# Mathematics

## SUBJECT overview

Mathematics can be applied in practical tasks, real life problems and within mathematics itself. The aim of the course is to develop mathematical vocabulary, improve mental calculation and use a range of methods of computation and apply these to a variety of problems.

The course of study should help you whether working individually or collaboratively to reason logically, plan strategies and improve your confidence in solving complex problems.

During Maths lessons you will learn how to:-

• Use and apply maths in practical tasks, real life problems and within mathematics itself.
• Develop and use a range of methods of computation and apply these to a variety of problems.
• Develop mathematical vocabulary and improve mental calculation.
• Consider how algebra can be used to model real life situations and solve problems.
• Explore shape and space through drawing and practical work using a range of materials and a variety of different representations.
• Use statistical methods to formulate questions about data, represent data and draw conclusions.

## UNIT OVERVIEW - AUTUMN 1 - MECHANICS

 Unit 5: Forces and Friction Skills Resolving forces in 2 dimensions. Problems may be set where forces need to be resolved. Understand and use the F ≤ µR model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and limiting equilibrium. Knowledge By the end of the unit, students should: understand the language relating to forces; be able to identify the forces acting on a particle and represent them in a force diagram; understand how to find the resultant force (magnitude and direction); be able to find the resultant of several concurrent forces by vector addition; be able to resolve a force into components and be able to select suitable directions for resolution. understand that a rough plane will have an associated frictional force, which opposes relative motion (i.e. the direction of the frictional force is always opposite to how the object is moving or ‘wants’ to move); understand that the ‘roughness’ of two surfaces is represented by a value called the coefficient of friction represented by µ; know that 0 ≤ µ but that there is no theoretical upper limit for µ although for most surfaces it tends to be less than 1 and that a ‘smooth’ surface has a value of µ = 0; be able to draw force diagrams involving rough surfaces which include the frictional force understand and be able to use the formula F ≤ µR. Rationale Newton’s laws summarise and govern the basic interactions between objects and forces. At their most fundamental level, forces tell us how different objects interact. This can be viewed on a macro scale (such as the forces between a car and the road), or on a micro scale (such as looking how the forces between atoms cause the atoms to form an object). As such, knowledge of forces and how they work has application to chemistry, physics, engineering, sport, architecture and building, and any other field the deals with one object interacting with another. For example, structural materials need to be tested to decide if they are appropriate for certain situations. As such they need to be subjected various shear and compressive/stretching forces to determine the grade or material.

## UNIT OVERVIEW - AUTUMN 1 - PURE

 Units 5, 6, and 7:   Radians and Trigonometry Skills Work with radian measure. Understand and use the standard small angle approximations of sine, cosine and tangent 2 i.e. sin θ ≈ θ, cos θ ≈ 1 – 8 , tan θ ≈ θ where θ is in radians. 2 Know and use exact values of sin and cos for 0 , n, n, n, n, π and multiples thereof, 6     4     3     2 and exact values of tan for 0 , n, n, n, n, π and multiples thereof. 6     4     3     2 Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains. Understand and use sec2 θ = 1 + tan2 θ and cosec2 θ = 1+ cot2 θ. Use of formulae for sin (A ± B), cos (A ± B) and tan (A ± B). Understand and use double angle formulae; use of formulae for sin (A ± B), cos (A ± B)                                                                                                                        and tan (A ± B); understand geometrical proofs of these formulae. Understand and use expressions for a cos θ + b  sin θ in the equivalent forms of R cos (θ ± α) or R sin (θ ± α) . Construct proofs involving trigonometric functions and identities. Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces. Knowledge By the end of the unit, students should: understand the definition of a radian and be able to convert between radians and degrees; know and be able to use exact values of sin, cos and tan. understand and be able to use the standard small angle approximations for sine, cosine and tangent. understand the secant, cosecant and cotangent functions, and their relationships to sine, cosine and tangent; be able to sketch the graphs of secant, cosecant and cotangent; be able to simplify expressions and solve involving sec, cosec and cot; be able to solve identities involving sec, cosec and cot; know and be able to use the identities 1 + tan2 x = sec2 x and 1 + cot2 x = cosec2 x to prove other identities and solve equations in degrees and/or radians be able to work with the inverse trig functions sin–1, cos–1 and tan–1; be able to sketch the graphs of sin–1, cos–1 and tan–1. be able to use compound angle identities to rearrange expressions; be able to use compound angle identities to rearrange equations into a different form and then solve.

