Syllabus
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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_statsTopic 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
- Establish and use logarithmic laws and definitions
- Interpret and use logarithmic scales such as decibels in acoustics, the Richter scale for earthquake magnitude, octaves in music, pH in chemistry
- Solve equations involving indices with and without technology
- Recognise the qualitative features of the graph of y = loga(x) (a > 1), including asymptotes, and of its translations y = loga(x) + b and y = loga(x + c)
- Solve equations involving logarithmic functions with and without technology
- Identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis.
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
- Estimate the limit of (a^h−1)/h as h → 0 using technology, for various values of a > 0
- Recognise that ee is the unique number aa for which the above limit is 1
- Define the exponential function eex
- Establish and use the formula d/dx e^x = e^x and d/dx e^f(x) = f'(x)e^(f(x))
- Identify contexts suitable for mathematical modelling by exponential functions and their derivatives and use the model to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis.
Calculus of logarithmic functions
view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of logarithmic functions
- Define the natural logarithm ln(x) = loge(x)
- Recognise and use the inverse relationship of the functions y = e^x and y = ln(x)
- Establish and use the formulas d/dx (ln(x)) = 1 x and d/dx (ln f(x)) = f'(x) f(x)
- Use logarithmic functions and their derivatives to solve practical problems.
Calculus of trigonometric functions
view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Calculus of trigonometric functions
- Establish the formulas d/dx sin(x) = cos(x), and d/dx cos(x) = − sin(x) by numerical estimations of the limits and informal proofs based on geometric constructions
- Identify contexts suitable for modelling by trigonometric functions and their derivatives and use the model to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
- Use trigonometric functions and their derivatives to solve practical problems; including trigonometric functions of the form y = sin(f(x)) and y = cos(f(x)).
Differentiation rules
view_agenda query_statsUnit 3: Further calculus > Topic 2: Further differentiation and applications 2 > Differentiation rules
- Select and apply the product rule, quotient rule and chain rule to differentiate functions; express derivatives in simplest and factorised form.
Topic 3: Integrals
view_agenda query_statsAnti-differentiation
view_agenda query_statsUnit 3: Further calculus > Topic 3: Integrals > Anti-differentiation
- Recognise anti-differentiation as the reverse of differentiation
- Use the notation for anti-derivatives or indefinite integrals
- Establish and use the formula ∫ x^n d/dx = (1/(n+1)) x^(n+1) + c for n ≠ −1
- Establish and use the formula ∫ e^x d/dx = e^x + c
- Establish and use the formulas ∫ 1 x d/dx = ln(x) + c , for x > 0 and ∫ 1/(ax+b) d/dx = (1/a)ln(ax + b) + c
- Establish and use the formulas of ∫sin(x) d/dx = −cos(x) + c and ∫cos(x)d/dx = sin(x) + cc
- Understand and use the formula for indefinite integrals of the form (f(x) + g(x)) d/dx = f(x) d/dx + g(x) d/dx
- Determine indefinite integrals of the form ∫f(ax + b)
- Determine f(x), given f'(x) and an initial condition f(a) = b
- Determine the integral of a function using information about the derivative of the given function (integration by recognition)
- Determine displacement given velocity in linear motion problems.
Fundamental theorem of calculus and definite integrals
view_agenda query_statsUnit 3: Further calculus > Topic 3: Integrals > Fundamental theorem of calculus and definite integrals
- Examine the area problem, and use sums to estimate the area under the curve y = f(x)
- Use the trapezoidal rule for the approximation of the value of a definite integral numerically
- Interpret the definite integral f(x) as area under the curve y = f(x) if f(x) > 0
- Recognise the definite integral f(x) as a limit of sums
- Understand the definite integral of a function f(x) over the interval [a, b] is equal to F(b) − F(a) and use it to calculate definite integrals.
Applications of integration
view_agenda query_statsUnit 3: Further calculus > Topic 3: Integrals > Applications of integration
- Calculate the area under a curve
- Calculate total change by integrating instantaneous or marginal rate of change
- Calculate the area between curves with and without technology
- Determine displacements given acceleration and initial values of displacement and velocity.
Unit 4: Further functions and statistics
view_agenda query_statsTopic 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
- Understand the concept of the second derivative as the rate of change of the first derivative function
- Recognise acceleration as the second derivative of displacement position with respect to time
- Understand the concepts of concavity and points of inflection and their relationship with the second derivative
- Understand and use the second derivative test for finding local maxima and minima
- Sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection
- Solve optimisation problems from a wide variety of fields using first and second derivatives, where the function to be optimised is both given and developed.
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
- Recall sine, cosine and tangent as ratios of side lengths in right-angled triangles
- Understand the unit circle definition of cos(x), sin(x) and tan(x) and periodicity using degrees and radians
- Establish and use the sine (ambiguous case is required) and cosine rules and the formula for the area of a triangle
- 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
- Use a Bernoulli random variable as a model for two-outcome situations
- Identify contexts suitable for modelling by Bernoulli random variables
- Recognise and determine the mean p and variance p(1 − p) of the Bernoulli distribution with parameter p
- Use Bernoulli random variables and associated probabilities to model data and solve practical problems.
Binomial distributions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 3: Discrete random variables 2 > Binomial distributions
- Understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes' in n independent Bernoulli trials, with the same probability of success p in each trial
- Identify contexts suitable for modelling by binomial random variables
- Determine and use the probabilities associated with the binomial distribution with parameters n and p
- Calculate the mean and variance of a binomial distribution using technology and algebraic methods
- Identify contexts suitable to model binomial distributions and associated probabilities to solve practical problems, including the language of ‘at most' and ‘at least'.
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
- Use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable
- Understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts
- Calculate the expected value, variance and standard deviation of a continuous random variable in simple cases
- Understand standardised normal variables (z-values, z-scores) and use these to compare samples.
Normal distributions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 4: Continuous random variables and the normal distribution > Normal distributions
- Identify contexts, such as naturally occurring variations, that are suitable for modelling by normal random variables
- Recognise features of the graph of the probability density function of the normal distribution with mean and standard deviation ?? and the use of the standard normal distribution
- Calculate probabilities and quantiles associated with a given normal distribution using technology and use these to solve practical problems.
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
- Understand the concept of a random sample
- Discuss sources of bias in samples, and procedures to ensure randomness
- Investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli, using graphical displays of real and simulated data.
Sample proportions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 5: Interval estimates for proportions > Sample proportions
- Understand the concept of the sample proportion as a random variable whose value varies between samples, and the formulas for the mean and standard deviation of the sample proportion
- Consider the approximate normality of the distribution of ? for large samples
- Simulate repeated random sampling, for a variety of values of p and a range of sample sizes, to illustrate the distribution of the sample proportion and the approximate standard normality of where the closeness of the approximation depends on both n and p.
Confidence intervals for proportions
view_agenda query_statsUnit 4: Further functions and statistics > Topic 5: Interval estimates for proportions > Confidence intervals for proportions
- Understand the concept of an interval estimate for a parameter associated with a random variable
- Use the approximate confidence interval as an interval estimate for p, where ?? is the appropriate quantile for the standard normal distribution
- Define the approximate margin of error and understand the trade-off between margin of error and level of confidence
- Use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain ?.