# 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.

Engage in practical and experimental activities in order to appreciate principles of probability. There is no coursework

## unit  overview - autumn 1 - mechanics

 Unit 8:   Quantities and units in mechanics Skills Understand and use fundamental quantities and units in the S.I. system: length, time, mass. Understand and use derived quantities and units: velocity, acceleration, force, weight. Knowledge By the end of the unit, students should: understand the concept of a mathematical model, and be able to abstract from a real- world situation to a mathematical description (model); know the language used to describe simplifying assumptions; understand the particle model; be familiar with the basic terminology for mechanics; be familiar with commonly-made assumptions when using these models; be able to analyse the model appropriately, and interpret and communicate the implications of the analysis in terms of the situation being modelled; understand and use fundamental quantities and units in the S.I. system: length, time and mass; Understand that units behave in the same way as algebraic quantities, e.g. meters per second is m/s = m × 1/s = ms-1. understand and use derived quantities and units: velocity, acceleration, force, weight; know the difference between position, displacement and distance; know the difference between velocity and speed, and between acceleration and magnitude of acceleration; know the difference between mass and weight (including gravity); understand that there are different types of forces. Rationale Having a firm knowledge of the quantities and units used in mechanics is a necessary part of mechanics as a whole. It allows us to judge if the result gained from a calculation is expected, reasonable or even possible. The interactions between the base units that create more complicated units and variables deepen our understanding of the physical processes in place and how they are interconnected.

## unit overview - autumn 1  - pure

 Units 1 - 4: Algebra and Functions Skills Understand and use the laws of indices for all rational exponents. Use and manipulate surds, including rationalising the denominator. Work with quadratic functions and their graphs. The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. Solution of quadratic equations, including solving quadratic equations in a function of the unknown. Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation. Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Express solutions through correct use of ‘and’ and ‘or’, or through set notation. Represent linear and quadratic inequalities such as y > x + 1 and y > ax2 + bx + c graphically. Manipulate polynomials algebraically, including expanding brackets, collecting like terms and factorisation and simple algebraic division; use of the factor theorem. Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, y = a and y = a (including their vertical and horizontal asymptotes) s                             s2 Interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand the effect of simple transformations on the graph of y = f(x) including sketching associated graphs: o  y = af(x),   y = f(x) + a,   y = f(x + a), y = f(ax) . Knowledge By the end of the unit, students should: be able to perform essential algebraic manipulations, such as expanding brackets, collecting like terms, factorising etc; understand and be able to use the laws of indices for all rational exponents; be able to use and manipulate surds, including rationalising the denominator. be able to solve a quadratic equation by factorising;

## unit overview - autumn 1 - statistics

 Unit 1:  Statistical sampling Skills Understand and use the terms ‘population’ and ‘sample’. Use samples to make informal inferences about the population. Understand and use sampling techniques, including simple random sampling and opportunity sampling. Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population. Knowledge By the end of the unit, students should: understand and be able to use the terms ‘population’ and ‘sample’; know how to use samples to make informal inferences about the population; be able to describe advantages and disadvantages of sampling compared to census; understand and be able to use sampling techniques; be able to describe advantages and disadvantages of sampling techniques; be able to select or critique sampling techniques in the context of solving a statistical problem; understand that different samples can lead to different conclusions about the population. Rationale Statistical sampling can be a valuable tool to collect and evaluate information about a large population, or universe, when it would otherwise be impractical (or impossible) to collect that information from the entire population. When done properly, statistical samples enable reasonable inferences to be drawn about the population based on information about the sample. Additionally, one will have an objective measure of the possible variation between samples and of the sample's relationship to the population. However, because “statistics by their very nature present an incomplete and potentially misleading description of the population,” to be reliable and useful, a sample must be designed, executed and analysed using appropriate statistical analysis techniques. Reference  surviving-the-hazards-6

## unit overview - autumn 2 - mechanics

 Unit 9:  Kinematics 1 (constant acceleration) Skills Understand and use the language of kinematics: position; displacement; distance travelled; velocity; speed; acceleration. Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph. Understand, use and derive the formulae for constant acceleration for motion in a straight line. Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in S.I. units to varying degrees of accuracy. Knowledge By the end of the unit, students should: be able to draw and interpret kinematics graphs, knowing the significance (where appropriate) of their gradients and the areas underneath them. recognise when it is appropriate to use the suvat formulae for constant acceleration; be able to solve kinematics problems using constant acceleration formulae; be able to solve problems involving vertical motion under gravity. Rationale Kinematics is the field of mechanics that deals with moving objects. As such, any object that moves can be modelled (at least partially) using a kinematics equation. Kinematics is therefore a core part of mechanics as a whole, and to the entire field of physics. Kinematics is also a logical and consistent framework while helps to develop problem solving and modelling skills. Kinematics has direct application to engineering and sport, along with many other fields. For example, is sport kinematics can be used to calculate trajectories of balls and to analyse the movement of players in order to identify areas of improvement. In engineering, kinematics is used in designing and testing cars, bikes, and other forms of transport. It is also necessary to calculate the required forces needed to accelerate and decelerate an object.

