 Adding The areavectors under a curve Students and vectors in both Students add move onsubtract to calculate definitive iintegrals j and column notation. They are and are introduced to the introduced to the three types of vectors notation. This enables them to calculate (equal, opposite and parallel) and learn the area under curves. Students complete that parallel vectors contain a common the first 6 questions of exercise 9.2A factor. 4 The area under a curve continued Recap knowledge from this and the previous chapter (differentiation) ensuring students clearly understand the link between differentiation and 3 Vector geometry integration. Students finish exercise 9.2A. Students add vectors using the triangle 5 Chapter lessonto the law and arereview introduced parallelogram law which shows that that Recap keyinpoints chapter. the order whichofvectors areStudents added isto complete the exam-style questions, not important. Students go on to be extension or theorem online end-ofintroducedquestions, to the ratio and the chapter test © HarperCollinsPublishers Ltd 2017 Make use of the connections between Learning objectives differentiation and integration Write a vector in i j or as a column vector Calculate the magnitude and direction of a vector as a bearing in two dimensions Identify a unit vector and scale it up as appropriate Find a velocity vector Find the equation of a curve when you know the gradient and point on a curve Add Use avectors technique called integration to Multiply a vector byaacurve scalar find the area under Use a technique called integration to find the area under a curve Apply the triangle law to add vectors Apply the ratio and midpoint theorems Identify areas of strength and development within the content of the chapter AS Paper 1/A-level Papers 1 & 2: Specification content 8.1 – Know and use the Fundamental AS Paper 1: Theorem of Calculus 8.2 – use Integrate 𝑥 𝑛in(excluding 𝑛= 9.1 vectors two −1), and related sums, differences dimensions and–constant 9.2 calculatemultiples the magnitude and direction of a vector and convert between component form and magnitude/direction form A-level Papers 1 & 2: AS Paper Papers 10.1 – use1/A-level vectors in two 1 & 2: 8.1 – Know and use the dimensions Fundamental Theorem of Calculus 10.2 – calculate the magnitude 𝑛 8.2 – Integrate 𝑥 (excluding and direction of a vector and 𝑛 = −1), andbetween related sums, differences convert component form and constant multiples and magnitude/direction form AS AS Paper Paper 1: 1/A-level Papers 1 & 2: 9.3 vectors 8.2 –– add Integrate 𝑥 𝑛 (excluding 𝑛 = diagrammatically and perform the −1), and related sums, differences algebraic operations of vector and constant multiples addition and multiplication by 8.3 – Evaluate definite integrals; scalars use a definite integral to find the area under a curve A-level Papers 1 & 2: 10.3 – add1/A-level vectors Papers 1 & 2: AS Paper diagrammatically perform 𝑛the 8.2 – Integrate 𝑥 𝑛and (excluding = algebraic operations of differences vector −1), and related sums, addition and multiplication by and constant multiples scalars 8.3 – Evaluate definite integrals; usePaper a definite AS 1: integral to find the area– under a curve 9.3 add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars and understand their geometric interpretations The fundamental theorem Topic links of calculus is used in 15.4 Displacement-time and Trigonometric bearings velocity-time graphs were introduced in 6.2 The sine rule and the cosine rule 9.1 Student Book Exreferences 9.1A 10.1 Ex 10.1A 9.1 Ex 9.1B Adding vectors is used to find resultant forces in Chapter 16 Forces 10.2 9.2 Ex Ex 10.2A 9.2A 9.2 Ex 9.2A 10.3 Ex 10.3A Ex 10.3B See below midpoint rule (a special case of the Assessment Exam-styletheorem) midpoint questions 9 in Student Book Exam-style extension questions 9 in Student Book End-of-chapter test 9 on Collins Connect 4 Position vectors Students learn that the position vector of a point is its position relative to the origin. They use this to calculate the relative displacement between two position vectors by subtracting the position vectors. Students calculate the distance between the two points by finding the magnitude of the relative displacement using Pythagoras’ theorem. 