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Boosting achievement in A2 Core Mathematics: Supporting lower ability students through the C3 and C4 modules Phil Chaffé 2012

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10.00 – 11.15am: Hitting the ground running: successful transition to A2 level 11.15 – 11.30am: DISCUSSION: COFFEE BREAK 11.30 – 12.30pm: Picking up the problems: identifying when and where students struggle 12.30 – 1.30pm: LUNCH AND INFORMAL DISCUSSION 1.30 – 2.45pm Materials and methods: teaching the difficult topics 2.45 – 3.00pm: DISCUSSION: AFTERNOON TEA 3.00 – 3.45pm: Preparing students for examinations

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Hitting the ground running: successful transition to A2 level What to do after AS levels are complete The skills that are needed to make a successful start to A2 level mathematics Developing the essential skills needed to start the A2 course Preparing students for the challenge of the A2 course Materials and ideas that ensure a good start (some moved to the afternoon session)

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Putting things in context The three post 16 transitions GCSE to AS level GCSE algebraic manipulation techniques are expected to be used with more fluency. Mathematical terms are expected to be a part of a student’s vocabulary. GCSE knowledge is expected to be applied efficiently (and quickly). Less guidance is given for solving problems. A limited number of new techniques are introduced.

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AS level to A2 level Students are expected to recall, select and use their knowledge of mathematical facts, concepts and techniques with fluency in a variety of contexts. Mathematical arguments now have to be rigorous, logical and precise. There is more emphasis on proof. Manipulation of mathematical expressions is expected to be fluent and precise. Students need to be able to handle substantial problems presented in an unstructured form.

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A2 level to university Mathematical arguments have to be concise and relevant. Mathematics has to be used creatively to solve complex problems. Students are expected to question the techniques that they use. Manipulation of mathematical expressions is expected to be fluent and precise. There is a high emphasis on proof and an expectation that students have a number of techniques at their disposal.

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AS level to A2 level Students are expected to recall, select and use their knowledge of mathematical facts, concepts and techniques with fluency in a variety of contexts. Mathematical arguments now have to be rigorous, logical and precise. There is more emphasis on proof. Manipulation of mathematical expressions is expected to be fluent and precise. Students need to be able to handle substantial problems presented in an unstructured form.

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The aims of an A level mathematics course (paraphrased from 4 specifications) To develop a deeper understanding of the way that mathematics and mathematical processes work. To promote confidence and foster enjoyment. To develop a student’s ability to reason logically. To give students the skills to recognise incorrect reasoning. To teach students how to generalise and to construct mathematical proofs.

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To extend the range of mathematical skills and techniques available to a student. To give students the opportunity to use their mathematical skills in more difficult, unstructured problems. To help students develop an understanding of coherence and progression in mathematics and of how different areas of mathematics can be connected. To develop a student’s ability to communicate effectively with mathematics. The help students acquire the skills needed to use technology effectively and recognise when this may be inappropriate and where there are limitations. To encourage students to take more responsibility for their own learning and the evaluation of their mathematical development.

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Preparing students for the challenge of the A2 course Students are expected to recall, select and use their knowledge of mathematical facts, concepts and techniques with fluency in a variety of contexts. “What we do in most traditional classrooms is require students to commit bits of knowledge to memory in isolation from any practical application—to simply take our word that they "might need it later." For many students, "later" never arrives. This might well be called the freezer approach to teaching and learning. In effect, we are handing out information to our students and saying, "Just put this in your mental freezer; you can thaw it out later should you need it." With the exception of a minority of students who do well in mastering abstractions with little contextual experience, students aren't buying that offer. The neglected majority of students see little personal meaning in what they are asked to learn, and they just don't learn it.” DALE PARNELL, Oregon State University From: High School Mathematics at Work: Essays and Examples for the Education of All Students (1998)

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Preparation starts at AS level Each “strand” of the specification is made clear to students Connections between skills/techniques across strands and levels are made clear to students Students are aware that the skills they are using will link to many other areas Developing a toolkit mentality Skills/techniques are taught with an indication of why they are useful and the many ways in which they may be applied

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What to do after finishing the AS level course Prepare students for the expectations of the A2 course. Use activities that show the strong links between AS and A2 mathematics skills. There should be some time to look at AS skills, ideas and techniques and develop them along the lines needed for A2 mathematics. Skills practice exercises can be set to develop the fluency needed.

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What skills are needed?

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Activity: What skills are needed? Look through the worked questions from C3/C4 What ‘bits’ of mathematics can you identify? Look out for notation that you recognise ‘normal’ mathematical skills being used

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Transition Work This should be used to reinforce AS skills, develop some problem solving tenacity and introduce some of the basics of the A2 core. An example of a transition unit This example is designed for discussion. Some questions to ask when looking through the unit Is the content appropriate? Are the correct skills being reinforced? Is the quantity appropriate? Are there enough problem solving activities? Is the introduction to A2 appropriate?

