Our Mathematics KS5 Solution is tailored to ensure students develop confidence and competence in a variety of practical, mathematical and problem solving skills. Most students develop enthusiasm in further study and careers associated with mathematics and we support with such development.

# Mathematics KS5

£75.00

### Description

**Mathematics KS5**

## Objectives and aims:

Our Mathematics KS5 solution meets the needs to develop understanding, skills and knowledge required for students at KS5 level. We incorporate fun and intriguing technologies to teach some of the hardest science concepts. Our solution emphasises how mathematical ideas are interconnected and how mathematics can be applied to model situations mathematically using algebra and other representations, to help make sense of data and to understand the physical world and to solve problems in a variety of contexts, including social sciences and business. It prepares students for further study and

employment in a wide range of disciplines involving the use of mathematics.

Our Mathematics KS5 Solution is tailored to ensure students develop confidence and competence in a variety of practical, mathematical and problem solving skills. Most students develop enthusiasm in further study and careers associated with mathematics and we support with such development.

Our Mathematics KS5 solution helps learners reason logically and recognise incorrect reasoning. It helps them to construct mathematical proofs and represent situations mathematically and understand the relationship between problems in context and mathematical models that may be applied to solve problems. Along with this, student get vital understanding of how mathematics contribute towards the success of the economy and society.

**Mathematics KS5 Contents:**

### Algebra and Functions:

- 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
- Represent linear and quadratic inequalities such as y x > + 1 and > ++ 2 y ax bx c graphically.
- Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem.
- Understand and use graphs of functions; sketch curves defined by simple equations including polynomials], the modulus of a linear function, [ a y x = and 2 a y x = (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations.
- Understand and use proportional relationships and their graphs.
- Understand and use composite functions; inverse functions and their graphs.
- Understand the effect of simple transformations on the graph of y = f ( x)including sketching associated graphs.
- Decompose rational functions into partial fractions, denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear.
- Use of functions in modelling, including consideration of limitations and refinements of the models.

### Proof:

- 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.
- Proof by contradiction (including proof of the irrationality of √2 and the infinity of primes, and application to unfamiliar proofs.

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

- Understand and use the equation of a straight line.
- 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 ; completing the square to find the centre and radius of a circle; use of the following properties: • the angle in a semicircle is a right angle.
- the perpendicular from the centre to a chord bisects the chord.
- the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.
- Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.
- Use parametric equations in modelling in a variety of contexts.
- Understand and use the binomial expansion of ( ) + n a bx for positive integer n; the notations n! and nCr; link to binomial probabilities.
- Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form xn+1 = f(xn); 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.
- Use sequences and series in modelling.

### Trigonometry

- Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form 1/2absinC
- Work with radian measure, including use for arc length and area of sector.
- Understand and use the standard small angle approximations of sine, cosine and tangent.
- Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity.
- Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains
- Understand and use double angle formulae; use of formulae for sin(A +- B), cos( A +- B ) tan( A +-B ) ; understand geometrical proofs of these formulae.
- Understand and use expressions for acosθ + bsinθ in the equivalent forms of rcos(θ+-α ) or sin(
- θ+-α).
- Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle.
- Construct proofs involving trigonometric functions and identities.
- Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

### Sequences and series:

- Understand and use the binomial expansion of ( a + bx) for positive integer n; the notations n! and nCr; link to binomial probabilities.
- Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form xn+1 = f(xn); 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.
- Use sequences and series in modelling.

### Exponentials and logarithms:

- Know and use the definition of loga x.
- Know and use the function lnx and its graph.
- Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models.

### Differentiation:

- 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; second derivatives; differentiation from first principles for small positive integer powers of x and for sinx cos x.
- Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection.
- Understand and use the derivative of ln x.
- Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection. Identify where functions are increasing or decreasing.
- Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.
- Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.
- Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand).

### Integration:

- Know and use the Fundamental Theorem of Calculus.
- [Evaluate definite integrals; use a definite integral to find the area under a curve] and the area between two curves.
- Understand and use integration as the limit of a sum.
- Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively.
- Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae.
- Integrate using partial fractions that are linear in the denominator.
- Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions.
- Separation of variables may require factorisation involving a common factor.
- Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics.

### Numerical Methods:

- Locate roots of f(x ) = 0 by considering changes of sign of f( x) in an interval of x on which f( x) is sufficiently well-behaved.
- Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams.
- Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between.
- Use numerical methods to solve problems in context.

### Vectors

- [Use vectors in two dimensions] and in three dimensions
- [Calculate the magnitude and direction of a vector and convert between 12 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] and kinematics.

### Statistical Sampling

- [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.

### Data Presentation and Interpretation

- 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.
- 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.

### Probability

- Understand and use mutually exclusive and independent events when calculating probabilities.
- Link to discrete and continuous distributions.
- Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables.
- Understand and use the conditional probability formula.
- Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.

### Statistical Distributions

- 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.
- Understand and use the Normal distribution as a model; find probabilities using the Normal distribution.
- Link to histograms, mean, standard deviation, points of inflection and the binomial distribution.
- Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate.

### Statistical Hypothesis Testing

- 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]; extend to correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given p-value or critical value (calculation of correlation coefficients is excluded.
- 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.
- Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.