Mathematics

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This subtopic explores the foundational concepts of functions and their graphical representations, essential for modelling real-world relationships in mathematics. It covers domain and range restrictions, composite and inverse functions, and the application of transformations to sketch and analyse graphs. Mastery of these skills enables students to solve complex equations and interpret graphical data in contexts such as kinematics and economics.

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Overarching themes

Mathematical modelling involves translating real-world situations into mathematical representations by making simplifying assumptions. Students must be able to use these models to explore scenarios, interpret outputs in the original context, and evaluate the appropriateness of the model through refinement.

OT3: Mathematical modelling, OT2: Mathematical problem solving, OT1: Mathematical argument, language and proof

I: Numerical methods

Numerical methods involve finding approximate solutions to equations that cannot be solved analytically. This includes locating roots of f(x) = 0 using sign changes, applying iterative methods like the Newton-Raphson formula, and performing numerical integration using the trapezium rule.

J: Vectors

This topic covers the fundamental principles of vectors in two and three dimensions, including their representation, magnitude, and direction. Students learn to perform algebraic operations such as addition and scalar multiplication, understand position vectors, and apply these concepts to solve problems in pure mathematics, kinematics, and forces.

K: Statistical sampling

This topic covers the fundamental concepts of statistical sampling, including the distinction between populations and samples. Students learn to use samples to make informal inferences about populations and explore various sampling techniques, such as simple random sampling and opportunity sampling, while evaluating their appropriateness in context.

L: Data presentation and interpretation

This topic focuses on the interpretation and analysis of statistical data using various graphical and numerical methods. Students are required to interpret diagrams for single-variable data, understand bivariate data through scatter diagrams and regression lines, and calculate and interpret measures of central tendency and variation, including standard deviation.

M: Probability

This topic covers the fundamental principles of probability, including the use of mutually exclusive and independent events. It extends to conditional probability, utilizing tools such as tree diagrams, Venn diagrams, and two-way tables, alongside the formal conditional probability formula.

N: Statistical distributions

This topic covers the application of discrete and continuous probability distributions as models for real-world data. Students learn to calculate probabilities using the binomial and Normal distributions, and develop the ability to select and justify the use of an appropriate model for a given context.

O: Statistical hypothesis testing

This topic covers the principles of statistical hypothesis testing, including the use of null and alternative hypotheses, significance levels, and critical regions. Students learn to conduct tests for proportions in a binomial distribution and for the mean of a Normal distribution, as well as interpreting correlation coefficients using p-values or critical values.

P: Quantities and units in mechanics

This topic covers the fundamental quantities and units used within the SI system for mechanics. It establishes the essential building blocks for physical analysis, specifically focusing on length, time, mass, velocity, acceleration, force, weight, and moments.

Q: Kinematics

This topic covers the fundamental principles of kinematics, focusing on motion in a straight line and extending to two dimensions using vectors. It requires students to interpret displacement-time and velocity-time graphs, apply constant acceleration formulae, and use calculus to relate position, velocity, and acceleration.

R: Forces and Newton’s laws

This topic covers the fundamental principles of forces, including Newton's three laws of motion and their application to particles. It involves resolving forces in two dimensions, understanding weight, and applying the model of friction to objects on rough surfaces.

A: Proof

This topic covers the fundamental structure of mathematical proof, requiring students to progress from given assumptions through logical steps to a valid conclusion. It includes specific methods such as proof by deduction, proof by exhaustion, disproof by counter-example, and proof by contradiction, with applications including the irrationality of root 2 and the infinity of primes.

S: Moments

This topic covers the concept of moments in simple static contexts. Students must understand how to calculate the turning effect of a force about a pivot and apply this to solve problems involving equilibrium.

Use of data in statistics

Students must become familiar with a specific large data set provided by AQA to explore statistical concepts and skills in real-world contexts. This involves using technology to analyze the data, interpret summary or graphical forms, and investigate questions arising from the data set.

Large data set

B: Algebra and functions

This topic covers advanced algebraic techniques and the study of functions, including indices, surds, and quadratic theory. It extends to the manipulation of polynomials, rational expressions, partial fractions, and the analysis of composite and inverse functions, alongside transformations of graphs.

C: Coordinate geometry in the ( x , y ) plane

This topic covers the analytical geometry of straight lines and circles in the Cartesian plane. It extends to the use of parametric equations to describe curves and their application in mathematical modelling.

D: Sequences and series

This topic covers the study of sequences and series, including binomial expansions for positive integer and rational powers. It encompasses arithmetic and geometric progressions, the use of sigma notation, and the application of these concepts to mathematical modelling.

E: Trigonometry

This topic covers advanced trigonometric functions, identities, and their applications in solving equations and modelling. It extends beyond basic right-angled trigonometry to include radian measure, secant, cosecant, cotangent, inverse functions, and complex identities such as double angle and harmonic forms.

F: Exponentials and logarithms

This topic covers the properties and graphs of exponential functions, including base 'a' and the natural exponential 'e'. It also introduces logarithms as the inverse of exponential functions, focusing on the laws of logarithms and solving exponential equations.

G: Differentiation

This topic covers the fundamental principles of differentiation, including the derivative as a gradient and rate of change. Students learn to differentiate various functions, apply rules such as the product, quotient, and chain rules, and use these techniques to solve problems involving tangents, normals, stationary points, and connected rates of change.

H: Integration

This topic covers the fundamental principles of integration, including the Fundamental Theorem of Calculus and techniques for evaluating definite and indefinite integrals. Students learn to apply integration to find areas under and between curves, solve first-order differential equations using separation of variables, and employ advanced methods such as integration by substitution and integration by parts.