5 Chapter review lesson Find the position vector of a point Find the relative displacement between two position vectors A-level Papers 1 & 2: 10.3 – add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars and understand their geometric interpretations AS Paper 1: 9.4 – understand and use position vectors; calculate the distance between two points represented by position vectors 10.4 Ex 10.4A A-level Papers 1 & 2: 10.4 – understand and use position vectors; calculate the distance between two points represented by position vectors Identify areas of strength and development within the content of the chapter See below Recap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-ofchapter test Assessment Exam-style questions 10 in Student Book Exam-style extension questions 10 in Student Book End-of-chapter test 10 on Collins Connect CHAPTER 11 – Proof (4 Hours) Prior knowledge needed CHAPTER 12 – Data presentation and interpretation (11 Hours) Proofs in context are flagged with boxes in every chapter. It is recommended to teach this chapter as a summation of proof and to ensure tested skills have been mastered. Prior knowledge Students will needneeded well developed algebra skills to write and understand proofs. Chapter 1 Technology Chapter 4 A spreadsheet package can be used to generate values. Technology © HarperCollinsPublishers Ltd 2017 Calculators can be used to calculate many useful statistics – put the calculator into statistics mode. One hour lessons Learning objectives Specification content Use frequency tables on calculators. Use a calculator to calculate the mean and standard using the frequency table option. AS Paper 1/A-level Papers 1 & 2: 1 Proof by deduction Use deviation proof by deduction Use graph-drawing software to plot cumulative frequencies. 1.1 – understand and use the Students are reminded that proofhistograms. by Use graphing software to create structure of mathematical proof, deduction was used when proving the set activities. Use spreadsheet packages for the data proceeding from given sine and cosine rules in Chapter 6 assumptions through a series of One hour lessons Learning objectives Specification content Trigonometry. They work through logical steps to a conclusion examples where conclusions are drawn by Use discrete or continuous data 1.1 – use methods proof; AS Paper 2/A-level of Paper 3: 1 Measures of central tendency including proof by deduction 2.3 – interpret measure of central using general rules in mathematics and Use measures of central tendency: Students review types of data, the tendency and variation working through logical steps. mean, median, mode median, the mode and the mean. should bewhich madeisaware that this Students consider the best is the most used method of measure of commonly central tendency to use and proof throughout this specification why. They usebyData set activity one to calculate 2 Proof exhaustion the median and mean and comment Students learn that when they complete a critically on findings. proof by exhaustion they need to divide the statement a finite number of 2 Measures of into central tendency and cases, prove that the list of cases spread identified is exhaustive and then prove Students review measures of spread; the that the statement is true for all the cases. range and interquartile range I.e. they need to try all the options They use frequency table to calculate 3 Disproof by counter example measure of central tendency and spread and estimate fromafrequency Students needthe to mean show that statement is false by finding one exception, a counter tables. example 3 Variance and standard deviation Students are introduced to variance and standard deviation and the fact that these measures use all the data items. Students 4 Chapter review lesson go onto calculating variance and standard deviation using frequency tables. Recap key points of chapter. Students to complete the exam-style questions, © HarperCollinsPublishers Ltd 2017 Use proof by exhaustion Work systematically and list all possibilities in a structured and ordered way Use discrete, continuous, grouped or ungrouped data Use measures of variation: range and interpercentile ranges AS Paper 1/A-level Papers 1 & 2: 1.1 - understand and use the structure of mathematical proof, proceeding from given AS Paper 2/A-level Paper 3: of assumptions through a series 2.3 – interpret of central logical steps to measure a conclusion tendency and variation 1.1 – use methods of proof; including proof by exhaustion Use proof by counter example AS Paper 1/A-level Papers 1 & 2: 1.1 - understand and use the structure of mathematical proof, proceeding from given assumptions through a series of AS Paper 2/A-level Paper 3: logical steps to a conclusion 2.3 – interpret measures of 1.1 – use methods of proof; variation, extending to standard including proof by counter deviation example 2.3 – be able to calculate standard deviation, including from summary statistics Use measure of variation: variance and standard deviation Use the statistic 𝑠𝑥𝑥 = ∑(𝑥 − 𝑥̅ )2 = ∑ 𝑥 2 (∑ 𝑥)2 𝑛 Use Identify areasdeviation of strength and𝑆 standard 𝜎 = √ 𝑥𝑥 𝑛 development within the content of the chapter Topic links Note: Proofs in context are flagged with boxes in every chapter. It is recommended to teach this chapter as a summation of proof and to Topic links ensure tested skills have been mastered Links back to GCSE Student Book references 11.1 Ex 11.1A Student Book references 12.1 DS 1 11.2 Ex 11.2A Links back to GCSE 12.1 Ex 12.1A 11.3 Ex 11.3A This is covered further in Book 2, Chapter 13 Statistical distributions 12.2 Ex 12.2A See below extension questions, or online end-ofchapter test Assessment Exam-style questions 11 in Student Book Exam-style extension questions 11 in Student Book 4 Data cleaningtest 11 on Collins Connect End-of-chapter Use measure of variation: variance and standard deviation Recap calculating variance and standard Use the statistic deviation from frequency tables and then (∑ 𝑥)2 ∑(𝑥 − 𝑥̅ )2 = ∑ 𝑥 2 𝑠 = 𝑥𝑥 introduce data cleaning as involving the 𝑛 𝑆 identification and removal of any invalid Use standard deviation 𝜎 = √ 𝑛𝑥𝑥 data points from the data set. Students use Data set activity 2 to clean the data and give reasons for their omissions. 