## UNIT OVERVIEW - AUTUMN 2 - MECHANICS

 Unit 7:  Applications of forces Skills Understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2D vectors); extend to situations where forces need to be resolved (restricted to 2 dimensions). Understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line; application to problems involving smooth pulleys and connected particles; resolving forces in 2 dimensions; equilibrium of a particle under coplanar forces. Understand and use addition of forces; resultant forces; dynamics for motion of a particle in a plane. An understanding of F ≤ R in a situation of equilibrium. Moments: problems involving parallel and non-parallel coplanar forces e.g. ladder problems. Knowledge By the end of the unit, students should: understand that a body is in equilibrium under a set of concurrent (acting through the same point) forces is if their resultant is zero; know that vectors representing forces in equilibrium form a closed polygon; understand how to solve problems involving equilibrium of a particle under coplanar forces, including particles on inclined planes and 2D vectors; know and understand the meaning of Newton's second law; be able to formulate the equation of motion for a particle in 1-dimensional motion where the resultant force is mass × acceleration; be able to formulate the equation of motion for a particle in 2-dimensional motion where the resultant force is mass × acceleration; be able to formulate and solve separate equations of motion for connected particles, where one of the particles could be on an inclined and/or rough plane. be able to solve statics problems for a system of forces which are not concurrent (e.g. ladder problems), thus applying the principle of moments for forces at any angle. Rationale Newton’s laws summarise and govern the basic interactions between objects and forces. At their most fundamental level, forces tell us how different objects interact. This can be viewed on a macro scale (such as the forces between a car and the road), or on a micro scale (such as looking how the forces between atoms cause the atoms to form an object). As such, knowledge of forces and how they work has application to chemistry, physics, engineering, sport, architecture and building, and any other field the deals with one object interacting with another. For example, structural materials need to be tested to decide if they are appropriate for certain situations. As such they need to be subjected various shear and compressive/stretching forces to determine the grade of material.         ## unit overview - autumn 2 - pure

 Subject: Units 5, 6, and 7:   Radians and Trigonometry Skills Work with radian measure. Understand and use the standard small angle approximations of sine, cosine and tangent 2 i.e. sin θ ≈ θ, cos θ ≈ 1 – 8 , tan θ ≈ θ where θ is in radians. 2 Know and use exact values of sin and cos for 0 , n, n, n, n, π and multiples thereof, 6     4     3     2 and exact values of tan for 0 , n, n, n, n, π and multiples thereof. 6     4     3     2 Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains. Understand and use sec2 θ = 1 + tan2 θ and cosec2 θ = 1+ cot2 θ. Use of formulae for sin (A ± B), cos (A ± B) and tan (A ± B). Understand and use double angle formulae; use of formulae for sin (A ± B), cos (A ± B)                                                                                                                        and tan (A ± B); understand geometrical proofs of these formulae. Understand and use expressions for a cos θ + b  sin θ in the equivalent forms of R cos (θ ± α) or R sin (θ ± α) . Construct proofs involving trigonometric functions and identities. Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces. Knowledge By the end of the unit, students should: understand the definition of a radian and be able to convert between radians and degrees; know and be able to use exact values of sin, cos and tan. understand and be able to use the standard small angle approximations for sine, cosine and tangent. understand the secant, cosecant and cotangent functions, and their relationships to sine, cosine and tangent; be able to sketch the graphs of secant, cosecant and cotangent; be able to simplify expressions and solve involving sec, cosec and cot; be able to solve identities involving sec, cosec and cot; know and be able to use the identities 1 + tan2 x = sec2 x and 1 + cot2 x = cosec2 x to prove other identities and solve equations in degrees and/or radians be able to work with the inverse trig functions sin–1, cos–1 and tan–1; be able to sketch the graphs of sin–1, cos–1 and tan–1. be able to use compound angle identities to rearrange expressions; be able to use compound angle identities to rearrange equations into a different form and then solve.         Subject: Unit 11:  Integration Skills Integrate xn, (including 1 ) and integrate ekx, sin kx , cos kx and related sums, s differences and constant multiples. To include integration of standard functions such as                 sin 3x,                 sec2 2x,                 tan x,                 e5x,                    1 . 2s Students are expected to be able to use trigonometric identities to integrate, for example, sin2 x, tan2 x, cos2 3x. Use a definite integral to find the area under a curve and the area between two curves. Understand and use integration as the limit of a sum. Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively. Integrate using partial fractions that are linear in the denominator. Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions. Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics. Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between. Knowledge By the end of the sub-unit, students should: be able to integrate expressions by inspection using the reverse of differentiation; be able to integrate xn for all values of n and understand that the integral of 1 is s ln |x|; be able to integrate expressions by inspection using the reverse of the chain rule (or function of a function); be able to integrate trigonometric expressions; be able to integrate expressions involving ex . recognise integrals of the form óô  f ¢( x) dx   = ln |f(x)| + c; õ  f( x) be able to use trigonometric identities to manipulate and simplify expressions to a form which can be integrated directly. be able to integrate expressions using an appropriate substitution; be able to select the correct substitution and justify their choices.

