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#### Unit 1: Algebra, statistics and functions

##### Topic 1: Arithmetic and geometric sequences and series 1

Arithmetic sequences

Unit 1: Algebra, statistics and functions > Topic 1: Arithmetic and geometric sequences and series 1 > Arithmetic sequences

- Recognise and use the recursive definition of an arithmetic sequence

- Use the formula for the general term of an arithmetic sequence and recognise its linear nature

- Use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest

- Establish and use the formula for the sum of the first n terms of an arithmetic sequence

##### Topic 2: Functions and graphs

Functions

Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Functions

- Understand the concept of a relation as a mapping between sets, a graph and as a rule or a formula

that defines one variable quantity in terms of another

Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > that defines one variable quantity in terms of another

- Recognise the distinction between functions and relations and use the vertical line test to determine whether a relation is a function

- Use function notation, domain and range, and independent and dependent variables

- Examine transformations of the graphs of f(x), including dilations and reflections, and the graphs of y = af(x), y = f(bx), y = f(x+c), y = f(x) + d

- Recognise and use piece-wise functions as a combination of multiple sub-functions with restricted domains

- Identify contexts suitable for modelling piece-wise functions and use them to solve practical problems (taxation, taxis, the changing velocity of a parachutist).

Review of quadratic relationships

Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Review of quadratic relationships

- Examine examples of quadratically related variables

- Recognise and determine features of the graphs of quadratics, including their parabolic nature, turning points, axes of symmetry and intercepts

- Solve quadratic equations algebraically using factorisation, the quadratic formula (both exact and approximate solutions), and completing the square and using technology

- Identify contexts suitable for modelling with quadratic functions and use models to solve problems with and without technology; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

- Understand the role of the discriminant to determine the number of solutions to a quadratic equation

- Determine turning points and zeros of quadratic functions with and without technology.

Inverse proportions

Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Inverse proportions

- Examine examples of inverse proportion

- Recognise features of the graphs of inverses, including their hyperbolic shapes, their intercepts, their asymptotes and behaviour as approaches positive and negative infinity

Powers and polynomials

Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Powers and polynomials

- Identify the coefficients and the degree of a polynomial

- Expand quadratic and cubic polynomials from factors

- Recognise and determine features of the graphs of cubics, including shape, intercepts and behaviour as approaches positive and negative infinity

- Use the factor theorem to factorise cubic polynomials in cases where a linear factor is easily obtained

- Solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained

- Recognise and determine features of the graphs, including shape and behaviour

- Solve equations involving combinations of the functions above, using technology where appropriate.

Graphs of relations

Unit 1: Algebra, statistics and functions > Topic 2: Functions and graphs > Graphs of relations

- Recognise and determine features of the graphs of circles including their circular shapes, centres and radii

- Recognise and determine features of the graph of y^2 = x, including its parabolic shape and axis of symmetry.

##### Topic 3: Counting and probability

Language of events and sets

Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Language of events and sets

- Recall the concepts and language of outcomes, sample spaces and events as sets of outcomes

- Use set language and notation for events, including the complement of an event, the intersection of events A and B, and the union, and recognise mutually exclusive events

- Use everyday occurrences to illustrate set descriptions and representations of events, and set operations, including the use of Venn diagrams.

Review of the fundamentals of probability

Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Review of the fundamentals of probability

- Recall probability as a measure of ‘the likelihood of occurrence' of an event

- Recall the probability scale for each event

- Recall the rules the complement of an event and the intersection of events A and B

- Use relative frequencies obtained from data as point estimates of probabilities.

Conditional probability and independence

Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Conditional probability and independence

- Understand the notion of a conditional probability, and recognise and use language that indicates conditionality

- Use the notation P(A|B) and the formula for this to solve problems

- Understand and use the notion of independence of an event A from an event B, as defined by P(A|B) = P(A)

- Establish and use the formula for independent events A and B

- Use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events.

Binomial expansion

Unit 1: Algebra, statistics and functions > Topic 3: Counting and probability > Binomial expansion

- Understand the notion of a combination as an unordered set of r objects taken from a set of n distinct objects

- Recognise and use the link between Pascal's triangle and the notation nCr

- Expand (x + y)^n for small positive integers n.