5 Standard deviation and outliers Student look at data sets that contain extreme values, if a value is more than three standard deviations away from the mean it should be treated as an outlier. Use measure of variation: variance and standard deviation Use the statistic 𝑠𝑥𝑥 = ∑(𝑥 − 𝑥̅ )2 = ∑ 𝑥 2 (∑ 𝑥)2 𝑛 𝑆 Use standard deviation 𝜎 = √ 𝑛𝑥𝑥 Students use Data set activity 3 to apply what they have learnt so far, calculating the mean and standard deviation, and cleaning a section of the large data set. 6 Linear coding Students then use linear coding to make certain calculations easier. They apply a linear code to a section of large data in Data set activity 4. Learning from this and the previous three lessons is consolidated in exercise 12.2B © HarperCollinsPublishers Ltd 2017 Understand and use linear coding 2.4 – select or critique data presentation techniques in the context of a statistical problem AS Paper 2/A-level Paper 3: 2.4 – select or critique data presentation techniques in the context of a statistical problem AS Paper 2/A-level Paper 3: 2.3 – interpret measures of variation, extending to standard deviation 2.3 – be able to calculate standard deviation, including from summary statistics 2.4 – select or critique data presentation techniques in the context of a statistical problem AS Paper 2/A-level Paper 3: 2.3 – interpret measures of variation, extending to standard deviation 2.3 – be able to calculate standard deviation, including from summary statistics 2.4 – select or critique data presentation techniques in the context of a statistical problem 12.2 DS 2 Normal distributions are covered in more depth in Book 2, Chapter 13 Statistical distributions 12.2 DS 3 12.2 DS 4 Ex 12.2B 7 Displaying and interpreting data – box and whisker plots and cumulative frequency diagrams Box and whiskers plots are used to compare sets of data. The way of showing outliers on a box and whisker plot and identifying outliers using quartiles is explained. Cumulative frequency diagrams are used to plot grouped data. (0, 0) should be plotted and the cumulative frequencies should be plotted at the upper bound of the interval. Percentiles are estimated from grouped data. Interpolation is used to find values between two points. 8 Displaying and interpreting data – histograms Interpret and draw frequency polygons, box and whisker plots (including outliers) and cumulative frequency diagrams Use linear interpolation to calculate percentiles from grouped data AS Paper 2/A-level Paper 3: 2.1 – Interpret diagrams for singlevariable data, including understanding that area in a histogram represent frequency 2.1 – connect probability to distributions 12.3 Ex 12.3A Interpret and draw histograms 2.1 – Interpret diagrams for singlevariable data, including understanding that area in a histogram represent frequency 12.3 DS 5 Ex 12.3B Describe correlation using terms such as positive, negative, zero, strong and weak 2.2 – understand informal interpretation of correlation 2.2 – interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded) 2.2 – understand that correlation does not imply causation 12.3 DS 6 Ex 12.3C Students plot histograms and look at the distribution of the data. They apply what they have learnt to Data set activity 5, creating a histogram and performing linear interpolation. 9 Displaying and interpreting data – scatter graphs Students plot scatter diagrams with bivariate data and look for a correlation in the data. In Data set activity 6 students use data interpretation skills to investigate the evidence for global warming in the large data set. © HarperCollinsPublishers Ltd 2017 10 Displaying and interpreting data – Use and interpret the equation of a regression line using statistical regression lines, lines of best fit, CHAPTER 13 – Probability and statistical distributions (5 Hours) functions on your calculator and make interpolation and extrapolation predictions within the range of values Prior knowledge needed Having scatter diagrams in the of the explanatory variable and Chapterplotted 1 previous lesson students go on to look at understand the dangers of Chapter 12 the link between the line of best fit and extrapolation Technology regression line, 𝑦to=perform 𝑎 + 𝑏𝑥 exact wherebinomial b is the distribution calculations. Use a calculator Use a calculator perform cumulative binomial distribution calculations. gradient and a isto the y-intercept. Equations ofOne regression lines are then hour