 be able to integrate an expression using integration by parts; be able to select the correct method for integration and justify their choices. be able to integrate rational expressions by using partial fractions that are linear in the denominator; be able to simplify the expression using laws of logarithms. understand and be able to use integration as the limit of a sum; understand the difference between an indefinite and definite integral and why we do not need + c; be able to integrate polynomials and other functions to find definite integrals, and use these to find the areas of regions bounded by curves and/or lines; be able to use a definite integral to find the area under a curve and the area between two curves. be able to use the trapezium rule to find an approximation to the area under a curve; appreciate the trapezium rule is an approximation and realise when it gives an overestimate or underestimate. be able to write a differential equation from a worded problem; be able to use a differential equation as a model to solve a problem; be able to solve a differential equation; be able to substitute the initial conditions or otherwise into the equation to find + c and the general solution. Rationale Integration is the second half of calculus, with the first half being differentiation. Integration is essentially the inverse operation of differentiation, and as such is applicable and useful wherever differentiation is. At the most fundamental level, integration finds the area between the graph of the function and the variable axis. Depending on the variables involved in the function, this will tell us different information. For example, for a function of velocity over time, the integral will give us the distance travelled over a given time period. Integration is heavily used in the field of Finance, Physics, Engineering, Chemistry, and almost all other forms of science and economic fields.

 Subject: Unit 12:    Vectors (3D) Skills Use vectors in three dimensions. Knowledge of column vectors and i, j and k unit vectors in three dimensions. Knowledge By the end of the unit, students should: be able to extend the work on vectors from AS Pure Mathematics to 3D with column vectors and with the use of i, j and k unit vectors; be able to calculate the magnitude of a 3D vector; know the definition of a unit vector in 3D; be able to add 3D vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations; understand and use position vectors, and calculate the distance between two 3D points represented by position vectors; be able to use vectors to solve problems in pure mathematics and in contexts (e.g. mechanics). Rationale A vector is a quantity that has an intrinsic magnitude and direction. Quantities like displacement, velocity, acceleration, force, and momentum are all vectors. When vectors of the same type are applied to the same object they combine in specific ways.   Vectors are utilised in many different fields, such as navigation, computer graphics, and engineering. In Navigation, when bearings are combined with distance they create a vector. Total displacement from a series of directions can then be calculated using vector addition, as well as average speed if the time frame is available.   Vectors are used to resolve forces, and as such are a core part of structural architecture, as buildings need to be in equilibrium. Through vector addition of forces and dispersion of these forces stable structures can be designed.

## unit overview - spring 1 - mechanics

 Unit 2: Conditional Probability Skills Understand and use mutually exclusive and independent events when calculating probabilities. Link to discrete and continuous distributions. Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables. Understand and use the conditional probability formula P(A| B) = P(A Ç B) . P(B) Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions. Knowledge By the end of the unit, students should: understand and be able to use probability formulae using set notation; be able to use tree diagrams, Venn diagrams and two-way tables; understand and be able to use the conditional probability formula P(A| B) = P(A Ç B) . P(B) be able to model with probability; be able to critique assumptions made and the likely effect of more realistic assumptions. Rationale Probability theory is applied in everyday life in risk assessment and modelling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis (Reliability theory of aging and longevity), and financial regulation. A good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioural finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict. Reference https://en.wikipedia.org/wiki/Probability#Applications