##### Topic 4: Exponential functions 1

Indices and the index laws

Unit 1: Algebra, statistics and functions > Topic 4: Exponential functions 1 > Indices and the index laws

- Recall indices (including negative and fractional indices) and the index laws

- Convert radicals to and from fractional indices

- Understand and use scientific notation.

##### Topic 5: Arithmetic and geometric sequences and series 2

Geometric sequences

Unit 1: Algebra, statistics and functions > Topic 5: Arithmetic and geometric sequences and series 2 > Geometric sequences

- Recognise and use the recursive definition of a geometric sequence

- Use the formula for the general term of a geometric sequence and recognise its exponential nature

- Understand the limiting behaviour as n → ∞ of the terms ??n in a geometric sequence and its dependence on the value of the common ratio r

- Establish and use the formula for the sum of the first n terms of a geometric sequence

- Establish and use the formula for the sum to infinity of a geometric progression

- Use geometric sequences in contexts involving geometric growth or decay, including compound interest and annuities.

#### Unit 2: Calculus and further functions

##### Topic 1: Exponential functions 2

Introduction to exponential functions

Unit 2: Calculus and further functions > Topic 1: Exponential functions 2 > Introduction to exponential functions

- Recognise and determine the qualitative features of the graph of y = a^x (a > 0), including asymptotes, and of its translations (y = a^x + b and y = a^(x+c))

- Recognise and determine the features of the graphs of y = b. a^x and y = a^kx

- Identify contexts suitable for modelling by exponential functions and use models to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

- Solve equations involving exponential functions with and without technology

##### Topic 2: The logarithmic function 1

Introduction to logs

Unit 2: Calculus and further functions > Topic 2: The logarithmic function 1 > Introduction to logs

- Define logarithms as indices: a^x = b is equivalent to x = loga(b)

- Recognise the inverse relationship between logarithms and exponentials: y = a^x is equivalent to x = loga(y)

- Solve equations involving indices with and without technology

##### Topic 3: Trigonometric functions 1

Circular measure and radian measure

Unit 2: Calculus and further functions > Topic 3: Trigonometric functions 1 > Circular measure and radian measure

- Define and use radian measure and understand its relationship with degree measure

- Calculate lengths of arcs and areas of sectors in circles.

Introduction to trigonometric functions

Unit 2: Calculus and further functions > Topic 3: Trigonometric functions 1 > Introduction to trigonometric functions

- Understand the unit circle definition of cos(x), sin(x)and tan(x) and periodicity using radians

- Recall the exact values of sin(x), cos(x) and tan(x) at integer multiples of ?/6 and ?/4

- Sketch the graphs of y = sin(x), y = cos(x), and y = tan(x) on extended domains

- Investigate the effect of the parameters A, B, C and D on the graphs of y = A sin(B(x + C)) + D, y = A cos(B(x + C)) + D with and without technology

- Sketch the graphs of y = A sin(B(x + C)) + D, y = A cos(B(x + C)) + D with and without technology

- Identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

- Solve equations involving trigonometric functions with and without technology, including use of the Pythagorean identity sin^2(A) + cos^2(A) = 1.

##### Topic 4: Introduction to differential calculus

Rates of change and the concept of derivatives

Unit 2: Calculus and further functions > Topic 4: Introduction to differential calculus > Rates of change and the concept of derivatives

- Explore average and instantaneous rate of change in a variety of practical contexts

- Use a numerical technique to estimate a limit or an average rate of change

- Examine the behaviour of the difference quotient as an informal introduction to the concept of a limit

- Differentiate simple power functions and polynomial functions from first principles

- Interpret the derivative as the instantaneous rate of change

- Interpret the derivative as the gradient of a tangent line of the graph of y = f(x).

Properties and computation of derivatives

Unit 2: Calculus and further functions > Topic 4: Introduction to differential calculus > Properties and computation of derivatives

- Examine examples of variable rates of change of non-linear functions

- Establish the formula d/dx x^n = nx^(n-1) for positive integers

- Understand the concept of the derivative as a function

- Recognise and use properties of the derivative: d/dx (f(x) + g(x)) = d/dx f(x) + d/dx g(x)

- Calculate derivatives of power and polynomial functions.