lessons explored. This is then used for 1 Calculating and representing to make interpolation and extrapolation probability predictions. Finally, students look at the correlation between two variables in Data Basic probability is recapped and sample set activity 7. are used to determine the space diagrams Learning objectives Use both Venn diagrams and tree diagrams to calculate probabilities probability equally likely events 11 Chapter of review lesson Identify areas of strength and happening. Students go on to calculate development within the content of the probabilities of mutually exclusive andto Recap key points of chapter. Students chapter independent usingquestions, Venn diagrams complete theevents exam-style extension questions, or online end-ofand tree diagrams. chapter test 2 Discrete and continuous distributions – Identify a discrete uniform distribution Assessment part 1 Exam-style questions 12 in Student Book Exam-style extension 12data in Student Book The term discrete andquestions continuous is End-of-chapter test 12 on Collins Connect recapped. A discrete random variable is then defined. Probability distribution tables are used to show all the possible values a discrete random variable can take along with the probability of each value occurring. When all events are equally likely they form a uniform distribution. Students go on to probability distributions of a random variable. © HarperCollinsPublishers Ltd 2017 2.2 – interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded) 2.2 – understand that correlation does not imply causation Gradients and intercepts were covered in Chapter 4 Coordinate geometry 1: Equations of straight lines Specification content Topic links AS Paper 2/A-level Paper 3: 3.1 – understand and use mutually exclusive and independent events when calculating probability Links back to GCSE 12.3 DS 7 Ex 12.3D Student Book references 13.1 Ex 13.1A This will help you in Book 2, Chapter 12 Probability See below AS Paper 2/A-level Paper 3: 3.1 – link to discrete and continuous distributions You will use knowledge from Chapter 12 Data presentation and interpretation 13.2 Ex 13.2A 3 Discrete and continuous distributions – part 2 Identify a discrete uniform distribution AS Paper 2/A-level Paper 3: 3.1 – link to discrete and continuous distributions You will use knowledge from Chapter 12 Data presentation and interpretation 13.2 Ex 13.2B Use a calculator to find individual or cumulative binomial probabilities Calculate binomial probabilities using the notation 𝑋~B(𝑛, 𝑝) AS Paper 2/A-level Paper 3: 4.1 – understand and use simple, discrete probability distributions (calculation of mean and variance of discrete random variables excluded), including the binomial distribution, as a model; calculate probabilities using the binomial distribution You will need the skills and techniques learnt in Chapter 1 Algebra and functions 1: Manipulating algebraic expressions 13.3 Ex 13.3A Continuous random variables are considered and when graphed a continuous line is used. The graph is called a probability density function (PDF), the area under a pdf shows the probability and sums to 1. Students then go on to find cumulative probabilities. 4 The binomial distribution Binomial distribution is about ‘success’ and ‘failure’ as there are only two possible outcomes to each trail. Students identify situations which can be modelled using the binomial distribution. They then calculate probabilities using the binomial distribution. Several examples are worked through. 5 Chapter review lesson Identify areas of strength and development within the content of the chapter Recap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-ofchapter test Assessment Exam-style questions 13 in Student Book Exam-style extension questions 13 in Student Book End-of-chapter test 13 on Collins Connect CHAPTER 14 – Statistical sampling and hypothesis testing (6 Hours) Prior knowledge needed (10 Hours) CHAPTER 15 – Kinematics Chapter 12 © HarperCollinsPublishers Ltd 2017 Knowing how to calculate binomial probabilities will help you in Chapter 14 Statistical sampling and hypothesis testing See below Prior knowledge needed Chapter 13 Chapter 2 Technology Chapter 3 Use a calculator to calculate binomial cumulative probabilities. Chapter 4 Use spreadsheet packages for the data set activities. Chapter 8 Chapter 9 One hour lessons Learning objectives Chapter 10 1 Populations and samples Use sampling techniques, including Technology simple random, stratified sampling, Graphics packages can be used Discuss the different terms and to visualise problems. systemic sampling, quota sampling and terminology One within populations opportunity (or convenience) hour lessons including; Learning objectives sampling finite and infinite population, census, Select or critique sampling techniques 1 The language of kinematics Find SI units displacement, sample survey, sampling unit, sampling in thethe context ofof solving a statistical velocity and