## unit overview - spring 1 - pure

 Unit 4: The binomial theorem Skills Understand and use the binomial expansion of (a + bx)n for rational n, including its use for approximation; be aware that the expansion is valid for |bs| < 1 (proof not required). a Knowledge By the end of the unit, students should: be able to find the binomial expansion of (1 − 𝑥)–1  for rational values of n and |𝑥| < 1; be able to find the binomial expansion of (1 + 𝑥)n  for rational values of n and |𝑥| < 1; be able to find the binomial expansion of (1 + 𝑏𝑥)n  for rational values of n and |𝑥| <  1 ; |b| be able to find the binomial expansion of (𝑎 + 𝑥)n  for rational values of n and |𝑥| < 𝑎; be able to find the binomial expansion of (𝑎 + 𝑏𝑥)n for rational values of n and |bs| < a 1; know how to use the binomial theorem to find approximations (including roots). be able to use partial fractions to write a rational function as a series expansion. Rationale The Binomial theorem, and therefore the binomial expansion is a naturally occurring distribution based off of a system with a fixed probability of success and only two states (success or failure). Even if the system technically has more than two states, it is a helpful tool that can simplify that system down to a success/failure cut-off point. A simple example of this is a dice, which can roll a 1, 2, 3, 4, 5, or 6. If you need to roll a minimum of a 5 for a game, then the success state is 5 or 6, and failure is 1, 2, 3, or 4. At this point the exact number does not matter – only if you “win” the roll or not. Binomial distributions have immediate applications to games of chance, but also to natural occurrences. For example, farmers or animal breeders could use a binomial expansion to calculate the probability of breeding a certain number of offspring with a desired genetic trait.

## unit overview - spring 1 - statistics

 Unit 2: Conditional Probability Skills Understand and use mutually exclusive and independent events when calculating probabilities. Link to discrete and continuous distributions. Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables. Understand and use the conditional probability formula P(A| B) = P(A Ç B) . P(B) Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions. Knowledge By the end of the unit, students should: understand and be able to use probability formulae using set notation; be able to use tree diagrams, Venn diagrams and two-way tables; understand and be able to use the conditional probability formula P(A| B) = P(A Ç B) . P(B) be able to model with probability; be able to critique assumptions made and the likely effect of more realistic assumptions. Rationale Probability theory is applied in everyday life in risk assessment and modelling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis (Reliability theory of aging and longevity), and financial regulation. A good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioural finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict. Reference  https://en.wikipedia.org/wiki/Probability#Applications

## unit overview - spring 2 - pure

 Unit 10:  Numerical methods Skills Locate roots of f (x) = 0 by considering changes of sign of f (x) in an interval of x on which f (x) is sufficiently well-behaved. Understand how change of sign methods can fail. Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams. Solve equations using the Newton-Raphson method and other recurrence relations of the form xn+1 = g(xn). Understand how such methods can fail. Use numerical methods to solve problems in context. Knowledge By the end of the unit, students should: be able to locate roots of f(x) = 0 by considering changes of sign of f(x); be able to use numerical methods to find solutions of equations. understand the principle of iteration; appreciate the need for convergence in iteration; be able to use iteration to find terms in a sequence; be able to sketch cobweb and staircase diagrams; be able to use cobweb and staircase diagrams to demonstrate convergence or divergence for equations of the form x = g(x). be able to solve equations approximately using the Newton-Raphson method; understand how the Newton-Raphson method works in geometrical terms. be able to use numerical methods to solve problems in context. Rationale Numerical methods are processed based methods of find approximations of a solution to complex problems. These can be used to get an approximate answer in order to fact check longer solutions, or when automating a problem solving process. For example, in computing and computer science, numerical methods are relatively straight-forward to implement as they usually focus on iterative processes. In fact, calculators and mathematical software will often use numerical methods to solve more complicated mathematical operations that would otherwise require abstract or algebraic solutions.

## unit overview - spring 2 - statistics

 Unit 1:    Regression and correlation Skills Change of variable may be required e.g. using knowledge of logarithms to reduce a relationship of the form 𝑦 = 𝑎𝑥n or 𝑦 = 𝑘𝑏s into linear form to estimate 𝑎 and 𝑛 or 𝑘 and 𝑏. Understand and apply the language of statistical hypothesis testing, …., extend to correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given p-value or critical value (calculation of correlation coefficients is excluded). Knowledge By the end of the unit, students should: be able to change the variable in a regression line; be able to estimate values from regression line. understand correlation coefficients; be able to calculate the PMCC (calculator only); be able to interpret a correlation coefficient; be able to conduct a hypothesis test for a correlation coefficient. Rationale Linear regression is widely used in biological, behavioural and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines. Linear regression is the predominant empirical tool in economics. For example, it is used to predict consumption spending, fixed investment spending, inventory investment, and purchases of a country's exports, spending on imports, the demand to hold liquid assets, labour demand, and labour supply. The capital asset pricing model uses linear regression as well as the concept of beta for analysing and quantifying the systematic risk of an investment. This comes directly from the beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets. Reference  https://en.wikipedia.org/wiki/Linear_regression#Applications

