Introduction - CCEA



-923925-62166500GCSE Further MathematicsContentsPageIntroduction1Unit 1: Pure Mathematics7Unit 2: Mechanics27Unit 3: Statistics53Unit 4: Discrete and Decision Mathematics67Appendix 199Appendix 2103IntroductionThe purpose of this Planning Framework is to support the teaching and learning of GCSE Further Mathematics Unit 1: Pure Mathematics. The Planning Framework is based on specification content but should not be used as a replacement for the specification. It provides suggestions for a range of teaching and learning activities which provide opportunities for students to develop their:Knowledge and understandingSubject specific skillsThe Cross-Curricular SkillsThinking Skills and Personal CapabilitiesThe Planning Framework is not mandatory, prescriptive or exhaustive. Teachers are encouraged to adapt and develop it to best meet the needs of their students.Subject Skills Assessed through Further Mathematics Unit 1: Pure Mathematics.The following skills are assessed in GCSE Further Mathematics Unit 1: Pure Mathematics:Use and apply standard techniquesaccurately recall facts, terminology and definitionsuse and interpret notation correctlyaccurately carry out routine proceduresaccurately carry out sets tasks requiring multi-step solutionsReason, interpret and communicate mathematicallymake deductions to draw conclusions from mathematical informationmake inferences to draw conclusions from mathematical informationconstruct chains of reasoning to achieve a given resultinterpret information accuratelycommunicate information accuratelypresent argumentspresent proofsassess the validity of an argumentcritically evaluate a given way of presenting informationSolve problems within mathematics and in other contextstranslate problems in mathematical contexts into a processtranslate problems in mathematical contexts into series of processestranslate problems in non-mathematical contexts into a mathematical processtranslate problems in non-mathematical contexts into a series of mathematical processmake and use connections between different parts of mathematicsinterpret the results in the context of the given problemevaluate methods usedevaluate results obtainedevaluate solutions to identify how they may have been affected by assumptions madeSupporting the Development of Statutory Key Stage 4 Cross-Curricular Skills and Thinking Skills and Personal CapabilitiesThis specification builds on the learning experiences from Key Stage 3 as required for the statutory Northern Ireland Curriculum. It also offers opportunities for students to contribute to the aim and objectives of the Curriculum at Key Stage 4, and to continue to develop the Cross-Curricular Skills and the Thinking Skills and Personal Capabilities. The extent of the development of these skills and capabilities will be dependent on the teaching and learning methodology used.Cross-Curricular Skills at Key Stage 4CommunicationStudents should be able to:communicate meaning, feelings and viewpoints in a logical and coherent manner, by using appropriate mathematical language and notation in response to open‐ended tasks, problems, structured questions or examination questions;make oral and written summaries, reports and presentations, taking account of audience and purpose through the use of varied learning activities applied to a wide range of contexts that require students to justify choice of strategy to solve problems, articulate processes, proofs etc. and provide feedback from collaborative learning activities;participate in discussions, debates and interviews by sharing ideas, investigating misconceptions, exploring alternative strategies, justifying choice of strategy, negotiating decisions and listening to others;interpret, analyse and present information in oral, written and ICT formats by developing a mathematical solution to a problem and communicating ideas, strategies and solutions; and,explore and respond, both imaginatively and critically, to a variety of texts by using open‐ended tasks and activities.Using MathematicsStudents should be able to:use mathematical language and notation with confidence, for example calculus and matrices;use mental computation to calculate, estimate and make predictions in a range of simulated and real‐life contexts, for example simplifying expressions in algebra;select and apply mathematical concepts and problem‐solving strategies in a range of simulated and real‐life contexts, for example a simple optimisation problem where you could be asked to find the length which gives a maximum area; andpresent mathematical data in a variety of formats which take account of audience and purpose, for example numerical, graphical and algebraic representations.Using ICTStudents should be able to make effective use of information and communications technology in a wide range of contexts to access, manage, select and present information, including mathematical information, for example calculators, suitable software packages to explore geometry, algebraic functions or calculus.Thinking Skills and Personal Capabilities at Key Stage 4Self‐ManagementStudents should be able to:plan work by identifying appropriate strategies, working systematically and persisting with open‐ended tasks and problems;set personal learning goals and targets to meet deadlines such as identifying, prioritising and managing actions required to develop competence and confidence in more challenging mathematics;monitor, review and evaluate their progress and improve their learning by self-evaluating their performance, identifying strengths and areas for improvement and seeking support where required; andeffectively manage their time by planning, prioritising and minimising distractions in order to meet deadlines set by teachers.Working with OthersStudents should be able to:learn with and from others through co‐operation by engaging in discussions and explaining ideas, challenging and supporting one another, creating and solving each other’s questions and working collaboratively to share methods and results during small group tasks;participate in effective teams and accept responsibility for achieving collective goals by working together on challenging small group tasks with shared goals but individual accountability; andlisten actively to others and influence group thinking and decision‐making, taking account of others’ opinions by participating constructively in small group activities, articulating possible problem‐solving strategies and presenting a well thought out rationale for one approach.Problem SolvingStudents should be able to:identify and analyse relationships and patterns, for example make links between cause and effects by considering, for example, the rates of change of a function with respect to one of its variables through the study of calculus;propose justified explanations, for example by proving a mathematical statement is true;analyse critically and assess evidence to understand how information or evidence can be used to serve different purposes or agendas, for example recognising that if a definite integral has a negative value it follows that the part of the curve between the two given ordinates is below the x-axis;analyse and evaluate multiple perspectives, for example modelling real‐life scenarios as further mathematics problems;weigh up options and justify decisions, for example using the most appropriate method to solve complex mathematical problems; andapply and evaluate a range of approaches to solve problems in familiar and novel contexts, for example finding the coordinates of the turning point of a quadratic function can be done by completing the square or by finding where the gradient is equal to zero using differentiation.There is an emphasis on problem solving within this new specification which will require the development of all of these skills.Although not statutory at Key Stage 4 this specification also allows opportunities for further development of the Thinking Skills and Personal Capabilities of Managing Information and Creativity.Key FeaturesThe Planning Framework:Includes suggestions for a range of teaching and learning activities which are aligned to the GCSE Further Mathematics Unit 1: Pure Mathematics specification content.Highlights opportunities for inquiry-based learning.Indicates opportunities to develop subject knowledge and understanding and specific skills.Indicates opportunities to develop the Cross-Curricular Skills and Thinking Skills and Personal Capabilities.Provides relevant, interesting, motivating and enjoyable teaching and learning activities which will enhance the student’s learning experience.Makes reference to supporting resources.Unit 1Pure MathematicsContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal CapabilitiesAlgebraic FractionsStudents should be able to:add, subtract, multiply and divide rational algebraic fractions with linear and quadratic numerators and/or denominators.Begin with numerical fractions to link with work done at KS3 – algebraic fractions work in the same way.Discuss meaning of a common factor and how they can be cancelled. Look at dividing numerator and denominator by a common factor in order to simplify algebraic fractions. Expressions in numerator and/or denominator may need to be factorised in order to find common factors. Methods of factorising to include:taking out a common factor,grouping,difference of two squares,quadratic expressions of the form ax2 + bx + c.For a recap on methods of factorisation – matching equivalent expressions card game.You may find this link useful on to multiplying the numerator and denominator by the same term to produce equivalent fractions.Equivalent algebraic fractions families card game (Resource 1).When adding/subtracting algebraic fractions discussion will be needed on how to find the lowest common denominator.Activity to address finding the lowest common denominator (Resource 2)Further Mathematics for CCEA GCSE Level pages 4-5 Ex 1A and Ex 1BWOWOWOSMContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal CapabilitiesIt may be necessary to factorise the denominators in order to find the lowest common denominator.Further Mathematics for CCEA GCSE Level page 5 Ex 1CReview multiplying simple algebraic fractions by noting that you must multiply the numerators and the denominators. Encourage cancellation by common factors before multiplying numerators and denominators.Divide algebraic fractions by multiplying by the reciprocal.Further Mathematics for CCEA GCSE Level page 6 Ex 1DIt may be necessary to factorise expressions in the numerator and/or denominator before multiplying or dividing.Further Mathematics for CCEA GCSE Level page 7 Ex 1E and Ex 1FSolve equations involving the addition or subtraction of algebraic fractions. It may be necessary to revise solving quadratic equations by factorisation and by use of the quadratic formula first (CCEA Further Maths textbook pages 8-9 Ex 2A and Ex 2B).This may be a good time to introduce the concept of cross multiplying which can be done when one fraction is equal to another.Further Mathematics for CCEA GCSE Level pages 9-11 Ex 2C and Ex 2D.PSComm – T&LPSSMSMPSComm – T&LComm – WResourcesMatching equivalent expressions card game – MrBarton Tarsia – 1 – Equivalent algebraic fractions families card gameResource 2 – Activity to address finding the lowest common denominatorContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal Capabilities Algebraic ManipulationStudents should be able to:manipulate algebraic expressions, including the expansion of three linear brackets.Stress that when expanding two linear brackets we multiply all the terms in the first bracket by all the terms in the second bracket.To expand three linear brackets we first expand two of the brackets and then multiply all of the terms of this result by the terms in the third bracket.Worksheet on expanding three brackets (Resource 3).Could link this work with cubing a bracket and showing that Pascal’s triangle gives the coefficients of the terms in the expanded expression.PSResourcesResource 3 – Worksheet on expanding three bracketsLinks for extra practice:Worksheet of questions – of exam style questions – to worksheet of exam style questions – example – the SquareStudents should be able to:complete the square where the coefficient of x2 will always be 1;apply completing the square to solving quadratic equations and identifying minimum turning points.Begin by completing the square for quadratic expressions where the coefficient of x2 will always be 1. This could be done using mini whiteboards (Resource 4).Explain to pupils that the method of completing the square can be used to solve quadratic equations.Further Mathematics for CCEA GCSE Level page 12 Ex 2EExtend use of completing the square to finding the minimum value of a function and the value of x at which it occurs.Further Mathematics for CCEA GCSE Level page 12 Ex 2FActivity – Matching quadratic expressions with their completing the square equivalent, factorisation equivalent, minimum value of function and where it occurs, where the function crosses both axes, sketch of quadratic function (Resources 5).UICTPSComm – T&LPSPSResourcesResource 4 – Completing the square for quadratic expressions – PowerPoint activity which could be done using mini whiteboards.Activity – Matching quadratic expressions with their completing the square equivalent, factorisation equivalent, minimum value of function and where it occurs, where the function crosses both axes, sketch of quadratic function.Resource 5 – Completing the square activity.Simultaneous EquationsStudents should be able to:form and solve three equations in three unknowns;Extend two simultaneous equations in two unknowns to three simultaneous equations in three unknowns.Further Mathematics for CCEA GCSE Level page 13 Ex 3AFind the answer to a real-life problem by setting up three equations in three unknowns and solving them simultaneously.Further Mathematics for CCEA GCSE Level pages 14-16 Ex 3BSolution of simultaneous equations where one is linear and the other is quadratic. Consider both algebraic and graphical solution.Further Mathematics for CCEA GCSE Level page 17 Ex 3CComm – T&LPSPSComm – T&LPSQuadratic InequalitiesStudents should be able to:solve quadratic inequalities, which are restricted to quadratic expressions that factorise;Can show pupils three methods for solving quadratic inequalities:Sketching the graph of the quadratic functionExamining the signs of the factors of the quadratic functionCompleting the square for the quadratic functionResource 6 – Solving quadratic inequalities PowerPoint.When sketching the graph of the quadratic function we identify the range of values of x for which the function is either above or below the x-axis as appropriate.When examining the signs of the factors of the quadratic function we first factorise the quadratic function, consider whether each of the factors is positive or negative for different ranges of values of x and hence whether the quadratic function is positive or negative for different ranges of values of x. This can be done using a table.When completing the square for the quadratic function in order to solve a quadratic inequality, care must be taken when taking the square root of both sides of the inequality. When considering the negative square root it is necessary to reverse the inequality sign.Worksheet on solving quadratic inequalities. (Resource 7)Mathematics for CCEA at AS Level page 36 Ex 2J Q3-6Comm – T&LPSPSSMPSResourcesRecourse 6 – PowerPoint showing worked solutions of all three methods.Resource 7 – Quadratic Inequalities worksheet.ContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal Capabilities Trigonometric EquationsStudents should be able to:sketch the graphs of sin x, cos x and tan x, where the range of x is a subset of -360o ≤ x ≤ 360o;solve simple trigonometric equations that lead to a maximum of two solutions in a given range;Spend time looking at the graphs of sin x, cos x and tan x – in particular look at values of x for which sin x is positive and negative, values of x for which cos x is positive and negative and values of x for which tan x is positive and negative. Link the trigonometric graphs with the CAST diagram to show in which quadrant each of the trigonometric functions is positive and negative. (Resource 8).Use graph software to examine the symmetrical properties of the graphs of the trigonometric functions. Establish thatsin x = sin(180 – x),cos x = cos (–x) andtan x = tan (180 + x).Discuss that using these symmetrical properties in the context of a CAST diagram can allow us to find all the solutions to a trigonometric equation within a given range.By ignoring whether the trigonometric ratio is positive or negative, we can work out the related acute angle using a calculator. The sign indicates which quadrants the related acute angle should be drawn in. We can then find all the solutions to a trigonometric equation within a given range.Further Mathematics for CCEA GCSE Level page 18 Ex 4A and page 19 Ex 4BInclude equations such as sin2x=0.5 for 0≤x≤360 andcos(12x-30)=0.8 for -180≤x≤180where it is necessary to adjust the range accordingly.Further Mathematics for CCEA GCSE Level page 20 Ex 4CUICTUICTComm – T&LPSPSResourcesResource 8 – Handout involving the use of graphical software which allows pupils to connect the trigonometric graphs of sin x, cos x and tan x with a CAST diagram.ContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal CapabilitiesDifferentiationStudents should be able to:differentiate expressions that are restricted to integer powers of x;apply differentiation to:gradients;finding equations of tangents and normals at points on a curve;simple optimisation problems; andelementary curve sketching of a quadratic or cubic function.Differentiate powers and sums of powers of x to find dydx.This can be broken down to look at integer coefficients of the power first followed by fractional coefficients of the power. Pupils may need reminded that when we multiply a fraction by a whole number we only multiply the numerator of the fraction by the whole number.Further Mathematics for CCEA GCSE Level page 26 Ex 6A and page 27 Ex 6BDifferentiating expressions where the x term is in the denominator. Stress that the x term must be brought to the numerator (using rules of indices) before differentiating.Further Mathematics for CCEA GCSE Level page 27 Ex 6C and page 28 Ex 6DDifferentiation of dydx to find d2ydx2, known as the second derivative.Further Mathematics for CCEA GCSE Level page 28 Ex 6EUse differentiation to find the gradient of the tangent to a curve at a specific point.Further Mathematics for CCEA GCSE Level page 29 Ex 7AInclude finding the coordinate(s) of the point(s) where the gradient of the tangent to a curve is given.Further Mathematics for CCEA GCSE Level page 30 Ex 7B and Ex 7CFind the equation of a tangent using both forms for the equation of a straight line y=mx+c and y- y1=m (x- x1).Further Mathematics for CCEA GCSE Level page 31 Ex 7DFind the gradient and equation of a normal to a curve at any point. (Resource 9).Could provide pupils with a skeletal solution on finding the equation of a tangent and/or normal to a curve at a given point and they have to give a detailed explanation as to what is happening at each stage of the solution encouraging them to develop a chain of reasoning.Further Mathematics for CCEA GCSE Level page 32 Ex 7ERemind pupils again at this stage that where a line/curve meets the y-axis, x = 0 and where a line/curve meets the x-axis, y = 0. Also the point where two lines meet can be found by solving their equations simultaneously either by substitution or elimination.Further Mathematics for CCEA GCSE Level page 33 Ex 7FFinding the point(s) where a tangent to a curve is parallel or perpendicular to a straight line.Further Mathematics for CCEA GCSE Level page 35 Ex 7GDiscuss types of turning point – maximum and minimum – and how the gradient of a curve changes for both. Stress that where turning points occur the gradient is equal to zero.Look at finding the turning point of a quadratic curve and stating what type of turning point it is. Show both methods for distinguishing between maximum and minimum turning points, that is, by finding the gradients before and after the turning points or by calculating the second derivative. Explain to pupils that d2ydx2 measures the rate of change of the gradient dydx as x changes and therefore if d2ydx2 is negative, then the gradient is getting smaller or changing from positive to 0 to negative and therefore the turning point will be a maximum. Ifd2y dx2 is positive, then the gradient is getting bigger or changing from negative to 0 to positive and therefore the turning point will be a minimum.Further Mathematics for CCEA GCSE Level page 37 Ex 8ADiscuss finding the turning points of a cubic curve and stating the nature of the turning points (one will be a maximum and the other a minimum).Further Mathematics for CCEA GCSE Level page 38 Ex 8BExtend this to quadratic and cubic curve sketching and encourage pupils to find the following information when sketching a quadratic/cubic function:Where the curve crosses the y axis by putting x = 0Where the curve crosses the x axis by putting y = 0Find the coordinates and nature of the turning point(s)Mark these points on a grid and draw a smooth curve through these pointsFurther Mathematics for CCEA GCSE Level page 39 Ex 8C (sketching quadratic functions) and page 41 Ex 8D (sketching cubic functions)Use differentiation in optimising problems (Resource 10). Remind pupils that they must prove a maximum or minimum by finding the second derivative.Could provide pupils with a skeletal solution on using differentiation in optimising problems and they have to give a detailed explanation as to what is happening at each stage of the solution encouraging them to develop a chain of reasoning.Further Mathematics for CCEA GCSE Level pages 41-42 Ex 8ESMSMPSSMSMSMSMComm – R, WComm – T&LPSUM/SM/PSComm – T&LUM/SMUM/SMComm – T&LPSComm – R, WPSPSPSResourcesResource 9 – Finding the equation of a target and normal to a curve of a fixed point.Resource 10 – using differentiation in optimising problems.Other possible resources:Improving learning in mathematics: resource file for teaching 2 – C2 Exploring functions involving fractional and negative powers of x.Improving learning in mathematics: resource file for teaching 2 – C3 Matching functions and derivatives.Improving learning in mathematics: resource file for teaching 2 – C4 Differentiating fractional and negative powers.Improving learning in mathematics: resource file for teaching 2 – C5 Finding stationary points of cubic functions.ContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal CapabilitiesIntegrationStudents should be able to:demonstrate knowledge that integration is the inverse process to differentiation;integrate expressions that are restricted to integer powers of x, (x≠-1);form and evaluate definite integrals;apply integration to finding the area under a curve.Pupils should begin integrating basic functions involving positive indices. To include integrating sums of terms.Further Mathematics for CCEA GCSE Level page 43 Ex. 9AIntegrating expressions where the x term is in the denominator. Stress that the x term must be brought to the numerator (using rules of indices) before integrating.Activity – Rewrite sums of terms in the form for integrating.Further Mathematics for CCEA GCSE Level page 43 Ex. 9BWhen evaluating definite integrals show pupils that the constant of integration, c, will always cancel out.Further Mathematics for CCEA GCSE Level page 44 Ex. 9CSince integration is the inverse process to differentiation we can integrate dydx to find y. Look at questions where pupils are asked to find the equation of a curve given the gradient function of the curve and a point on the curve.Further Mathematics for CCEA GCSE Level page 45 Ex. 9DPupils should be able to use integration to find the area enclosed by a curve, two given x-ordinates x = a and x = b and the x-axis. Pupils will need to recognise that if a definite integral has a negative value it follows that the part of the curve between the two given ordinates is below the x-axis.Pupils could be provided with a skeletal solution on using integration to find the area enclosed by a curve and then asked to give a detailed explanation as to what is happening at each stage of the solution encouraging them to develop a chain of reasoning (Resource 11).Further Mathematics for CCEA GCSE Level page 46 Ex. 10APupils may have to find where a quadratic/cubic curve crosses the x-axis in order to find the limits of integration.Further Mathematics for CCEA GCSE Level page 46 Ex. 10BSMWOPSPSPSPSComm – R, WPSSMResourcesResource 11 – Skeletal solution on using integration to find the area enclosed by a curve.Improving learning in mathematics: resource file for teaching 2 – C4 Integrating fractional and negative powers.ContentLearning outcomesSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal Capabilities LogarithmsStudents should be able to:demonstrate understanding of logarithms as a natural evolution from indices;solve problems usingthe laws of logarithms andlog/log graphs in context;solve indicial equations using logarithms;Define the logarithm of a number as: The power to which you would raise the base to get the number.Be able to change from log notation to index notation and vice versa with confidence.Further Mathematics for CCEA GCSE Level pages 54-56 Ex. 12A, Ex. 12C, Ex.12D and Ex.12EIntroduce log button on the calculator as log10. Highlight that logs to base 10 are generally written without the base highlighted, for e.g. log5 means log105. All other bases must be specifically written.Derive the three basic laws of logarithms from the laws of indices and use these in simplifying and manipulating expressions involving logarithms:logpq=logp+logqlogpq=logp-logqlogpn=nlogpCan look at special cases such as when p=q we havelogpq=logp-logq or log1=0and logaa=1 follows from a1=a.Further Mathematics for CCEA GCSE Level page 55 Ex. 12BUse the three basic laws of logarithms to write logs of numbers in terms of logs of prime numbers. You may need to recap how to write a number as a product of prime factors.Further Mathematics for CCEA GCSE Level page 57 Ex. 12FUse the three basic laws of logarithms to write logs of numbers given to different bases in terms of logs of prime numbers.Further Mathematics for CCEA GCSE Level pages 57-58 Ex. 12GSolve equations of the form af(x)=bgx for simple functions f(x) and g(x).Further Mathematics