acceleration frame and target population. Move onto problem, including understanding that Students are introduced to motion in a looking at the different types of sampling; different sample can lead to different straight line and the vocabulary conclusions about population random sampling, stratified sampling, displacement, velocity and acceleration opportunity sampling, quota sampling and (vectors quantities) as opposed to systematic sampling. In Data set activity distance, time and speed (scalar 1, students compare random quantities). They use athe SI unitsample for with a systematic sample from the large distance (metres) and time (seconds) to data set and draw conclusions. derive other units. Time permitting introduce the testing five SUAT equations. 2 Hypothesis – part 1 2 Equations of constant acceleration The two different types of hypothesis –are part 1 explained, the null and alternative hypotheses. 1-tail and 2-tail hypothesis (𝑣−𝑢) The definition of acceleration (𝑎 = ) are explained in relation to a six-sided𝑡 die. is used to derive the SUVAT the null Having carried out anfive experiment equations. Students have to workorout hypothesis can only be accepted which of the equations to use to itsolve rejected based on the evidence, cannot problems. categorically be concluded from the findings. 3 Equations of constant acceleration – part 2 apply what they have learnt in Students this lesson in Data set equations activity 2.from the Recap the five SUVAT previous lesson. Questions become more complicated in ex 15.2B and students are © HarperCollinsPublishers Ltd 2017 Apply the language of statistical hypothesis testing, developed through Derive and use the equations of a binomial model: null hypothesis, constant acceleration alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value Have an informal appreciation that the expected value of a binomial distribution as given by np may be required for a 2-tail test Derive and use the equations of Conduct a statistical hypothesis test for constant acceleration the proportion in the binomial distribution and interpret that results in context Specification content Topic links AS Paper 2/A-level Paper 3: 1.1 – mathematical proof 2.4 – recognise and interpret possible outliers in data sets and Specification content statistical diagrams AS 2/A-level Paper 3: 2.4Paper – select or critique data 6.1 – understand and use presentation techniques in the fundamental quantitiesproblem and units context of a statistical in the S.I. system: length, time, 2.4 – be able to clean data, mass including dealing with missing 6.1 – understand and use derived data, errors and outliers quantities and units; velocity and acceleration 7.1 – understand and use the language of kinematics; position; displacement; distance travelled; AS Paperspeed; 2/A-level Paper 3: velocity; acceleration 5.1 – understand and apply AS Paper 2/A-level Paper 3: statistical hypothesis testing, 7.3 – understand, use and derive develop through a binomial the formulae for constant model: null hypothesis, alternative acceleration for motion in a hypotheses, significance level, test straight line statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value 5.2 – conduct a statistical hypothesis test for the proportion AS Paper 2/A-level Paper 3: and in the binomial distribution 7.3 – understand, use derive interpret the results inand context the formulae for constant acceleration for motion in a straight line You will need to know how to clean data for Chapter 12 Data presentation and interpretation Topic links Student Book references 14.1 DS 1 Ex 14.1A Student Book references 15.1 Ex 15.1A You will need to know Chapter 13 Probability and The SUVAT equations are statistical distributions used in Chapter 16 Forces 14.2 DS 2 15.2 Ex 15.2A 15.2 Ex 15.2B encouraged draw a–diagram 3 Hypothesistotesting part 2 to assist them in understanding the situation Recap previous lesson and students go on to complete exercise 14.2A 4 Vertical motion – part 1 Students continue to solve problems using the equations of constant acceleration but this time the acceleration is equal to gravity. They need to decide if gravity is influencing the situation in a positive or negative way. 5 Vertical motion – part 2 Recap the previous lesson and move on to look at when two objects are launched in 4 Significance – part different ways,levels students are1advised to look for what the journeys have in Students look at significance levels being common. the probability of mistakenly rejecting the 6 Displacement-time and velocity-time null hypothesis. The terms test statistic graphs – part 1 and critical region are explained and students move onto isfinding the critical Where acceleration constant, students region andgraphs criticalusing value. They work construct straight lines and through the examples up to theofend the concepts that the gradient a of example 5. displacement-time graphs is velocity and the gradient of a velocity-time graph