Year 12 – Mathematics

Surds and Indices 

Students will understand and use the laws of indices for all rational exponents; use and manipulate surds, including rationalising the denominator. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Algebra and Functions 

Students will learn about quadratic functions and their graphs including the discriminant of a quadratic function, completing the square; 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; manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Graph Transformations 

Students will understand the effect of simple transformations on the graph of y = f(x) including sketching associated graphs: y = af(x), y = f(x) + a, y = f(x + a) and y = f(ax). Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Differentiation 

Students will understand and use the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; differentiation from first principles for small positive integer powers of x; understand and use the second derivative as the rate of change of gradient; differentiate constant multiples, sums and differences, apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points and identify where functions are increasing or decreasing. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Binomial Expansion

Students will understand and use the binomial expansion of (a + bx)^n for positive integer n; the notations n! and nCr; link to binomial probabilities. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Statistics and Probability 

Students will use samples to make informal inferences about the population; understand and use sampling techniques; interpret diagrams for single-variable data; interpret scatter diagrams and regression lines for bivariate data; interpret measures of central tendency and variation; be able to calculate standard deviation; understand and use mutually exclusive and independent events when calculating probabilities and link to discrete and continuous distributions; understand and use simple, discrete probability distributions including the binomial distribution; calculate probabilities using the binomial distribution; understand and apply the language of statistical hypothesis testing, 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 and 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. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Proof 

Students will 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 and disproof by counter example; proof by contradiction (including proof of the irrationality of surds and the infinity of primes, and application to unfamiliar proofs). Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Binomial Expansion 

Students will be familiar with, and be able to use, the binomial expansion of (1 + x)^n. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Sequences and Series 

Students will work with sequences including those given by a formula for the nth term, increasing sequences,  decreasing sequences,  periodic sequences; understand and use sigma notation for sums of series; understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms; understand and work with geometric sequences and series including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r| < 1; modulus notation and use sequences and series in modelling. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Coordinate Geometry in the (x, y) plane 

Students will understand and use the equation of a straight line, gradient conditions for two straight lines to be parallel or perpendicular, be able to use straight line models in a variety of contexts; understand and use the coordinate geometry of the circle including using the equation of a circle by completing the square to find the centre and radius of a circle. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Trigonometry 

Students will understand and use the definitions of sine, cosine, and tangent for all arguments; the sine and cosine rules; the area of a triangle; understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity and basic trigonometrical identities. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Vectors 

Students will 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, understand their geometrical interpretations; understand and use position vectors; calculate the distance between two points represented by position vectors and use vectors to solve problems in pure mathematics and in context. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Integration 

Students will know and use the Fundamental Theorem of Calculus. Integrate x^n (excluding n = –1), and related sums, differences, and constant multiples.  Evaluate definite integrals and use a definite integral to find the area under a curve. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Exponentials and Logarithms 

Students will know and use the functions a^x and e^x and their graphs; understand why the exponential model is suitable in many applications; know and use the function ln x and its graph; know and use ln x as the inverse function of e^x; understand and use the laws of logarithms; solve equations of the form ax = b; use logarithmic graphs to estimate parameters in relationships of the form y = ax^n and y = kb^x, given data for x and y and understand and use exponential growth and decay, their use in modelling with consideration of limitations and refinements of exponential models. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Kinematics 

Students will 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; students will use calculus in kinematics for motion in a straight line when the acceleration of the body is not constant. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Forces and Newton’s Laws 

Students will understand the concept of a force; understand and use Newton’s first and second laws; understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in SI units to varying degrees of accuracy; understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Algebraic Methods 

Students will simplify rational expressions including by factorising, cancelling, and algebraic division and decompose rational functions into partial fractions. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Functions and Graphs 

Students will understand and use composite functions, inverse functions, and their graphs; apply two or more transformations to a function or describe a combination of two or more transformations that result in a given function and understand that applying transformations in a different order may result in two different functions. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test. 

Radians 

Students will work with radian measure, including use for arc length and area of sector. Assessment will be via continuous scrutiny of class and homework, coupled with an end of module test.