for CCEA GCSE Level pages 58-59 Ex. 13A and Ex. 13BLook at indicial equations where each side of the equation has an index with the same unknown.Further Mathematics for CCEA GCSE Level page 59 Ex. 13CUsing graphical software look at the exponential curve y= 10x. Since reflection in the line y=x results in the interchange of x and y in the equation of the curve it follows that reflection of y= 10x in the line y=x results in the graph of x= 10y which can also be written as log10x=y. Therefore y=log10x is a reflection of y= 10x in the line y=x.Stress that because the range of an exponential function is always greater than 0, it follows that we cannot take the log of a negative number and link this to the graphs.Reduce a law of the form P=kln to linear form using logarithms. Given a table of values of a quantity P at various values of a quantity l verify that a relationship of the form P=kln exists by drawing a suitable straight line graph and hence determine the values of k and n.The test which establishes a relationship of the form P=kln involves completing a table of values for logP and logl and plotting the resulting pairs. If the result is approximately linear, the law holds. Pupils discuss.We plot logP against logl to give a straight line y=mx+c where y=logP, x=logl, c=logk and m=n.Thus finding the gradient of the straight line gives n and finding the y intercept yields k.Additional Mathematics (Hugh Morrison) pages 64-67 Exercise 3.3SMComm – T&LSMSMSMPSComm – T&LPSUICTComm – T&LPSResourcesImproving learning in mathematics: resource file for teaching 1 – A13 Simplifying logarithmic expressions.ContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal Capabilities MatricesStudents should be able to:add, subtract and multiply matrices;find inverse of 2x2 matrices;solve matrix equations;use matrices to solve 2x2 simultaneous equations.Introduce language of matrices including order, row, column, square, elements.Add and subtract matrices. All calculations with matrices to include matrices of order 1x2, 2x1 or 2x2.Further Mathematics for CCEA GCSE Level pages 48-49 Ex. 11AMultiplication of a matrix by a scalar.Further Mathematics for CCEA GCSE Level page 49 Ex. 11BSolve simple matrix equations using the same rules as used to solve algebraic linear equations.Further Mathematics for CCEA GCSE Level page 50 Ex. 11CMultiplication of a matrix by a matrix to include matrices of order 1x2, 2x1 or 2x2. Demonstrate that matrix multiplication is non-commutative. Pupils should be able to explain that two matrices A and B can only be multiplied together to give AB if the number of columns in matrix A is equal to the number of rows in matrix B and that the order of the resulting matrix will be the number of rows in matrix A x the number of columns in matrix B.Demonstrate to pupils that in order to square a matrix it is not possible to square each of the elements.Further Mathematics for CCEA GCSE Level page 51 Ex. 11DCalculate the determinant of a matrix.If A=abcd then the determinant of A is given by detA=ad-bc. Stress that the determinant is a number and not a matrix.Further Mathematics for CCEA GCSE Level page 51 Ex. 11EDefine the unit matrix (or identity matrix) as I=1001 and show that when any matrix A is multiplied by I, or when I is multiplied by any matrix A the answer is always A. That is AI = IA = A.Calculate the inverse of a matrix.If A=abcd then the inverse of A is given by A-1= 1detAd-b-caIt follows that if the determinant of a matrix is 0 then the matrix will have no inverse because division by zero is not defined.Further Mathematics for CCEA GCSE Level page 52 Ex. 11FA matrix equation of the form AX = B can be solved by finding the inverse of A and multiplying both sides of the equation by the inverse of A. Note that this results in X = A-1B and that A-1 and B must be multiplied in this order.Further Mathematics for CCEA GCSE Level page 53 Ex. 11GSolve two linear simultaneous equations in two unknowns using matrices by rewriting the two equations as one matrix equation of the form AX = B where A is the matrix with the coefficients of x and y from each equation, X= xy and B is the matrix with the constants from each equation.Further Mathematics for CCEA GCSE Level pages 53-54 Ex. 11HSMSMSMComm – T&LComm – T&LComm – T&LSMPSComm – T&LPSUnit 2MechanicsIntroductionThe purpose of this Planning Framework is to support the teaching and learning of GCSE Further Mathematics Unit 2 Mechanics. The Planning Framework is based on specification content but should not be used as a replacement for the specification. It provides suggestions for a range of teaching and learning activities which provide opportunities for students to develop their:Knowledge and understandingSubject specific skillsThe Cross-Curricular SkillsThinking Skills and Personal CapabilitiesThe Planning Framework is not mandatory, prescriptive or exhaustive. Teachers are encouraged to adapt and develop it to best meet the needs of their students.Subject Skills Assessed through GCSE Further Mathematics Unit 2 Mechanics:Use and apply standard techniquesaccurately recall facts, terminology and definitionsuse and interpret notation correctlyaccurately carry out routine proceduresaccurately carry out sets tasks requiring multi-step solutionsReason, interpret and communicate mathematicallymake deductions to draw conclusions from mathematical informationmake inferences to draw conclusions from mathematical informationconstruct chains of reasoning to achieve a given resultinterpret information accuratelycommunicate information accuratelypresent argumentspresent proofsassess the validity of an argumentcritically evaluate a given way of presenting informationSolve problems within mathematics and in other contextstranslate problems in mathematical contexts into a processtranslate problems in mathematical contexts into series of processestranslate problems in non-mathematical contexts into a mathematical processtranslate problems in non-mathematical contexts into a series of mathematical processmake and use connections between different parts of mathematicsinterpret the results in the context of the given problemevaluate methods usedevaluate results obtainedevaluate solutions to identify how they may have been affected by assumptions madeThe following skills are assessed in GCSE Further Mathematics Unit 2 Mechanics:CommunicationUsing MathematicsUsing ICTSelf-ManagementWorking with OthersProblem SolvingSupporting the Development of Statutory Key Stage 4 Cross-Curricular Skills and Thinking Skills and Personal CapabilitiesThis specification builds on the learning experiences from Key Stage 3 as required for the statutory Northern Ireland Curriculum. It also offers opportunities for students to contribute to the aim and objectives of the Curriculum at Key Stage 4, and to continue to develop the Cross-Curricular Skills and the Thinking Skills and Personal Capabilities. The extent of the development of these skills and capabilities will be dependent on the teaching and learning methodology used.Cross-Curricular Skills at Key Stage 4CommunicationStudents should be able to:communicate meaning, feelings and viewpoints in a logical and coherent manner, by using appropriate mathematical language and notation in response to open‐ended tasks, problems, structured questions or examination questions;make oral and written summaries, reports and presentations, taking account of audience and purpose through the use of varied learning activities applied to a wide range of contexts that require students to organise and record data, justify choice of strategy to solve problems, articulate processes, proofs etc. and provide feedback from collaborative learning activities;participate in discussions, debates and interviews by sharing ideas, investigating misconceptions, exploring alternative strategies, justifying choice of strategy, negotiating decisions and listening to others;interpret, analyse and present information in oral, written and ICT formats by developing a mathematical solution to a problem and communicating ideas, strategies and solutions; and,explore and respond, both imaginatively and critically, to a variety of texts by using open‐ended tasks and activities.Using MathematicsStudents should be able to:use mathematical language and notation with confidence, for example force diagrams;use mental computation to calculate, estimate and make predictions in a range of simulated and real‐life contexts, for example working out weights given masses;select and apply mathematical concepts and problem‐solving strategies in a range of simulated and real‐life contexts, for example applying Newton’s laws to questions relating to dynamics; andpresent mathematical data in a variety of formats which take account of audience and purpose, for example graphical representations of forces.Using ICTStudents should be able to make effective use of information and communications technology in a wide range of contexts to access, manage, select and present information, including mathematical information, for example calculators, suitable software packages to explore mechanicsThinking Skills and Personal Capabilities at Key Stage 4Self‐ManagementStudents should be able to:plan work by identifying appropriate strategies, working systematically and persisting with open‐ended tasks and problems;set personal learning goals and targets to meet deadlines such as identifying, prioritising and managing actions required to develop competence and confidence in more challenging mathematics;monitor, review and evaluate their progress and improve their learning by self evaluating their performance, identifying strengths and areas for improvement and seeking support where required; andeffectively manage their time by planning, prioritising and minimising distractions in order to meet deadlines set by teachers.Working with OthersStudents should be able to:learn with and from others through co‐operation by engaging in discussions and explaining ideas, challenging and supporting one another, creating and solving each other’s questions and working collaboratively to share methods and results during small group tasks;participate in effective teams and accept responsibility for achieving collective goals by working together on challenging small group tasks with shared goals but individual accountability; andlisten actively to others and influence group thinking and decision‐making, taking account of others’ opinions by participating constructively in small group activities, articulating possible problem‐solving strategies and presenting a well thought out rationale for one approach.Problem SolvingStudents should be able to:identify and analyse relationships and patterns, for example make links between cause and effects by considering the resultant of forces;propose justified explanations, for example by proving whether a body will move;analyse and evaluate multiple perspectives, for example modelling real‐life scenarios as further mathematics problems;weigh up options and justify decisions, for example using the most appropriate method to solve complex mathematical problems; andapply and evaluate a range of approaches to solve problems in familiar and novel contexts, for example deciding to use either constant acceleration formulae or velocity/time graphs.There is an emphasis on problem solving within this new specification which will require the development of all of theseKey FeaturesThe Planning Framework:Includes suggestions for a range of teaching and learning activities which are aligned to the GCSE Further Mathematics Unit 2 Mechanics specification content.Highlights opportunities for inquiry-based learning.Indicates opportunities to develop subject knowledge and understanding and specific skills.Indicates opportunities to develop the Cross-Curricular Skills and Thinking Skills and Personal Capabilities.Provides relevant, interesting, motivating and enjoyable teaching and learning activities which will enhance the student’s learning experience.Makes reference to supporting resources.UnitLearning outcomesSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal CapabilitiesKinematicsStudents should be able to:draw, interpret and use displacement/time graphs and velocity/time graphs;use constant acceleration formulae;Review work from GCSE Maths on distance/time graphs and intersecting travel graphsWorksheet 1 with distance/time graphs. Pupils work in pairs to finddistances at specific times times when certain distances are reachedTeacher gets pupils to walk 20 steps down a corridor and then 15 backPupils sayHow many steps they have gone (distance defined)How many steps they are now from the start (displacement defined)Teacher gets pupils to walk 20 steps down a corridor and then 25 backPupils sayHow many steps they have gone (distance defined)How many steps they are now from the start (meaning of a negative displacement defined)Teacher works through example 1 with the class Example 1Lily leaves home at 0800 and drives 20 miles to Lurgan arriving at 0840She stays in Lurgan for 45 minutesShe then drives back towards home for 16 miles to a garage arriving there at 0949Teacher draws the displacement/time graphPupils work in pairs, one making up a question and the other drawing the displacement/time graphPupils discuss in pairs