is 5 SignificanceThe levels – part 2 a velocityacceleration. area under time is used to calculate Recapgraph previous lesson, with a focus on displacement. Students focus on how to calculate values using calculators displacement-time graphs. instead of statistical tables. After working 7 Displacement-time graphs and velocitythrough the rest of the chapter from time graphs – part 2 example 6 onwards, students go on to complete 14.2B. Recap theexercise facts from last lesson and students go on to answer questions focusing on velocity-time graphs. © HarperCollinsPublishers Ltd 2017 Apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, region, Derive and use thecritical equations of acceptance region, p-value constant acceleration, including for situation Have an informal appreciation that the with gravity expected value of a binomial distribution as given by 𝑛𝑝 may be required for a 2-tail test Conduct a statistical hypothesis test for the proportion in the binomial Derive and use equations distribution andthe interpret thatofresults in constant acceleration, including for context situation with gravity Appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis Draw and interpret displacement-time graphs and velocity-time graphs Appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis Draw and interpret displacement-time graphs and velocity-time graphs AS Paper 2/A-level Paper 3: 5.1 – understand and apply statistical hypothesis testing, develop through a binomial model: null hypothesis, alternative hypotheses, significance AS Paper 2/A-level Paperlevel, 3: test statistic, 1-tail test,use 2-tail 7.3 – understand, andtest, derive critical value, for critical region, the formulae constant acceptance region, p-value acceleration for motion in a 5.2 – conduct straight line a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context AS Paper 2/A-level Paper 3: 7.3 – understand, use and derive the formulae for constant acceleration for motion in a 5.2 – understand that a sample is straight line being used to make an inference about the population and appreciate that they significance AS Paper 2/A-level Paper 3: level– is the probability of 7.2 understand, use and incorrectlygraphs rejecting the null for interpret in kinematics hypothesis motion in a straight line; displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph 5.2 – understand that a sample is being used to make an inference about the population and appreciate that they significance level is the2/A-level probability of 3: AS Paper Paper incorrectly rejecting the null 7.2 – understand, use and hypothesis interpret graphs in kinematics for motion in a straight line; displacement against time and interpretation of gradient; velocity 14.2 Ex 14.2A 15.3 Ex 15.3A 15.3 Ex 15.3B 14.2 15.4 Ex 15.4A 14.2 Ex 14.2B 15.4 Ex 15.4B 6 Chapter review lesson Identify areas of strength and development within the content of the chapter CHAPTER – Forces (8 Hours) Recap key16 points of chapter. Students to complete the exam-style Prior knowledge needed questions, 8 Variable acceleration – part 1end-ofextension or online Use calculus to find displacement, Chapter 10questions, chapter test velocity and acceleration when they are Chapter 15 Start the lesson with a quiz on functions of time Assessment Technology differentiation and can integrations. Exam-style questions 14beinused Student Book problems. Graphics packages to visualise Students use differentiation and14 in Student Book Exam-style extension questions One hour lessons Learning objectives integration to solve problems where End-of-chapter test 14 on Collins Connect acceleration is not constant. This lesson 1 Forces Find a resultant force by adding vectors focuses on the use of differentiation. Find the resultant force in a given Students told the SI unit of 2force is the Use calculus to find displacement, 9 Variableare acceleration – part direction velocity and acceleration when they are newton and mass is the kilogram. They Students use differentiation and functions of time use vectors, Pythagoras’ theorem and integration to solve problems where trigonometry to find resultant forces. acceleration notface constant. lesson They also useisthe that if This the resultant focuses on thetouse of then integration. force is equal zero an object is in 10 Chapter review lesson Identify areas of strength and equilibrium. development within the content of the Recap key points of chapter. Students to chapter complete the exam-style questions, extension questions, or online end-ofchapter test Assessment Exam-style questions 15 in Student Book Exam-style questions in StudentBook 2 Newton’sextension laws of motion part15 one Apply Newton’s laws and the equations End-of-chapter test 15 on Collins Connect of constant acceleration to problems Students are told Newton’s laws of involving forces motion. They apply these to problems. Distinguish between the forces of Students are then introduced to the five reaction, tension, thrust, resistance