## unit overview - autumn 2 - pure

 Unit 13: Integration Skills Know and use the Fundamental Theorem of Calculus. Integrate 𝑥n (excluding n = −1), and related sums, differences and constant multiples. Evaluate definite integrals; use a definite integral to find the area under a curve. Knowledge By the end of the unit, students should: know and be able to use the Fundamental Theorem of Calculus; be able to integrate 𝑥n (excluding n = −1), and related sums, differences and constant multiples. be able to evaluate definite integrals; be able to use a definite integral to find the area under a curve. 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.

## unit overview - autumn 2 - statistics

 Units 2 and 3: Data presentation and interpretation Skills Interpret measures of central tendency and variation, extending to standard deviation Be able to calculate standard deviation, including from summary statistics. Recognise and interpret possible outliers in data sets and statistical diagrams Select or critique data presentation techniques in the context of a statistical problem Be able to clean data, including dealing with missing data, errors and outliers. Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency. Connect to probability distributions. Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded). Understand informal interpretation of correlation. Understand that correlation does not imply causation. Knowledge be able to calculate measures of location, mean, median and mode; be able to calculate measures of variation, standard deviation, variance, range and interpercentile range; be able to interpret and draw inferences from summary statistics. know how to interpret diagrams for single variable data; know how to interpret scatter diagrams and regression lines for bivariate data; recognise the explanatory and response variables; be able to make predictions using the regression line and understand its limitations; understand informal interpretation of correlation; understand that correlation does not imply causation; recognise and interpret possible outliers in data sets and statistical diagrams; be able to select or critique data presentation techniques in the context of a statistical problem; be able to clean data, including dealing with missing data, errors and outliers. Rationale Learning about data presentation and interpretation has many uses. For example, it is used for all kinds of questions where statistical inference is appropriate, among them hypothesis testing and experimentation of all kinds, machine learning algorithms. It is useful when trying to come up with a prediction for some time series data like financial predictions, weather predictions. In manufacturing plants it is used for quality control / screening. Reference:

## unit overview - spring 1 - mechanics

 Unit 10: Forces and Motion Skills Understand the concept of a force; understand and use Newton’s first law. 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 (i, j) vectors). 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. Knowledge By the end of the unit, students should: understand the concept of a force; understand and use Newton’s first law. understand and be able to 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 (i, j) vectors.); 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. 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 - spring 1 - pure

 Unit 11: Vectors (2D) Skills Use vectors in two dimensions. Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form. Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations. Understand and use position vectors; calculate the distance between two points represented by position vectors. Use vectors to solve problems in pure mathematics and in context, (including forces). Knowledge By the end of the sub-unit, students should: be able to use vectors in two dimensions; be able to calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form; be able to add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations. understand and be able to use position vectors; be able to calculate the distance between two points represented by position vectors; be able to use vectors to solve problems in pure mathematics and in context, (including forces). 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 - statistics

 Unit 6: Statistical distributions Skills Understand and use simple, discrete probability distributions (calculation of mean and variance of discrete random variables is excluded), including the binomial distribution, as a model; calculate probabilities using the binomial distribution. Knowledge understand and be able to use simple, discrete probability distributions, including the binomial distribution; be able to identify the discrete uniform distribution; be able to calculate probabilities using the binomial distribution. Rationale Every time you try to describe a large set of observations with a single indicator you run the risk of distorting the original data or losing important detail. For example, the batting average does not tell you whether the batter is hitting home runs or singles. It does not tell whether she's been in a slump or on a streak. The GPA does not tell you whether the student was in difficult courses or easy ones, or whether they were courses in their major field or in other disciplines. Even given these limitations, descriptive statistics provide a powerful summary that may enable comparisons across people or other units. Reference