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The skills needed for a successful transition to A level Personal skills Retention of previously acquired information and skills Initiative in solving problems Perseverance in solving problems Willingness to overcome the desire for a quick trick or formula Overcoming the aversion to ‘wordy’ problems An understanding of why the skills are useful

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Mathematical Skills EdexcelCore Mathematics 3Previously acquired skills Algebra and functions Simplification of rational expressions including factorising and cancelling, and algebraic division. Factorising a quadratic expression The difference of two squares Multiplying out brackets Equating coefficients Simple algebraic division The factor and remainder theorems Definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs. Changing the subject of a formula Substitution into an expression Substituting an algebraic expression The modulus function. Graphs of linear functions Graphical transformation Combinations of the transformations y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax). Graphical transformation Substitution into an expression

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Focus on learning rather than teaching Discovery activities in lessons (groups or individually) at home Use lesson objectives that tantalise ‘Eureka’ moments, fine tuning and deep understanding

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Developing understanding Working towards examination questions Text book or activity led Build in discussion time Link to previous knowledge ‘Eureka’ moments, fine tuning and deep understanding

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Encourage thinking more about the maths Differentiation by outcome Challenge both able and weaker students ‘Eureka’ moments, fine tuning and deep understanding

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Picking up the problems: identifying when and where students struggle Anticipating problems and preparing for difficulties Spotting problems by everyday classroom monitoring Diagnostic activities and instant troubleshooting Dealing with deep seated problems

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Anticipating problems and preparing for difficulties Look at the C3 and C4 examination questions. Pick a problem - analyse the skills required to solve the problem. Discussion questions What are the mathematical skills required? Where are these skills taught (at what stage)? Where do you think your students will have difficulty with the question? If they would simply not be able to start, what is stopping them? What would they need to be able to get started with the question?

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Spotting problems by everyday classroom monitoring Direct questioning Working with groups/pairs “Culture of explanation” Activities/exercises that can be used to monitor understanding Focused on a sensible number of things in a topic. Key questions asked – promote thinking.

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Say as much about …. as you can Example You have introduced function and taught most of the initial skills including the main definitions.

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Asking the right questions – domain and range Be upfront about what you are doing. Let the students know that you will be assessing their responses to your questions. Make sure that the weaker students have a some chance of answering – the idea is to find out what they do know rather that prove that they know nothing. Have a series of options available that the student can choose from. Use these to get past the “I don’t know” response. When supplying options, give possibilities that are at least partially correct as well as the real answer. This allows the student to show how they understand something even if they did pick the wrong option. Have a balance of questions. Don’t keep things to easy all of the time; ask questions that will stretch the understanding of the most able students. Think about how you will deal with zero or negative responses. Even though you are assessing them, remember to be liberal with praise.

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Asking the right questions activity Domain and range

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What are the skills that are being tested? How do you know that a student has understood those skills? What does a student need to say to indicate that they have those skills? What are you going to ask to check that the student really has understood? How many questions is enough?

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Instant troubleshooting Small group activities Groups of >2 allow you to help one of the students while the others get on with the task Occasionally social engineering helps when arranging groups Deal with the immediate problem whilst trying to assess if it is deep seated or a “quick fix” Keep trying to tie the explanation in to what the student does know.

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Dealing with deep seated problems Find out what the student is thinking first. Ask questions to break down the steps that they think in. Avoid saying directly that the student has it all wrong. Build the explanation from the ground up. Don’t be afraid of going back to the very basics. Involve the student in the explanation by getting them to take you through it. It does take time and patience. Don’t try to do it all at once.

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Intervention Mentoring, where possible, is an effective method. Student study pairing has also been used effectively in schools. BUT It can be “after the horse has bolted”. It almost always involves staff giving up their own time.

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This question requires a number of skills that the students find difficult. How would you build a ground up explanation to help a student overcome a deep seated problem with it.

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Materials and methods: teaching the difficult topics Algebra and Functions, Trigonometry, Vectors Teaching to promote confidence and fluency in algebra Providing focused support for those struggling with algebra Teaching to promote confidence and fluency in trigonometry Providing focused support for those struggling with trigonometry Introducing vectors to lower ability students Materials and ideas to develop the key skills

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Algebra and Functions OCR Specification – Core Mathematics 3

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The modulus function Introduction using Geogebra – link to transforming graphs Get students to identify what is happening and why. Follow up activities Modulus graph matching activity How many solutions – modulus equations activity

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Providing focused support for students struggling with functions 1.Terminology The terminology is very important so make sure the student is as confident as possible with the language of functions. 2.Transformations Students need to be very familiar with the effect of transformations and the links to what they have done before. The order in which transformations is applied needs to be very clear.