the difference between cars travelling at 30 mph outside school in one direction and at 30 mph outside school in the opposite directionTeacher defines velocity, explaining the difference between velocity and speedTeacher calculates the average velocity for each part of the journey in Example 120/40 X 60 =30mph0 mph-16/24 X 60 = -40 mphPupils discuss in pairs how the average velocities could have been found using the displacement/time graphPupils work in pairs finding the average velocities for each part using their displacement/time graphsWorksheet 2 with (i) displacement/time graphs drawn. Pupils to findDisplacementsTimesVelocitiesAnd (ii) journeys described and pupils to draw displacement/time graphsThey then individually work through TEXTBOOK P69 Ex 15BTeacher defines acceleration, uniform acceleration and decelerationTeacher works through example 2 with the class Example 2An object accelerates uniformly from 2m/s to 6m/sIt then moves at 6m/s for 2 secIt then decelerates to rest in a further 8 secDraw the velocity/time graph and find the acceleration in each partPupils work in pairs finding the acceleration in each part of their graphsReview work from GCSE Maths on finding areas of rectangle, triangle and trapeziumTeacher explains how to find the total distance travelledExample continuedFind the total distancePupils in pairs find the total distance travelled for their graphsThey then individually work through TEXTBOOK P69 Ex 15BTeacher puts example pg. 71 from textbook on whiteboardPupils work in pairs to decide how they would solve this problemClass discussion consideringwhat do we know when they pass each otherwhat variable would be the samePupils work in pairs to answer the question with teacher support as appropriateThey then individually work through TEXTBOOK P72 Ex 15CReview substitution into formulae and using formulae to solve equationsPupils work in pairs through a worksheet using the 4 constant acceleration formulae in turn with 3 values explicitly givenE.g. v = u + atFind u when v = 20, a = 5, t=2Teacher derives the 4 constant acceleration formulae (ref pg. 71 textbook)Teacher works through the following exampleEXAMPLE 1A body travels 21m in 3sec at a constant acceleration of 2 m/s2Find its initial velocityPupils work in pairs and list explicitly each variable they know and then choose which equation to usePupils work in pairs using cards with similar questionsThey then individually work through TEXTBOOK P73 Ex 16A,16BTeacher introduces gravity (Link with Science??)E.g. dropping 2 coins together on to floor, then a pen and a page together, asking class to explain what and why is happeningg defined as 10 m/s2Teacher works through the following exampleEXAMPLE 2A pen drops to the ground from a height of 3m. What is its velocity on hitting the ground.Pupils work in pairs and list explicitly each variable they know and then choose which equation to usePupils work in pairs using cards with similar questionsThey then individually work through TEXTBOOK P76 Ex 16C, P78 Ex 16DReview solving simultaneous linear equations from GCSE MathsTeacher works through the following exampleEXAMPLE 3A body moving at a constant acceleration reaches a velocity of 6m/s in 4sIt reaches a velocity of 10m/s in a further 4sFind the total distance travelledPupils work in pairs answering the following questionswhat variables do you knowwhat variables will be the same in 2 parts of the motionThey then form 2 equations and solve them with teacher assistance as appropriateThey then individually work through a worksheet on similar questionsUMWOUMWOComm – T&LWOSMComm – T&LSMWOWOSMUICTWOSMWOWOSMWOWOSMPSSMComm – T&LSMComm – T&LPSResourcesTextbook:Further Mathematics for CCEA GCSE LevelWorksheets and cards – resources 1 to 9UnitLearning outcomesSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal CapabilitiesVectorsStudents should be able to:demonstrate understanding of the definition of vector and scalar quantities;calculate the magnitude and direction of a vector; use i and j vectors in calculations;Teacher discusses magnitude and direction with classTeacher defines vectors and scalarsPupils work in pairs sorting sets of cards with different quantities written on them into vectors and scalars (force, distance, displacement, mass etc.)Review Pythagoras from GCSE MathsWorksheet on finding the hypotenuse in right-angled trianglesTeacher defines i and j and draws the vector a =2i -3j on a squared gridPupils work in pairs, one defining a vector and the other drawing itTeacher draws the vector b =i +5j on the same squared gridTeacher then asks class how could you draw a + b on another squared gridWorksheet on drawing 2 vectors at a timeTeacher then goes back to a asks each pair how they could find its magnitudePupils work in pairs finding the magnitude of the vectors they have drawnReview trigonometry from GCSE MathsWorksheet on finding an angle in right-angled triangles using tanTeacher then goes back to a asks each pair how they could find its directionPupils work in pairs finding the direction of the vectors they have drawnThey then individually work through TEXTBOOK P88 Ex 19ATeacher asks each pair how they could find the speed if they knew the velocity and the distance if they knew the displacementTeacher works through the following exampleEXAMPLE 1A body has a velocity of (4i -3j) m/sFind its speedThey then individually work through TEXTBOOK P89 Ex 19BTeacher defines resultant forceTeacher works through the following exampleEXAMPLE 2The forces (4i -3j)N, (i -2j)N and (-7i +6j)N act on a body. Find the resultant forceThey then individually work through TEXTBOOK P89 Ex 19CTeacher works through the following exampleEXAMPLE 3The forces PN, (4i -3j)N and (i -2j)N act on a body. The resultant force is3i -9j. Find PThey then individually work through TEXTBOOK P89 Ex 19DAfter Newton's Law and equations of motion have been taughtTeacher works through the following exampleEXAMPLE 4The forces (4i -3j)N and (i -2j)N act on a body of mass 5 kg. Find the accelerationThey then individually work through TEXTBOOK P90 Ex 19ETeacher gives the following exampleEXAMPLE 5A body of mass 5 kg accelerates from (4i -3j) m/s to (6i -j) m/s in 2sFind (i) the acceleration (ii) the force causing the acceleration (iii) the displacementPupils work in pairs to plan how to do this question and then work through their solutionThey then individually work through TEXTBOOK P91 Ex 19FTeacher gives the following exampleEXAMPLE 6A body moves parallel to the vector (12i -5j) with a speed of 52 m/s.Find its velocityPupils work in pairs to plan how to do this question and then work through their solutionThey then individually work through TEXTBOOK P91 Ex 19GComm – T&LWOUMSMWOUICTSMComm – T&LWOUMSMWOSMWOComm – T&LUMSMPSSMPSSMPSSMPSWOSMPSWOSMResourcesTextbook:Further Mathematics for CCEA GCSE LevelWorksheets and cards – Resources 10 to 15UnitLearning outcomesSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal CapabilitiesForcesStudents should be able to:demonstrate understanding that force is a vector; identify all forces acting on a body;resolve forces into components;find the resultant of a set of forces;demonstrate understanding of and apply the concept of equilibrium;Pupils work in pairs:One pupil attaches string to a box, places it on the desk, lifts desk up and pulls box up the desk.The other pupil attaches string to a box, places it on the desk, lifts desk up until the?box moves down the desk.They then discuss and decide whether force is a vector or scalarThey define the difference between the mass and the weight of a bodyThey discuss friction, determining its direction in different scenarios (link with Science?)Class discussion on the different types of forces (potential for cross curricular link with Science)Pupils work in pairs to mark on all the forces acting on a body in different scenarios. They have sheets showing these scenarios and different cards listing different forces e.g. Weight, Reaction, Resistance, Tension, Tractive Force etc.Pupils individually work through a worksheet marking all the forces acting on a bodyReview trigonometry from GCSE MathsPupils individually work through a worksheet finding sides in a right-angled triangle given the hypotenuse and an angleTeacher draws a force of 36N acting at 20? to the horizontal on the whiteboardPupils work in pairs discussing?? Which directions would it be easy to split the force into (resolving)?? How can each resolved part be found (component)Pupils are given sets of workcards with diagrams of forces acting at an angleThey work in pairs to split these forces into their componentsThey then individually work through textbook P81 Ex 18ATeacher puts the example on pg. 82 on the whiteboardPupils work in pairs, one resolving the left side and the other resolving the right sideThey draw a diagram showing the resolved forces, one finding the resultant horizontal force and the other finding the resultant vertical forceThey then plan together how to find the magnitude and direction of the resultant force and individually work this out, checking each other’s answersThey then individually work through textbook P83 Ex 18BTeacher then draws the rectangle PQRS on the whiteboardPupils individually mark on the following forces:6N acting along the side SR4N acting along the side SR7N acting along the side SR3N acting along the side SRThey then discuss how to find the magnitude and direction of the resultant forceThey then finish the question and check in pairs each other’s workThey then individually work through textbook P84 Ex 18CTeacher puts the example on pg. 84 on the whiteboardPupils work in pairs and discuss which other 2 perpendicular directions they could resolve the forces in to make the solution easier than resolving horizontally and verticallyThey then plan together how to find the magnitude and direction of the resultant force and individually work this out, checking each other’s answersThey then individually work through textbook P85 Ex 18D Pupils in pairsdescribe scenarios when forces are in equilibrium (potential for cross-curricular link with Science)discuss and try to define ‘equilibrium of forces’Teacher puts the example on pg. 85 on the whiteboardPupils work in pairs and discuss how to find Q and R, with prompting from the teacher if needed)They then individually work through the solution, checking each other’s answersThey then individually work through textbook P86 Ex 18ETeacher puts the example on pg. 86 on the whiteboardPupils work in pairs and write down how to find Q and RThey then individually work through textbook P86 Ex 18FThey then in pairs work through textbook P87 Ex 18GWOComm – T&LWO Comm – T&L SM SMWOComm – T&LWOComm – T&LPSWOComm – T&LWOSMSMWOWOUICTWOWOSMWOWOSMPSResourcesTextbook:Further Mathematics for CCEA GCSE LevelWorksheet and cards – Resources 16 to 19UnitLearning outcomes Suggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal Capabilities Newton’s Laws of MotionStudents should be able to:apply F = ma to the following scenarios:a body in horizontal or vertical motion;a body on an inclined plane;two connected bodies in rectilinear motion;(F = μ R will not be tested – if included, friction will be given as a value or as a value per unit mass where appropriate.)Teacher shows videos about Newton’s laws (potential for cross-curricular link with Science)Review direct proportion from GCSE MathsTeacher then shows that ‘F is proportional to a’ equates to F = maPupils write down Newton’s 3 lawsTeacher gives sets of pairs of cards, one showing 2 values (of F, m, a) and the other showing the other value e.g. card 1 with F = 8N and mass = 4kg and card 2 having a = 2 m/s2Pupils work in pairs to match the cardsThey then match cards showing mass and weight e.g. 6kg matched with 60NThey then individually work through textbook P79 Ex 17AAfter equations of motion have been taughtTeacher puts the following question on the whiteboard:Find the force needed to accelerate a mass of 8kg from 5m/s to 17m/s in travelling 33mPupils work in pairs and discusswhat must you find firsthow can you find thiswhich equation do you useThey then individually work through the solution, checking each other’s answersThey then individually work through textbook P79 Ex 17BTeacher puts the following question on the whiteboard:Find the force needed to bring a body mass 3kg travelling at 12m/s to rest in 2sPupils work in pairs and discusswhat must you find firsthow can you find thiswhich equation do you useThey then individually work through the solution, checking each other’s answersThey then individually work through textbook P80 Ex 17CTeacher puts the following question on the whiteboard:A block mass 7.6kg is initially at rest. It is pulled along a rough horizontal plane by a horizontal force of 58N. The frictional force on the block is 45.6NFind (i) its acceleration(ii) the time it takes to reach a velocity of 4m/sPupils will work in pairs anddiscuss which direction friction will act inplan how to do this question, with teacher prompting if necessaryThey then individually work through the solution, checking each other’s answersThey then individually work through worksheetTeacher puts the following question on the whiteboard:A block mass 7.2kg rests on a rough plane inclined at 36° to the horizontal. A force of 64N