## unit overview - spring 2 - mechanics

 Unit 11: Kinematics 2 (variable acceleration) Skills Use calculus in kinematics for motion in a straight line. Knowledge By the end of the unit, students should: be able to use calculus (differentiation) in kinematics to model motion in a straight line for a particle moving with variable acceleration; understand that gradients of the relevant graphs link to rates of change; know how to find max and min velocities by considering zero gradients and understand how this links with the actual motion (i.e. acceleration = 0). be able to use calculus (integration) in kinematics to model motion in a straight line for a particle moving under the action of a variable force; understand that the area under a graph is the integral, which leads to a physical quantity; know how to use initial conditions to calculate the constant of integration and refer back to the problem. Rationale Kinematics is the field of mechanics that deals with moving objects. As such, any object that moves can be modelled (at least partially) using a kinematics equation. Kinematics is therefore a core part of mechanics as a whole, and to the entire field of physics. Kinematics is also a logical and consistent framework while helps to develop problem solving and modelling skills. Kinematics has direct application to engineering and sport, along with many other fields. For example, is sport kinematics can be used to calculate trajectories of balls and to analyse the movement of players in order to identify areas of improvement. In engineering, kinematics is used in designing and testing cars, bikes, and other forms of transport. It is also necessary to calculate the required forces needed to accelerate and decelerate an object.

## unit overview - spring 2 - pure

 Units 7 and 8:  Further algebra Skills Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including: proof by deduction, proof by exhaustion, disproof by counter-example. Understand and use the binomial expansion of (𝑎 + 𝑏𝑥)n for positive integer n; the notations n! and  n𝐶r; link to binomial probabilities. Knowledge By the end of the unit, students should: understand and be able to use the binomial expansion of (a + bx)n for positive integer n; be able to find an unknown coefficient of a binomial expansion. be able to use algebraic division; know and be able to apply the factor theorem; be able to fully factorise a cubic expression; understand and be able to use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; be able to use methods of proof, including proof by deduction, proof by exhaustion and disproof by counter-example. Rationale Algebra is the basis of all higher mathematics. It allows for mathematics to be done with variables in place of numerical values, and so allows for solving and the expression of relationships with regard to these variables. In addition, it is often quicker for more complicated numerical problems to be solved algebraically instead. Algebra has many applications over a wide range of fields. For instance, it is utilized in finance, chemistry, physics and environmental science. In particular, proof develops strong reasoning skills. This improves critical thinking and causes the learner to more readily critically analyse situations. This has wide applications to almost any field of study or work that a learner may wish to pursue. Algebra also develops modelling, logic, and rationalisation skills. These can be widely applied to other areas that do not have a direct application of algebra.

## unit overview - spring 2 - statistics

 Unit 7: Statistical hypothesis testing Skills Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value. Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis. Knowledge By the end of the sub-unit, students should: understand and be able to apply the language of statistical hypothesis testing, developed through a binomial model. be able to conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context; understand that a sample is being used to make an inference about the population; appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis. Rationale There are many real world applications of hypothesis testing and some of these are; Testing whether more men than women suffer from nightmares Establishing authorship of documents Evaluating the effect of the full moon on behaviour Determining the range at which a bat can detect an insect by echo Deciding whether hospital carpeting results in more infections Selecting the best means to stop smoking Checking whether bumper stickers reflect car owner behaviour Testing the claims of handwriting analysts Reference:  https://en.wikipedia.org/wiki/Statistical_hypothesis_testing#Use_and_importance

## unit overview - summer 1 - pure

 Units 7 and 8: Further algebra Skills Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including: proof by deduction, proof by exhaustion, disproof by counter-example. Understand and use the binomial expansion of (𝑎 + 𝑏𝑥)n for positive integer n; the notations n! and  n𝐶r; link to binomial probabilities. Knowledge By the end of the unit, students should: understand and be able to use the binomial expansion of (a + bx)n for positive integer n; be able to find an unknown coefficient of a binomial expansion. be able to use algebraic division; know and be able to apply the factor theorem; be able to fully factorise a cubic expression; understand and be able to use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; be able to use methods of proof, including proof by deduction, proof by exhaustion and disproof by counter-example. Rationale Algebra is the basis of all higher mathematics. It allows for mathematics to be done with variables in place of numerical values, and so allows for solving and the expression of relationships with regard to these variables. In addition, it is often quicker for more complicated numerical problems to be solved algebraically instead. Algebra has many applications over a wide range of fields. For instance, it is utilised in finance, chemistry, physics and environmental science. In particular, proof develops strong reasoning skills. This improves critical thinking and causes the learner to more readily critically analyse situations. This has wide applications to almost any field of study or work that a learner may wish to pursue. Algebra also develops modelling, logic, and rationalisation skills. These can be widely applied to other areas that do not have a direct application of algebra.

## knowledge organisers

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.