and different types of force. These are weight represented diagrammatically so that they Represent problems involving forces in are clearly distinguishable. Students then a diagram go on to use simplified models to represent real-life situation and solve problems © HarperCollinsPublishers Ltd 2017 against time and interpretation of gradient and area under the graph AS Paper 2/A-level Paper 3: 7.4 – use calculus in kinematics for motion in a straight line: … Specification content AS Paper 1: 9.5 – use vectors to solve AS Paper 2/A-level Paper 3: problems in pure mathematics 7.4 – use calculus in kinematics and in context, including forces for motion in a straight line: … A-level Papers 1 & 2: 10.5 – use vectors to solve problems in pure mathematics and in context, including forces AS Paper 2/A-level Paper 3: 6.1 – understand and use derived quantities and units; force and weight 8.1 – understand the concept of a force AS Paper 2/A-level Paper 3: 8.1 – understand the concept of a force; understand and use Newton’s first law 8.2 – understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors) See below Links back to Chapter 8 Differentiation and Chapter 9 Integration Topic links Builds on Chapter 10 Vectors 15.5 Ex 15.5A Student Book references 16.1 EX 16.1A 15.5 Ex 15.5B See below 16.2 Ex 16.2A 3 Newton’s laws of motion part two Students are told Newton’s laws of motion. They apply these to problems. Students are then introduced to the five different types of force. These are represented diagrammatically so that they are clearly distinguishable. Students then go on to use simplified models to represent real-life situation and solve problems 4 Vertical motion Students move from particles travelling horizontally to particles travelling in vertically. When particles travel vertically they have to consider the effects of gravity and weight. 5 Connected particles – part 1 More complex problems are solved using Newton’s laws and simultaneous equations due to the consideration of resistive forces. Modelling assumptions are made and shown with diagrams when answering questions. The connected particles considered are particles suspended on ropes, vehicles towing caravans and lifts. 6 Connected particles – part 2 © HarperCollinsPublishers Ltd 2017 Apply Newton’s laws and the equations of constant acceleration to problems involving forces Distinguish between the forces of reaction, tension, thrust, resistance and weight Represent problems involving forces in a diagram Distinguish between the forces of reaction, tension, thrust, resistance and weight Model situations such as lifts, connected particles and pulleys Model situations such as lifts, connected particles and pulleys Represent problems involving forces in a diagram Model situations such as lifts, connected particles and pulleys 8.3 – understand and use weight and motion in a straight line under gravity AS Paper 2/A-level Paper 3: 8.1 – understand the concept of a force; understand and use Newton’s first law 8.2 – understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors) 8.3 – understand and use weight and motion in a straight line under gravity AS Paper 2/A-level Paper 3: 8.3 – understand and use weight and motion in a straight line under gravity; gravitational acceleration , g, and its value in S>I> units to varying degrees of accuracy 8.4 – understand and use Newton’s third law AS Paper 2/A-level Paper 3: 8.4 - understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line; application to problems involving smooth pulleys and connected particles AS Paper 2/A-level Paper 3: 8.4 - understand and use Newton’s third law; equilibrium of 16.2 Ex 16.2B 16.3 Ex 16.3A 16.4 Ex 16.4A 16.4 Ex 16.4B Recap on the chapter so far. Students complete exercise 16.4B. Represent problems involving forces in a diagram 7 Pulleys Model situations such as lifts, connected particles and pulleys Represent problems involving forces in a diagram Connected particles are now considered in the form of pulleys. The main difference being that when an object is connected via a pulley it will be travelling in a different direction. Students should again draw diagrams representing the situation in the question. They solve problems using Newton’s laws of motion, the SUVAT equations and simultaneous equations. 8 Chapter review lesson Identify areas of strength and development within the content of the chapter Recap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-ofchapter test Assessment Exam-style questions 16 in Student Book Exam-style extension questions 16 in Student Book End-of-chapter test 16 on Collins Connect © HarperCollinsPublishers Ltd 2017 forces on a particle and motion in a straight line; application to problems involving smooth pulleys and connected particles AS Paper 2/A-level Paper 3: 8.4 - understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line; application to problems involving smooth pulleys and connected particles 16.5 Ex 16.5A See below