parallel to, and acting up, the plane is applied to the body and it begins to accelerate reaching 6m/s in 3s.Find the frictional force.Pupils will work in pairs anddiscuss which direction friction will act inplan how to do this question, with teacher prompting if necessaryThey then individually work through the solution, checking each other’s answersThey then individually work through worksheetTeacher puts the following question on the whiteboard:A van of mass 1100kg tows a trailer of mass 560kg by means of a lighthorizontal tow barPupils work in pairs marking on all the forcesTeacher then completes the question on the whiteboard:The tractive force produced by the van’s engine is 3960N. The van and trailer travel along a straight horizontal road and accelerate at1.6 m/s2The resistance to motion of the van is 950NFind (i) the resistance to motion of the trailer(ii) the magnitude of the tension in the tow barPupils work in pairs to plan a strategy for doing this question and then complete itReview solving simultaneous equations from GCSE MathsPupils individually do a worksheet on simultaneous equations of the type 20 – x = 2y and x – 2 = 4yTeacher puts the following question on the whiteboard:A van of mass 1050kg tows a trailer of mass 520kg by means of a light horizontal tow bar. The tractive force produced by the van’s engine is 3750N. The van and trailer travel along a straight horizontal roadThe resistance to motion of the van is 860N and the resistance to motion of the trailer is 425NFind (i) the acceleration(ii) the magnitude of the tension in the tow barPupils work in pairs to plan a strategy for doing this question and then complete itThey then individually work through textbook P97 Ex 21AReview proportion and compound measures from GCSE MathsTeacher puts the following question on the whiteboard:The resistance to motion of a car mass 950kg is 0.76N per kgand asks pupils to write down how to find the total resistancePupils work in pairs. They are given cards with 2 of 3 variables on one and the other variable on another card (e.g. resistance is 0.6N per kg, mass 1000kg, total resistance 600N). They then match the cardsTeacher puts the following question on the whiteboard:A van of mass 1125kg tows a trailer of mass 540kg by means of a light horizontal tow bar. The tractive force produced by the van’s engine is 3250N. The van and trailer travel along a straight horizontal roadThe resistance to motion of the van is 0.86N per kg of mass and the resistance to motion of the trailer is 0.82N per kg of mass.Find (i) the acceleration(ii) the magnitude of the tension in the tow barPupils work in pairs to plan a strategy for doing this question and then complete itThey then individually work through textbook P99 Ex 21BThe class work in groups. Teacher gives a pulley and different sets of masses to each group. They investigate what happens putting 1 mass on each pulley and repeat their work with different sets of massesEach group reports back re e.g. direction of motion, tension in the string, force acting on the pulley (potential to link with Science)Teacher puts a diagram on the whiteboard showing masses of 6kg and 3kg connected over a smooth light pulley. Pupils individually mark on all the forces, indicating the direction of motionThe pupils work in pairs. They are given cards showing different pulleys and different masses. They mark on all forcesTeacher then asks the pupils individually to write down how to find(i) the tension in the string,(ii)the acceleration of each body and(iii)the force acting on the pulley (with guidance as needed)The pupils work in pairs going back to the cards and finding(i) the tension in the string,(ii)the acceleration of each body and(iii)the force acting on the pulleyThey then individually work through textbook P102 Ex 21DThey then work in pairs through textbook P104 Ex 21ETeacher draws example on p106 in textbookPupils discuss how to solve this responding to the following questions:what is the same this time as the previous questionswhat is different this time from the previous questionswhat direction will the system move in when releasedwhat forces do not affect the motionwhat strategy should be usedThe teacher then works through the solutionThey then individually work through textbook P106 Ex 21FUICTWOSMUICTWOComm – T&LSMSMWOComm – R&WSMUICTWOComm – T&LPSUICTWOComm – R&WPSUICTWOWOUICTSMSMPSSMComm – T&LWOWOSMSMWOPSWOSMSMResourcesTextbook:Further Mathematics for CCEA GCSE LevelVideo linkswatch?v=mn34mnnDnKUwatch?v=NYVMlmL0BPQwatch?v=5JiR3rD0FvcWorkshops and cards – resources 20 27MomentsStudents should be able to:demonstrate understanding of the Principle of Moments and equilibrium of a rigid body (restricted to a horizontal uniform rod supported by one or two pivots).Teacher shows videos about moments (potential to link with Science)Teacher defines a momentTeacher draws a diagram on the whiteboard showing a uniform rod AB weight 35N length 10m with a weight of 20N marked on 3m from AIn pairs pupils write down the moment of(i) 35N from A(i) 20N from A(i) 20N from BPupils then match pairs of cards showing rods and momentsTeacher shows videos about principle of moments (potential to link with Science)Teacher draws a diagram on the whiteboard showing a uniform rod AB mass 6kg length 8m with supports at C and D where AC = 3m and DB = 2mIn pairs pupils mark on all the forcesThey then in turn make up similar questions for the other to draw(i) 35N from A(i) 20N from A(i) 20N from BThe pupils then go back to the previous question on the whiteboard and discuss in pairs:how to find the reactionswhy equating forces as a first step won’t workwhat other strategy they could useThe teacher works through this exampleThey then individually work through textbook P111 Ex 22ATeacher writes example bottom right on p111 in textbookClass discuss strategy neededPupils individually work through this and then check each other’s workThey then individually work through textbook P112 Ex 22BTeacher writes example top right on p112 in textbook on whiteboardPupils individually mark on all the forcesThey then work in pairs marking on all the forces on diagrams shown on cardsThe teacher works through this example. Pupils then individually work through textbook P112 Ex 22CPupils in pairs set a metre stick across 2 supports. One moves their fingers along the stick until it just starts to tiltThey then discuss and write down what is happening at that momentTeacher writes example 1 top right on p113 in textbook on whiteboardPupils in pairsdiscuss the reaction at Cmark on all the forcesdecide on a strategyThe teacher works through this exampleTeacher writes example 2 bottom right on p113 in textbook on whiteboardPupils in pairsdiscuss the reaction at Cmark on all the forcesdecide on a strategyThe teacher works through this examplePupils then individually work through textbook P114 Ex 22UICTWOComm – T&LUICTWOCommSMComm – T&LSMWOSMPSWOSMWOComm – T&LWOComm – T&LComm – T&LSMResourcesTextbook:Further Mathematics for CCEA GCSE LevelVideo linkswatch?v=UoCWHoHR8IUmeenng/turning-effect-of-forceswatch?v=j71rLH-NpPwworksheets and cards – Resources 28 – 29Unit 3StatisticsIntroductionThe purpose of this Planning Framework is to support the teaching and learning of GCSE Further Mathematics Unit 3: Statistics. The Planning Framework is based on specification content but should not be used as a replacement for the specification. It provides suggestions for a range of teaching and learning activities which provide opportunities for students to develop their:Knowledge and understandingSubject specific skillsThe Cross-Curricular SkillsThinking Skills and Personal CapabilitiesThe Planning Framework is not mandatory, prescriptive or exhaustive. Teachers are encouraged to adapt and develop it to best meet the needs of their students.Subject Skills Assessed through Further Mathematics:The following skills are assessed in GCSE Further Mathematics Unit 3: Statistics:Use and apply standard techniquesaccurately recall facts, terminology and definitionsuse and interpret notation correctlyaccurately carry out routine proceduresaccurately carry out sets tasks requiring multi-step solutionsReason, interpret and communicate mathematicallymake deductions to draw conclusions from mathematical informationmake inferences to draw conclusions from mathematical informationconstruct chains of reasoning to achieve a given resultinterpret information accuratelycommunicate information accuratelypresent argumentspresent proofsassess the validity of an argumentcritically evaluate a given way of presenting informationSolve problems within mathematics and in other contextstranslate problems in mathematical contexts into a processtranslate problems in mathematical contexts into series of processestranslate problems in non-mathematical contexts into a mathematical processtranslate problems in non-mathematical contexts into a series of mathematical processmake and use connections between different parts of mathematicsinterpret the results in the context of the given problemevaluate methods usedevaluate results obtainedevaluate solutions to identify how they may have been affected by assumptions madeSupporting the Development of Statutory Key Stage 4 Cross-Curricular Skills and Thinking Skills and Personal CapabilitiesThis specification builds on the learning experiences from Key Stage 3 as required for the statutory Northern Ireland Curriculum. It also offers opportunities for students to contribute to the aim and objectives of the Curriculum at Key Stage 4, and to continue to develop the Cross-Curricular Skills and the Thinking Skills and Personal Capabilities. The extent of the development of these skills and capabilities will be dependent on the teaching and learning methodology used.Cross-Curricular Skills at Key Stage 4CommunicationStudents should be able to:communicate meaning, feelings and viewpoints in a logical and coherent manner, by using appropriate mathematical language and notation in response to open‐ended tasks, problems, structured questions or examination questions;make oral and written summaries, reports and presentations, taking account of audience and purpose through the use of varied learning activities applied to a wide range of contexts that require students to organise and record data, justify choice of strategy to solve problems, articulate processes, proofs etc. and provide feedback from collaborative learning activities;participate in discussions, debates and interviews by sharing ideas, investigating misconceptions, exploring alternative strategies, justifying choice of strategy, negotiating decisions and listening to others;interpret, analyse and present information in oral, written and ICT formats by developing a mathematical solution to a problem and communicating ideas, strategies and solutions; and,explore and respond, both imaginatively and critically, to a variety of texts by using open‐ended tasks and activities.Using MathematicsStudents should be able to:use mathematical language and notation with confidence, for example conditional probability, Binomial, Normal and correlation coefficient;use mental computation to calculate, estimate and make predictions in a range of simulated and real‐life contexts, for example finding differences in ranks;select and apply mathematical concepts and problem‐solving strategies in a range of simulated and real‐life contexts, for example Spearman’s rank to investigate relationship between results achieved in 2 subjects;interpret and analyse a wide range of mathematical data, for example calculating an estimate for the mean and standard deviation from a grouped frequency distribution;assess probability and risk in a range of simulated and real‐life contexts, for example tree diagrams or Venn diagrams; andpresent mathematical data in a variety of formats which take account of audience and purpose, for example numerical, graphical and algebraic representations.Using ICTStudents should be able to make effective use of information and communications technology in a wide range of contexts to access, manage, select and present information, including mathematical information, for example calculators, suitable software packages to research data online, analysing data and working with formulae in spreadsheets.Thinking Skills and Personal Capabilities at Key Stage 4Self‐ManagementStudents should be able to:plan work by identifying appropriate strategies, working systematically and persisting with open‐ended tasks and problems;set personal learning goals and targets to meet deadlines such as identifying, prioritising and managing actions required to develop competence and confidence in more challenging mathematics;monitor, review and evaluate their progress and improve their learning by self-evaluating their performance, identifying strengths and areas for improvement and seeking support where required; andeffectively manage their time by planning, prioritising and minimising distractions in order to meet deadlines set by teachers.Working with OthersStudents should be able to:learn with and from others through co‐operation by engaging in discussions and explaining ideas, challenging and supporting one another, creating and solving each other’s questions and working collaboratively to share methods and results during