## unit overview - summer 2 - mechanics

 Unit 4:      Moments Skills Understand and use moments in simple static contexts. Knowledge By the end of the unit, students should: realise that a force can produce a turning effect; know that a moment of a force is given by the formula force × distance giving Nm and know what the sense of a moment is; understand that the force and distance must be perpendicular to one another; be able to draw mathematical models to represent horizontal rod problems; realise what conditions are needed for a system to remain in equilibrium; be able to solve problems when a bar is on the point of tilting. Rationale Moments are created when a force causes an object to rotate. The size of this moment depends on the size of the force, the angle of application and the distance that the fore was applied from the axis of rotation. Moments are a core concept to the idea of equilibrium, which is when an object is in a completely balanced state. This idea is applied to construction and architecture, where if a building gin not in equilibrium it may become unstable and collapse. Moments are also used in engineering, when a rotating wheel (on a car, for example) will have a moment applied to the axle which will cause the wheel to spin, hence accelerating the car. The size of the moment will determine how fast the car can accelerate.

## unit overview - summer 2 - pure

 Unit 1 (Part 1):   Algebraic and partial fractions Skills Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only). Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear). Knowledge By the end of the unit, students should: be able to add, subtract, multiply and divide algebraic fractions; know how to use the factor theorem to shown a linear expression of the form (𝑎 + 𝑏𝑥) is a factor of a polynomial; know how to use the factor theorem for divisors of the form (𝑎 + 𝑏𝑥); be able to simplify algebraic fractions by fully factorising polynomials up to cubic. be able to split a proper fraction into partial fractions; be able to split an improper fraction into partial fractions, dividing the numerator by the denominator (by polynomial long division or by inspection). Rationale Algebraic manipulation is a key skill in any further mathematical studies, as well as fields that make frequent use of mathematical models. This includes finance, economics, engineering, and most forms of science. Algebraic and partial fractions is an important part of this algebraic manipulation toolkit. Being able to split and simplify fractions can in turn reduce complicated fractional functions into much easier to solve in implement mathematical models, which can be introduced into Full models linearly in sections rather than all at once. In the field of computer programming (or applied programming to finance or science) it is essential to be able to individually test and isolate different parts of the function. This can be achieved through partial fractions.

## unit overview - summer 2 - statistics

 Subject: Unit 3: The Normal distribution Skills Understand and use the Normal distribution as a model; find probabilities using the Normal distribution Link to histograms, mean, standard deviation, points of inflection and the binomial distribution. Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or the Normal model may not be appropriate. Conduct a statistical hypothesis test for the mean of the Normal distribution with known, given or assumed variance and interpret the results in context. Knowledge By the end of the unit, students should: understand the properties of the Normal distribution; be able to find probabilities using the Normal distribution; know the position of the points of inflection of a Normal distribution. be able to find the mean and variance of a binomial distribution; understand and be able to apply a continuity correction; be able to use the Normal distribution as an approximation to the binomial distribution. be able to conduct a statistical hypothesis test for the mean of the Normal distribution; be able to interpret the results in context Rationale Measurement errors in physical experiments are often modelled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.   Many scores are derived from the normal distribution, including percentile ranks ("percentiles" or "quantiles"), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, some behavioural statistical procedures assume that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores. Reference:    https://en.wikipedia.org/wiki/Normal_distribution#Occurrence_and_applications

## knowledge Organiser

A knowledge organiser is an important document that lists the important facts that learners should know by the end of a unit of work. It is important that learners can recall these facts easily, so that when they are answering challenging questions in their assessments and GCSE and A-Level exams, they are not wasting precious time in exams focusing on remembering simple facts, but making complex arguments, and calculations.

We encourage all pupils to use them by doing the following:

• Quiz themselves at home, using the read, write, cover, check method.
• Practise spelling key vocabulary
• Further researching people, events and processes most relevant to the unit.