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3.Composite functions The order for applying each function should be clearly understood. Substituting numbers into one then the other should be done first before moving on to algebraic substitution. 4.Inverse functions These need to be though of first as “undoing” something. “I think of a number multiply by 5 and add 6 the result is..” type of questions work well initially. The links between the graph of the function and that of its inverse should be made very clear.

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Trigonometry OCR Specification – Core Mathematics 3

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1 A starting point….

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1 1 An alternative….

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1

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A Geogebra activity that uses links to graphical transformation.

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Providing focused support for students struggling with trigonometry 1.Periodicity Students really need to know how to use the periodicity of the trigonometric functions so they can calculate all of the required solutions to a trigonometric equation. Sketch graphs using both degrees and radians are essential. 2.Using the formula booklet Weaker students should have (at least a copy) of the formula booklet page with the given trig identities from the start. It makes it clear what they do need to learn and gets them used to looking in the correct place.

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Vectors OCR Specification – Core Mathematics 4

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Representation of Vectors The card set ‘Representation of Vectors’ shows the different ways in which vectors can be represented. Students have to link one of each form together. Reinforces the need to be flexible in the way vector information is recorded. As a follow up students could be given one piece of information and asked to construct the other forms.

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Target Grid Match the cards to the appropriate cell of the target grid. You are trying to cover all of the grid. Some cells require more than one card. If you have any cards left over…. Is there a cell you could put each one in? Could you write a description of a cell that would contain all of the leftover cards?

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Vector Equations of lines Match the vector equations to the descriptions. Now match the vector equations to the lines on the graph.

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Providing focused support for students struggling with vectors 1.Multiple representations and meanings Students really need to be confident with all of the ways of representing vectors. The different meanings are very important for weaker students to understand and time should be taken to make it clear how each meaning can be identified (vector equation of a line). 2.‘Simple’ skills should be firmly embedded Skills such as finding the vector connecting two points should be done fluently and not be a chore even for weaker students.

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3.Terminology The terminology should be clearly understood and not confused e.g. ‘dot product’ should always be accompanied by ‘scalar product’.

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Preparing students for examinations Breaking revision down into suitable sections for weaker students Providing appropriate revision materials and support for weaker students Preparing students for the ‘unexpected’ exam questions

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Breaking revision down into suitable sections for weaker students Two useful (although not necessarily quick) methods: Exam question analysis Examiners’ reports

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Work through the question List all the skills required – put them in a sensible order

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Examiners’ reports Functions 1. Students need to know what all of the terminology means Check that students know the meaning of all the terminology relating to mappings and functions, and in particular, when a mapping is a function. 2. Students need to know what effect a transformation has on the equation and graph Make sure that students know the effect on the equation of a graph of translations, stretches and reflections. 3. Students need to take care when doing multiple transformations Make sure students are careful when using more than one transformation. Students need to realise that changing the order can sometimes give a different result.

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4. For composite functions, make sure students are applying the functions in the right order Students need to be careful to apply functions in the correct order when finding composite functions. They must remember that the function fg means “first apply g, then apply f to the result”. 5. Students need to remember that only a one-to-one function has an inverse function Sometimes a function can be defined with a restricted domain so that it does have an inverse function: for example, f(x) = x² is a many-to-one function for x ∈ R, and so does not have an inverse, but if the domain is restricted to x ≥ 0, then the function is one-to-one and the inverse function f −1 (x) = √x

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6. When finding the domain or range for f -1, students should look at the limits of the original function Students need to notice that the domain of an inverse function f -1 is the same as the range of f, and the range of f -1 is the same as the domain of f. 7. Students must check that they have the right number of solutions They need to be careful when solving equations involving a modulus function that they have the correct number of solutions. Sketching a graph is always helpful. They should also check their solution(s) by substituting back into the original equation.

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8. Students need to take care with inequality signs, especially when they involve negative numbers When solving inequalities involving a modulus sign, students need to be very careful with the inequality symbol. They need to remember to reverse it if they are multiplying or dividing through by a negative number. Students should check their answer by substituting a number from within the solution set into the original inequality. 9. Students need to learn and be confident using the laws of indices and logarithms Make sure that students know the rules of logarithms and of indices so they can manipulate expressions involving exponentials and logarithms confidently. 10. Make sure that students remember that the exponential and logarithm functions are the inverses of each other Students need to remember that the exponential function and the natural logarithm function are inverse functions; so they can “undo” an exponential function by using natural logarithms, and “undo” a natural logarithm by using exponentials.