small group tasks;participate in effective teams and accept responsibility for achieving collective goals by working together on challenging small group tasks with shared goals but individual accountability; andlisten actively to others and influence group thinking and decision‐making, taking account of others’ opinions by participating constructively in small group activities, articulating possible problem‐solving strategies and presenting a well thought out rationale for one approach.Problem SolvingStudents should be able to:identify and analyse relationships and patterns, for example make links between sets of data to identify correlation using Spearman’s Rank Correlationpropose justified explanations, for example by proving a mathematical statement is true;analyse critically and assess evidence to understand how information or evidence can be used to serve different purposes or agendas, for example using data to make predictions about the likelihood of future events occurring;analyse and evaluate multiple perspectives, for example modelling real‐life scenarios as further mathematics problems;weigh up options and justify decisions, for example using the most appropriate method to solve complex mathematical problems; andapply and evaluate a range of approaches to solve problems in familiar and novel contexts, for example by choosing the appropriate diagrammatic method in probability.There is an emphasis on problem solving within this new specification which will require the development of all these skills.Although not statutory at Key Stage 4 this specification also allows opportunities for further development of the Thinking Skills and Personal Capabilities of Managing Information and Creativity.Key FeaturesThe Planning Framework:Includes suggestions for a range of teaching and learning activities which are aligned to the GCSE Further Mathematics Unit 3: Statistics specification content.Highlights opportunities for inquiry-based learning.Indicates opportunities to develop subject knowledge and understanding and specific skillsIndicates opportunities to develop the Cross-Curricular Skills and Thinking Skills and Personal Capabilities.Provides relevant, interesting, motivating and enjoyable teaching and learning activities which will enhance the student’s learning experience.Makes reference to supporting resources.ContentLearning outcomesSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal CapabilitiesCentral Tendency and DispersionStudents should be able tocalculate the mean and standard deviation from data or estimates of these from grouped data;calculate the mean and standard deviation for combined sets of data;demonstrate knowledge of the effect on the mean and standard deviation of a linear transformation on a set of data;Prior knowledge: Discussion on the purpose of the mean and range. Revision of calculating the mean from a grouped data set.Introduction to standard deviation and its purpose.Collect and sort data appropriately forShoe sizes of 20 Year 9 pupilsHeights of Year 10 pupilsUse the formulae to calculate the mean (or estimate for the mean) and standard deviationUse extracts of published data such as census data about the population of Northern Ireland to calculate the mean and standard deviation. Extracting data on, for example, qualifications to compare means and standard deviations. Data could also be used to find the mean and standard deviation of combined data sets, for example, occupations for males and females. Presentation of results to peers to demonstrate knowledge and understanding of findings.Use an appropriate software package to explore the effects on the mean and standard deviation by transforming data in a systematic way.SMSMWOComm – RUICTComm – WUICTResourcesNI Census data .uk/Census/2011Census.htmlGCSE Level Further Mathematics for CCEA GCSE level.ProbabilityStudents should be able tocalculate combined probabilities using the addition rule to include events which may not be mutually exclusive; calculate and interpret conditional probabilities using expected frequencies, two-way tables, tree diagrams and Venn diagrams;use the most appropriate method to solve complex problems, including the construction and use of Venn diagrams and tree diagrams;Prior knowledge: Calculating basic probabilities possibly through a snap card game or use of mini whiteboards. Consolidate language of probability already assumed.Discussion on types of events, recap meaning on dependent/independent/mutually exclusive events.Use Venn diagrams to revisit ‘addition’ ruleProvide opportunities to create a two way table or Venn diagram to manage/sort information and calculate probability/expected frequenciesUse Venn diagrams to demonstrate not mutually exclusive eventsUse tree diagrams to display probabilities and demonstrate the ‘multiplication rule’.Using the language/symbols associated with addition/multiplicationUsing tree diagrams to introduce the concept of conditional probability further exploration through Venn diagramsGroup activity: Pupils given data for example, number of pupils studying Biology, Chemistry, Physics and/or combinations of these subjects and asked to present in the most appropriate manner. Using their diagram to calculate probabilities/problem solve algebraically with the introduction of an unknown valueComm – T&LWOPSComm – WPSWOResourcesGCSE Level Further Mathematics for CCEA GCSE level.Resources 1 to 3Mini whiteboardsContentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal Capabilities Binomial DistributionStudents should be able touse Pascal’s triangle to expand (p+q)n where n ≤ 8;understand and use the binomial expansion to calculate probabilities in real life contexts;Prior knowledge: Generating sequences and exploring patterns; rules of multiplication with indices, expanding double brackets, independent probability.Research in groups the mathematician Blaise Pascal, this could be directed by the class teacher with specific information to be foundExplore the generation of successive rows in Pascal’s triangle and the number patterns contained within the triangle, this can be done in pairsLinking to ‘Expanding three brackets’ in Unit 1, pupils can then investigate coefficients of powers of the variables in polynomial expansions of the form recording their observations and drawing conclusions from findings.This then leads to the introduction of a binomial expansion and the binomial distribution. It might be useful to discuss the sum of, for example, being 1 and so the sum of the probabilities for =1 with each term representing a probability, to link the 2 areas.The conditions for a binomial distribution given to the pupils;The number of trials is fixedTrials are independentThere are only 2 possible outcomes described as ‘success’ and ‘failure’With the probability of ‘success’ for each trial remaining the same and conversely the probability of ‘failure’ for each trial remaining the same and can now be used to represent the probabilities of ‘success’ and ‘failure’Pupils may work on paired/group activities to identify and extract probabilities using Pascal’s triangleThis is also opportunity to link to ‘Combinations’ in Unit 4UICTComm – T&LWOComm – T&LWOSMResourcesGCSE Level Further Mathematics for CCEA GCSE plete Advanced level Mathematics Pure Mathematics, Stanley ThornesResources – 4 to 9ContentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal Capabilities Normal DistributionStudents should be able torecognise that the distribution of many real world variables takes the shape of a bell curve;calculate a single probability from the normal distribution using tables and where the mean and standard deviation will be given;Prior knowledge: Pupils already have experience with histograms and box plots. Patterns in their shape can be explored to introduce the concept of a ‘normal curve’/‘bell curve’, the mean and the standard deviation.Pupils also have basic knowledge of central tendency and dispersionDiscussion on how there are a number of examples of variables that are normally distributed in everyday life e.g. height of 100 Year 10 pupils, scores achieved in an examination. Explore why this distribution is important in everyday life.Explore the normal curve;Area under the curve is 1The curve is symmetrical about the mean(mu), therefore the mean, mode and median are the sameThe standard deviation (sigma), is the proximity to the meanUse examples of normal curves to relate how approximately 68% of data lies within 1 standard deviation of the mean, 95% within 2 standard deviations and 99.7% within 3 standard deviationStandardising variablesWhy this is necessary, 0,1)Areas under the curve represent probabilitiesStandardising a variable, ‘z’ values, to calculate a probability, (z)Encouraging pupils to draw clear diagrams to help calculate the probabilityUsing the Normal Probability table to read correct probability, this could be practised through card matching activity.WOComm – T&LPSPSComm – WComm – RResourcesImproving Learning in Mathematics Statistics unitHeinemann Modular Mathematics for London AS and A-Level Statistics 1 (T1)Resources 10 to 13ContentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal Capabilities Bivariate AnalysisStudents should be able tocalculate and interpret Spearman’s Rank Correlation coefficient;draw the line of best fit by eye passing through (x,y);calculate and use the equation of the line of best fit.Prior knowledge: Scatter graphs, correlation and line of best fit. Calculating mean of a data set. Finding the equation of a straight line using 2 coordinates.Ask pupils to discuss if they think certain pairs of variables are related. Explain why they think the variables are or are not related. These can be in written form or in graph form. This can be extended to refine understanding of strength in correlation. Think pair share activities could be used here.Introduction to Spearman’s Rank Correlation coefficient, being the relationship between the ranks of 2 sets of variables. Pupils given 2 sets of data, for example, scores achieved by 10 pupils in a maths and physics exam. It is useful to discuss what they think the relationship is before working through the activity. A guided activity through ranking their data sets, calculating differences, differences squared and using the formula. This activity could be adapted to investigate positive/negative (weak and strong) and perfect correlationsExtend the previous activity to plotting data on x-y axes, line of best fit passing through (x,y)There is an opportunity to use an appropriate software package to plot data and calculate the correlation coefficientWOSMComm – T&LWOSMPSUICTResourcesGCSE Level Further Mathematics for CCEA GCSE level.Unit 4Discrete and Decision MathematicsIntroductionThe purpose of this Planning Framework is to support the teaching and learning of GCSE Further Mathematics Unit 4: Discrete and Decision Mathematics. The Planning Framework is based on specification content but should not be used as a replacement for the specification. It provides suggestions for a range of teaching and learning activities which provide opportunities for students to develop their:Knowledge and understandingSubject specific skillsThe Cross-Curricular SkillsThinking Skills and Personal CapabilitiesThe Planning Framework is not mandatory, prescriptive or exhaustive. Teachers are encouraged to adapt and develop it to best meet the needs of their students.Subject Skills Assessed through Further Mathematics Unit 4: Discrete and Decision Mathematics:The following skills are assessed in GCSE Further Mathematics Unit 4: Discrete and Decision Mathematics:Use and apply standard techniquesaccurately recall facts, terminology and definitionsuse and interpret notation correctlyaccurately carry out routine proceduresaccurately carry out sets tasks requiring multi-step solutionsReason, interpret and communicate mathematicallymake deductions to draw conclusions from mathematical informationmake inferences to draw conclusions from mathematical informationconstruct chains of reasoning to achieve a given resultinterpret information accuratelycommunicate information accuratelypresent argumentspresent proofsassess the validity of an argumentcritically evaluate a given way of presenting informationSolve problems within mathematics and in other contextstranslate problems in mathematical contexts into a processtranslate problems in mathematical contexts into series of processestranslate problems in non-mathematical contexts into a mathematical processtranslate problems in non-mathematical contexts into a series of mathematical processmake and use connections between different parts of mathematicsinterpret the results in the context of the given problemevaluate methods usedevaluate results obtainedevaluate solutions to identify how they may have been affected by assumptions madeResource: Planning template for Further Mathematics Unit 4: Discrete and Decision Mathematics and the Subject Skills at Key Stage 4The cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.Supporting the Development of Statutory Key Stage 4 Cross-Curricular Skills and Thinking Skills and Personal Capabilities through Further Mathematics Unit 4: Discrete and Decision Mathematics:This specification builds on the learning experiences from Key Stage 3 as