Applications of derivatives

Unit 2: Calculus and further functions > Topic 4: Introduction to differential calculus > Applications of derivatives

- Determine instantaneous rates of change

- Determine the gradient of a tangent and the equation of the tangent

- Construct and interpret displacement-time graphs, with velocity as the slope of the tangent

- Sketch curves associated with power functions and polynomials up to and including degree 4; find stationary points and local and global maxima and minima with and without technology; and examine behaviour as x approaches positive and negative infinity

- Identify contexts suitable for modelling optimisation problems involving polynomials up to and including: degree 4 and power functions on finite interval domains, and use models to solve practical problems, with and without technology; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis.

##### Topic 5: Further differentiation and applications 1

Differentiation rules

Unit 2: Calculus and further functions > Topic 5: Further differentiation and applications 1 > Differentiation rules

- Understand and apply the product rule and quotient rule for power and polynomial functions

- Understand the notion of composition of power and polynomial functions and use the chain rule for determining the derivatives of composite functions

- Select and apply the product rule, quotient rule and chain rule to differentiate power and polynomial functions, express derivative in simplest and factorised form.

##### Topic 6: Discrete random variables 1

General discrete random variables

Unit 2: Calculus and further functions > Topic 6: Discrete random variables 1 > General discrete random variables

- Understand the concepts of a discrete random variable and its associated probability function, and its use in modelling data

- Use relative frequencies obtained from data to determine point estimates of probabilities associated with a discrete random variable

- Recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes

- Examine simple examples of non-uniform discrete random variables

- Recognise the mean or expected value of a discrete random variable as a measurement of centre, and evaluate it in simple cases

- Recognise the variance and standard deviation of a discrete random variable as a measure of spread, and evaluate these in simple cases

- Use discrete random variables and associated probabilities to solve practical problems.

#### Unit 3: Further calculus

view_agenda query_stats##### Topic 1: The logarithmic function 2

view_agenda query_statsLogarithmic laws and logarithmic functions

view_agenda query_statsUnit 3: Further calculus > Topic 1: The logarithmic function 2 > Logarithmic laws and logarithmic functions

##### Topic 2: Further differentiation and applications 2

view_agenda query_statsCalculus of exponential functions

view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of exponential functions

Calculus of logarithmic functions

view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of logarithmic functions

Calculus of trigonometric functions

view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of trigonometric functions

Differentiation rules

view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Differentiation rules

##### Topic 3: Integrals

view_agenda query_statsAnti-differentiation

view_agenda query_statsFundamental theorem of calculus and definite integrals

view_agenda query_statsUnit 3: Further calculus > Topic 3: Integrals > Fundamental theorem of calculus and definite integrals

Applications of integration

view_agenda query_stats#### Unit 4: Further functions and statistics

view_agenda query_stats##### Topic 1: Further differentiation and applications 3

view_agenda query_statsThe second derivative and applications of differentiation

view_agenda query_statsUnit 4: Further functions and statistics > Topic 1: Further differentiation and applications 3 > The second derivative and applications of differentiation

##### Topic 2: Trigonometric functions 2

view_agenda query_statsCosine and sine rules

view_agenda query_statsUnit 4: Further functions and statistics > Topic 2: Trigonometric functions 2 > Cosine and sine rules

- Construct mathematical models using the sine and cosine rules in two- and three-dimensional contexts (including bearings in two-dimensional context) and use the model to solve problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis.

##### Topic 3: Discrete random variables 2

view_agenda query_statsBernoulli distributions

view_agenda query_statsUnit 4: Further functions and statistics > Topic 3: Discrete random variables 2 > Bernoulli distributions

Binomial distributions

view_agenda query_statsUnit 4: Further functions and statistics > Topic 3: Discrete random variables 2 > Binomial distributions

##### Topic 4: Continuous random variables and the normal distribution

view_agenda query_statsGeneral continuous random variables

view_agenda query_statsUnit 4: Further functions and statistics > Topic 4: Continuous random variables and the normal distribution > General continuous random variables

Normal distributions

view_agenda query_statsUnit 4: Further functions and statistics > Topic 4: Continuous random variables and the normal distribution > Normal distributions

##### Topic 5: Interval estimates for proportions

view_agenda query_statsRandom sampling

view_agenda query_statsUnit 4: Further functions and statistics > Topic 5: Interval estimates for proportions > Random sampling

Sample proportions

view_agenda query_statsUnit 4: Further functions and statistics > Topic 5: Interval estimates for proportions > Sample proportions

Confidence intervals for proportions

view_agenda query_stats