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Trigonometry 1. Students must make sure solutions to an equation are in the right range When solving an equation make sure that students check: what range the solutions should lie in whether the solutions should be in radians or degrees. 2. Students should never cancel a factor in an equation In an equation such as sinθ − sinθ cosθ = 0 students should never cancel out the term sinθ because they will lose the roots to the equation sinθ = 0. They should never cancel – always factorise. 3. Students should work from one side of the identity which they are trying to prove When trying to prove an identity students should only ever work with one side of the identity. They should never try to rearrange it and cancel out terms. 4. Students should read the question carefully Students should always check which form of r sin(θ ± α) or r cos(θ ± α) the question is looking for

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Differentiation 1. Students must make sure they don’t mix up the derivative of e x with that of x n 2. Students must make sure they don’t mix up the integral and differential of e kx 3. Students should remember that they cannot integrate across an asymptote when evaluating a logarithmic integral 4. Make sure students remember the du/dx part of the chain rule 5. Make sure students recognise situations when the chain rule should be used Students should know that the chain rule is used for functions which can be written in the form y = f(u), where u is a function of x. They should be clear that it cannot be used to differentiate functions which are a product of two functions – and that requires the product rule.

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6. Make sure students use the product rule correctly 7. Make sure students use the quotient rule correctly They must make sure they don’t get ‘u’ and ‘v’ mixed up and remember the negative sign in the numerator 8. Students must be careful when finding stationary points of quotient functions 9. Students must remember that when differentiating trigonometric functions the derivative results rely on measuring x in radians 10. Students must be careful not to mix up the derivatives and integrals of sin x and cos x 11. Students must make sure that they understand the process of differentiating an equation implicitly

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Integration 1. When using the integration by parts formula, students must remember to integrate to find ‘v’ rather than differentiating. 2. Students must be careful with signs when using the integration by parts formula 3. Students need to remember to substitute for dx in the integral when integrating by substitution 4. Students must remember to change the limits of a definite integral when making a substitution When students change the variable in an integration (from x to u say) by making a substitution, they must change the limits of the integration from values of x to the equivalent values of u. 5. Students need to be careful with signs when substituting values into definite integrals

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6. Students should always check their integration by differentiating It is easy for students to make mistakes when integrating. Differentiating the result is a quick and comparatively easy way of checking their work. 7. Students should learn to look out for the standard patterns Students should look for any integrals which they should be able to integrate by inspection. They should make sure that they adjust any constants if necessary. 8. Students need to remember when to use logarithms in integration Some students make the mistake of wrongly using logarithms when integrating inverse powers of linear functions of x. 9. Students should be careful to use the correct integration technique when dealing with products Some products require integration by substitution, other need integration by parts.

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10. Students should remember the ‘π’ in the volume of revolution formula 11. Students must make sure that they use the correct limits of integration for volumes of rotation Students need to remember that if they are rotating about the x-axis, the limits of integration must be x-coordinates, and if they are rotating about the y-axis, the limits of integration must be y-coordinates. 12. Students must remember to integrate with respect to the correct variable for volumes of revolution They need to correctly substitute for x² or y² to do this.

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Vectors 1. Students must make sure they use vector notation correctly They should remember that in handwriting they should underline vectors, or in the case of a vector joining two points, use an arrow above, e.g. AB 2. Students must make sure they know how to find the resultant of two vectors 3. Students must know how to find the vector joining two points 4. Students should know how to find a unit vector To find a unit vector in the same direction as a given vector, a, they should divide by the magnitude of a

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5. Students need to understand the relationship between vector and cartesian equations of lines 6. Students should always read the question carefully They should check whether the question is asking for the angle or the cosine of the angle. 7. Students should know how to find the angle between two lines They should know that to find the angle between two lines simply find the angle between the two direction vectors. 8. Students need to remember that the scalar product of perpendicular vectors is zero To show that two vectors are perpendicular they should just show that the scalar (or dot) product of the vectors is 0. 9. Students should draw diagrams to make sure that you are using the right vectors

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10. Students should be careful with signs when converting between the vector and cartesian equations of a line. 11. Students must be careful when writing down the Cartesian equation of a line which has one or two zeros in the direction vector.

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Providing appropriate revision materials for weaker students Past papers have always been some of the best preparation for examinations BUT Weaker students often expect exactly the same questions to turn up each time! Weaker students often can’t really start or get through the questions

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Final activity Back to the exam questions Pick a key question Re-write that question so that it is broken up into much smaller sections that guide the student through the process Change the question. What else could be asked from that set up? Change some of the information. Re-write the new question so that it is broken up into steps as before but reduce the guidance.

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Helping students deal with the unexpected Adapting exam paper question by …. What else could you ask? How could you change the question?

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Helping students deal with the unexpected 2. (a) Use the duocodification doctrine to aggrandize

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Syllabus Additional Mathematics Senior II Cambridge O Level Additional Mathematics Syllabus code 4037 For examination in June and November 2012.

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