required for the statutory Northern Ireland Curriculum. It also offers opportunities for students to contribute to the aim and objectives of the Curriculum at Key Stage 4, and to continue to develop the Cross-Curricular Skills and the Thinking Skills and Personal Capabilities. The extent of the development of these skills and capabilities will be dependent on the teaching and learning methodology used.Cross-Curricular Skills at Key Stage 4Communication skillsTeachers should enable pupils to develop skills in:communicating meaning, feelings and viewpoints in a logical and coherent manner;making oral and written summaries, reports and presentations that take account of audience and purpose;participating in discussions, debates and interviews;interpreting, analysing and presenting information in oral, written and ICT forms; andexploring and responding, both imaginatively and critically, to a variety of texts.Using Mathematics skillsTeachers should enable pupils to develop skills in:using mathematical language and notation with confidence;using mental computation to calculate, estimate and make predictions in a range of simulated and real-life contexts;selecting and applying mathematical concepts and problem solving strategies in a range of simulated and real-life contexts;interpreting and analysing a wide range of mathematical data;assessing probability and risk in a range of simulated and real-life contexts; andpresenting mathematical data in a variety of formats that take account of audience and purpose.Using ICT skillsTeachers should enable pupils to make effective use of information and communications technology (ICT) Pupils should use ICT in a range of contexts to access, manage, select and present information, including mathematical information.Resource: Discrete and Decision Mathematics and the Cross Curricular Skills at Key Stage 4Thinking Skills and Personal Capabilities at Key Stage 4Self-Management skillsTeachers should enable pupils to develop the capability to:plan work;set personal learning goals and targets to meet deadlines;monitor, review and evaluate their progress and improve their learning; andeffectively manage their time.Although not statutory at Key Stage 4 this specification also allows opportunities for further development of the Thinking Skills and Personal Capabilities of Managing Information and Creativity.Working with Others skillsTeachers should enable pupils to develop the capability to:learn with and from others through co-operation;participate in effective teams and accept responsibility for achieving collective goals; andlisten actively to others and influence group thinking and decision-making, taking account of others’ opinions.Problem Solving skillsTeachers should enable pupils to develop skills to:identify and analyse relationships and patterns;propose justified explanations;analyse critically and assess evidence to understand how information or evidence can be used to serve different purposes or agendas;analyse and evaluate multiple perspectives;weigh up options and justify decisions;apply and evaluate a range of approaches to solve problems in familiar and novel contexts.Resource: Discrete and Decision Mathematics and the Thinking Skills and Personal Capabilities at Key Stage 4Key FeaturesThe Planning Framework:Includes suggestions for a range of teaching and learning activities which are aligned to the GCSE Further Mathematics Unit 4: Discrete and Decision Mathematics specification content.Highlights opportunities for inquiry-based learning.Indicates opportunities to develop subject knowledge and understanding and specific skillsIndicates opportunities to develop the Cross-Curricular Skills and Thinking Skills and Personal Capabilities.Provides relevant, interesting, motivating and enjoyable teaching and learning activities which will enhance the student’s learning experience.Makes reference to supporting resources.Unit/option contentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal CapabilitiesCountingStudents should be able to:demonstrate understanding of and use the addition and multiplication principles to count events in series and parallel respectively;calculate the number of ways of arranging r objects from n objects;calculate the number of ways of choosing r objects from n objects;Teaching thinking through Discrete and Decision Mathematics activitiesDefine keywordsFrom memory write a definition of a keyword related to this topic.For example write a definition for the keyword PermutationFor example write a definition for the keyword CombinationExplain keywordsIn pairs explain the difference between the keywords Permutation and Combination to your partner. Use a list of the possible outcomes when tossing a coin 3 times to help you explain the difference between the two terms.Explain by example the multiplication principle for pare and Contrast the efficiency of the following counting methods: Listing, two way tables, tree diagrams, Pascal’s triangle. Use a list of the possible outcomes when tossing a coin 4 times to help you compare and contrast the methods.Images and StatementsProvide a suitable image with several statements which relate to it. Learners have to decide which statements best describe the image. For example an image of Pascal’s triangle with the 6 in the fourth row circled. Learners are asked to choose from a list of statements of the form: How many ways can ‘x’ items be chosen from ‘y’ items? and match the correct statement to the circled number. How many ways can ‘2’ items be chosen from ‘4’ items? corresponds to the circled 6.Worked examples and repeated practicePresent several worked examples on using the boxes method for solving a staged variety of permutations problems. Set pupils a series of problems to solve using the knowledge and understanding modelled in the worked examples. Ask the pupils to reflect on any patterns or common procedures that they followed when they were producing solutions to the set problems and to consider if and how their solutions were improving as they worked through the set of problems. Ask the pupils (in pairs) to write and solve a problem that could be solved using their knowledge about permutations. They should be able to explain and justify their solution to a larger group within the class.7 questionsUse the internet to research Blaise Pascal and his contribution to Decision and Discrete Mathematics. Use the seven questions What? When? Where? Which? Who? Why? and How? to focus research. At least 5 out of the 7 questions should be answered. Active/Passive LearningUse active learning strategies to approach past paper practice. For example ask learners to write a further question based on the source material m – T&LWOComm – T&LPSPSUICTPSResourcesResources 1 to 15Planning template for Further Mathematics Unit 4: Discrete and Decision Mathematics and the Subject Skills at Key Stage 4LO 1 Starter ActivitiesLO 1 Main Lesson Activities Discrete and Decision Mathematics and the Cross Curricular Skills at Key Stage 4Discrete and Decision Mathematics and the Thinking Skills and Personal Capabilities at Key Stage 4LO 1 Plenary ActivitiesLO 1 Assess Learning ActivitiesCountingLO 1.1 ASBAT demonstrate understanding of and use the addition and multiplication principles to count events in series and parallel respectivelylower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO1.1aLO1.1bLO1.1c…CountingLO 1.2 ASBAT calculate the number of ways of arranging r objects from n objectslower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO1.2aLO1.2bLO1.2c…CountingLO 1.3 ASBAT calculate the number of ways of choosing r objects from n objectslower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO1.3aLO1.3bLO1.3c…Unit/option contentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal CapabilitiesLogicAll should be able to:demonstrate understanding of the concept of Boolean variables, including forming compound expressions using logical operators AND, OR and NOT;use truth tables to prove the equivalence of propositional statements (involving no more than 3 variables);Teaching thinking through Discrete and Decision Mathematics activitiesDefine keywordsFrom memory write a definition of a keyword related to this topic.For example write a definition for the keyword propositionFor example write a definition for the keyword connectiveFor example write a definition for the keywords Boolean variableExplain keywordsIn pairs explain the difference between the keywords simple proposition and compound proposition to your partner. Use a list of statements to help you explain the difference between the two terms.Sort and classify statements into groups and explain the criteria you used to your partner.This should be monitored by the pare and Contrast the language of sets and propositions.For example considerSets PropositionsComplementNegationVenn diagramsTruth tablesIntersectionConjunctionUnionAlternationImages and StatementsProvide an image with several statements which relate to it. Learners have to decide which statements best describe the information presented in the image. For example an image of two truth tables that represent logical expressions is presented to learners. Learners are asked to decide if the logical expressions are equivalent.Worked examples and repeated practicePresent several worked examples covering a range of truth tables. Set pupils a series of problems to solve using the knowledge and understanding modelled in the worked examples. Ask the pupils to reflect on any patterns or common procedures that they followed when they were producing solutions to the set problems and to consider if and how their solutions were improving as they worked through the set of problems. Ask the pupils (in pairs) to write and solve a problem that could be represented using their knowledge about truth tables. They should be able to explain and justify their solution to a large group within the class.7 questionsUse the internet to research George Boole and his contribution to Decision and Discrete Mathematics. Use the seven questions What? When? Where? Which? Who? Why? and How? to focus research. At least 5 out of the 7 questions should be answered.Active/Passive LearningUse active learning strategies to approach past paper practice. For example ask learners to write a parallel problem that corresponds to an original m – T&LSMWOSMPSPSWOUICTPSResourcesResources 16 to 17Planning template for Further Mathematics Unit 4: Discrete and Decision Mathematics and the Subject Skills at Key Stage 4LO 2 Starter ActivitiesLO 2 Main Lesson ActivitiesDiscrete and Decision Mathematics and the Cross Curricular Skills at Key Stage 4Discrete and Decision Mathematics and the Thinking Skills and Personal Capabilities at Key Stage 4LO 2 Plenary ActivitiesLO 2 Assess Learning ActivitiesLogicLO 2.1 ASBAT demonstrate understanding of the concept of Boolean variables, including forming compound expressions using logical operators AND, OR and NOTlower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO2.1aLO2.1bLO2.1c…LogicLO 2.2 ASBAT use truth tables to prove the equivalence of propositional statements (involving no more than 3 variables)lower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO2.2aLO2.2bLO2.2c…Unit/option contentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal CapabilitiesLinear ProgrammingAll should be able to:model real life scenarios as linear programming problems;use graphical methods to maximise or minimise an expression involving up to five inequalities in one or two variables. (Solutions may be real numbers or restricted to integers);Teaching thinking through Discrete and Decision Mathematics activitiesDefine keywordsFrom memory write a definition of a keyword related to this topic.For example write a definition for the keyword regionFor example write a definition for the keyword optimisingExplain keywordsIn pairs explain the difference between the keywords maximising and minimising to your partner. Use images of a region bounded by inequalities to help you explain the difference between the two terms.Graphs and StatementsProvide a suitable graph with several statements which relate to it. Learners have to decide which statements best describe the information presented in the graph. A solution to a Linear Programming question is given to learners along with several True/False statements that relate to the graph. Learners have to decide the Truth or falsehood of the statements given.Provide a suitable graph, for example a graphical solution to a Linear Programming question is given to learners. The learners have to identify the optimum solution from the possibilities presented in the graph. Learners should justify their findings to their partner or group. Responses should be monitored by the teacher.Worked Examples and Guided PracticePresent several worked examples for using linear programming to represent and solve a staged variety of optimisation problems. Set pupils a series of problems to solve using the knowledge and understanding modelled in the worked examples.Ask the pupils to reflect on any patterns or common procedures that they followed when they were producing solutions to the set problems and to consider if and how their solutions were improving as they worked through the set of problems.ICTUse graphing software to demonstrate a solution to a linear programming problem.Active/Passive LearningUse active learning strategies to approach past paper practice. For example ask learners to consider ‘what would happen if ‘one of the constraints was m – T&LSMComm – T&LComm – RPSPSUICTPSResourcesResources 18 to 21Planning template for Further Mathematics Unit 4: Discrete and Decision Mathematics and the Subject Skills at Key Stage 4LO 3 Starter ActivitiesLO 3 Main Lesson ActivitiesDiscrete and Decision Mathematics and the Cross Curricular Skills at Key Stage 4Discrete and Decision Mathematics and the Thinking Skills and Personal Capabilities at Key Stage 4LO 3 Plenary ActivitiesLO 3 Assess Learning ActivitiesLinear ProgrammingLO 3.1 ASBAT model real life scenarios as linear programming problemslower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO3.1aLO3.1bLO3.1c…Linear ProgrammingLO 3.2 ASBAT use graphical methods to maximise or minimise an expression involving up to five inequalities in one or two variableslower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO3.2aLO3.2bLO3.2c…Unit/option contentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activities Supporting Key Stage 4 Statutory Skills and Personal CapabilitiesTime SeriesAll should be able to:demonstrate understanding of why we smooth data;calculate appropriate moving averages (using three, four or five points);draw a trend line and use it to make predictions;(The plotting of original data will be given)Teaching thinking through Discrete and Decision Mathematics activitiesDefine keywordsFrom memory write a definition of a keyword related to this topic.For example write a definition for the keyword smoothingFor example write a definition for the keyword trend lineExplain keywordsIn pairs explain the keywords moving average to your partner. Use images of a time series graph to help you explain the term.Graphs and StatementsProvide a relevant graph. Learners have to decide which statements best describe the information presented in the graph. Provide a Time Series Graph representing sales over a period of time is given to learners along with several True/False statements that relate to the graph. Learners have to decide the Truth or falsehood of the statements given.Provide a graph for interpretation. For example a Time Series Graph with a trend line drawn. The learners have to identify the point value of the moving average and predict the value for the next period based on the information in the graph. Learners should justify their findings to their partner or group. Responses should be monitored by the teacher.Learners compare and contrast several time series graphs that represent sales over a given period of time.Worked Examples and Guided PracticePresent several worked examples involving the analysis of a time series. Set pupils a series of problems to solve using the knowledge and understanding modelled in the worked examples. Ask the pupils to reflect on any patterns or common procedures that they followed when they were producing solutions to the set problems and to consider if and how their solutions were improving as they worked through the set of problems.ICTUse a spreadsheet to produce a time series graph for ‘sales over a specified period of time’ and calculate the appropriate moving averages.Use a spreadsheet to carry out a ‘what would happen if analysis’ for example by asking the question what would happen to the estimated sales for the next quarter if the sales figures for the last quarter had a different value?Active/Passive LearningUse active learning strategies to approach past paper practice. For example ask learners to consider what would happen if the data values where all increased by the same m – T&LSMWOComm – T&LPSSMPSUICTPSResourcesResources 22 to 31Planning template for Further Mathematics Unit 4: Discrete and Decision Mathematics and the Subject Skills at Key Stage 4LO 4 Starter ActivitiesLO 4 Main Lesson ActivitiesDiscrete and Decision Mathematics and the Cross Curricular Skills at Key Stage 4Discrete and Decision Mathematics and the Thinking Skills and Personal Capabilities at Key Stage 4LO 4 Plenary ActivitiesLO 4 Assess Learning ActivitiesTime SeriesLO 4.1 ASBAT demonstrate understanding of why we smooth datalower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeLO4.1aLO4.1bLO4.1c…Time SeriesLO 4.2 ASBAT calculate appropriate moving averages (using three, four or five points)lower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeLO4.2aLO4.2bLO4.2c…Time SeriesLO 4.3 ASBAT draw a trend line and use it to make predictionslower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO4.3aLO4.3bLO4.3c…Unit/option contentLearning outcomes or Elaboration of contentSuggestions for teaching and learning activitiesSupporting Key Stage 4 Statutory Skills and Personal CapabilitiesCritical Path AnalysisAll should be able to:demonstrate understanding of how an activity network represents a project; (using activities on an arc)construct an activity network from a precedence table;identify the critical path by finding the earliest and latest event times;calculate float times; perform basic scheduling using Gantt charts.Teaching thinking through Discrete and Decision Mathematics activitiesDefine keywordsFrom memory write a definition of a keyword related to this topic.For example write a definition for the keyword early timeFor example write a definition for the keyword late timeFor example write a definition for the keyword float timeExplain keywordsIn pairs explain the keywords Critical activity and Critical path to your partner. Use images of an activity network diagram to help you explain the terms.To Do Next or Not To Do Next? (individual → pair → group activity)This activity focuses on problem solving and decision making by using the question prompt ‘next or not next’?OutlineGenerate a list of activities that form part of a project plan and decide which activities must be completed before another activity can be started. For example in creating a precedence table for the activities listed in a project plan ask ‘is this activity an immediate predecessor’ or is this activity an immediate successor’.Choose one of the options and think through the advantages and disadvantagesReview each and come up with the best optionAsk each group individual/pair/group to explain and justify their decision.Learners should reflect on how this approach to decision making was or was not useful and think of other situations where they could use this approach.Worked Examples and Guided PracticePresent several worked examples involving the creation of a precedence table. Set pupils a series of problems to solve using the knowledge and understanding modelled in the worked examples. Ask the pupils to reflect on any patterns or common procedures that they followed when they were producing solutions to the set problems and to consider if and how their solutions were improving as they worked through the set of problems.Diagrams and StatementsProvide a suitable diagram. Learners have to decide which statements best describe the information presented in the diagram. A diagram of an activity network is given to learners along with several True/False statements that relate to the diagram. Learners have to decide the Truth or falsehood of the statements given.A diagram is provided and learners have to interpret the diagram. For example an Activity Network diagram is given to learners. The learners have to identify the critical path and the shortest time to complete the task represented by the activity network diagram. Learners should justify their findings to their partner or group. Responses should be monitored by the teacher.Design a Gantt Chart for a project to show how the project can be completed.Design a Gantt Chart for a project to show how the project can be completed as efficiently as possible when there is a constraint on the number of workers available.7 questions activityUse the internet to research Henry Gantt and his contribution to Decision and Discrete Mathematics. Use the seven questions What? When? Where? Which? Who? Why? and How? to focus research. At least 5 out of the 7 questions should be answered.Active/Passive LearningUse active learning strategies to approach past paper practice. For example ask learners to consider the same scenario with some altered activity m – T&LSMWOPSPSPSComm – T&LSMPSPSPSUICTPSResourcesResources 32 to 34Planning template for Further Mathematics Unit 4: Discrete and Decision Mathematics and the Subject Skills at Key Stage 4LO 5 Starter ActivitiesLO 5 Main Lesson ActivitiesDiscrete and Decision Mathematics and the Cross Curricular Skills at Key Stage 4Discrete and Decision Mathematics and the Thinking Skills and Personal Capabilities at Key Stage 4LO 5 Plenary ActivitiesLO 5 Assess Learning ActivitiesAppendix 1Critical Path AnalysisLO 5.1 ASBAT demonstrate understanding of how an activity network represents a project (using activities on an arc)lower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO5.1aLO5.1bLO5.1c…Critical Path AnalysisLO 5.2 ASBAT construct an activity network from a precedence tablelower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO5.2aLO5.2bLO5.2c…Critical Path AnalysisLO 5.3 ASBAT identify the critical path by finding the earliest and latest event timeslower order thinking skills ------------------------------------------------------------------------→ higher order thinking skills1Remembering2Understanding3Applying4Analysing5Evaluating6Creatingconceptual knowledge -----------→ abstract knowledgeAfLPrior KnowledgeAFactual KnowledgeBConceptual KnowledgeCProceduralKnowledgeDMetacognitiveKnowledgeThe cells in the planning template can be populated with relevant objectives and activities for teaching, learning and assessing.LO5.3aLO5.3bLO5.3c…Appendix 2More Ideas for Teaching and Learning ActivitiesDiscrete and DecisionDiscrete and Decision MathematicsCountingStarter:Making up a band from a number of people – e.g. One Direction on the X Factor.Arranging the apps on a phone screen – how many different ways can apps be arranged, this can be done on interactive whiteboard using click and drag.Development:Gritting roads in winter – pupils must determine which routes the gritters should take to cover the most roads without going down a road multiple times.LogicStarter:Police investigation game – pupils will be given pictures of suspects of a crime. The teacher will read out witness statements that include the ideas of and, or, not etc. and pupils must eliminate suspects until the criminal is caught.There is a scene in Shrek 3 where Pinokio tries not to give away Shrek’s whereabouts. He uses double and triple negatives in his efforts.Development:What shop has what I need? – different shops each sell a specific number and type of products. The teacher will read out statements that include and, or, not etc. and pupils must decide where to shop to get exactly what they need.Linear ProgrammingDevelopment:Shopping – different combinations of clothes that someone wants to buy, restrictions can be set based on how many products they want or the amount of money they want to spend.Making revision productive – in science, calculations and written questions each take a certain amount of time. How many of each can a pupil do in an hour? The pupil only has a certain number of pages in a book left and questions take up a specific amount of space. Limitations can be set based on their ability in a specific type of question, e.g. they are good at calculations so only want to do three.Time SeriesDevelopment:iPhone sales – determine the general trend for sales of iPhones, provide reasoning for any peaks or apparent ‘decreases’ (statistics availablestatistics/263401/global-apple-iphone-sales-since-3rd-quarter-2007/)Sports team statistics – monitor the progress of a sports team within the school as they have got older.Critical Path AnalysisDevelopment:Plan a day-trip – visit specific tourist locations within a city but the people want to split up and visit different places but still want to be together for parts of the day (e.g. travelling, meals).Planning a Technology project – cross-curricular links with what the GCSE pupils are designing and building in technology, pupils will make a plan for managing their time.Web links to Teaching and LearningResourcesDiscrete and DecisionDiscrete and Decision Mathematics LinksGeneralActivities – various problem-solving activities, accompanied by PowerPoints and guidance for teachers.fsmqs/level-3-decision-mathsCountingStarter – counting the number of ways a coat can be buttoned up and predicting results for more buttons. –finding the best routes. – revision on inequalities and shading the appropriate regions, extra support.teaching-resource/a-level-maths-decision-1-inequality-regions-sheet-6147446Linear ProgrammingPowerPoint – different examples of linear programming questions set in real-life scenarios, useful for revision or extra support.teaching-resource/linear-programming-problems-carpenter-problem-6229434Time SeriesCritical Path AnalysisActivity – create a schedule for cooking a meal. documents including critical path diagrams, cascade charts and questionsteaching-resource/critical-path-analysis-6147452 ................
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