###### A-Level Mathematics

#### Passgenaue Hilfe für A-Level Mathematics im Edexcel Programm.

Aktuell biete ich Unterstützung in den Edexcel Modulen Core 1 – Core 4, sowie S1 & S2.

##### Unterrichtssprache

Der Unterricht findet in Deutsch statt.

##### Unterrichtsort

Sofern es von der Entfernung passt, mache ich Hausbesuche. Alternativ kann der Unterricht in meinen Nachhilferäumen statt finden (Haltestelle Zoo oder S-Bahn Station Ostendstr.).

##### Dauer einer Unterrichteinheit

Unterrichtseinheiten am Nachmittag / Abend dauern 60 oder 90 Minuten. Für Unterricht am Vormittag oder in den Ferien sind individuelle Absprachen möglich.

##### Hilfe über Skype

Um die Kontinuität der Hilfe auch außerhalb der Ferien zu gewährleisten, unterstütze ich Schüler im Ausland über Skype.

##### Individuelles Lernprogramm

Nach zwei bis drei Stunden Unterricht mit Ihrem Kind kann ich Ihnen sagen, mit welchem zeitlichen und finanziellen Aufwand ein realistisch gesetztes Ziel zu erreichen ist.

##### Nicht im Edexcel Programm?

Für Schüler in anderen Programmen (beispielsweise OCR, AQA, MEI oder CIE) kann ich ebenfalls Hilfe anbieten. Da ich aber noch keine Schüler aus diesen Programmen hatte, würde ich den Mehraufwand für Einarbeitung und Materialbeschaffung zusätzlich berechnen.

## Bei folgenden Themen kann ich helfen:

### Core 1 Mathematics

### Algebra and functions

- Simplifying expressions by collecting like terms
- The rules of indices
- Expanding an expression
- Factorising expressions
- Factorising quadratic expressions
- The rules of indices for all rational exponents
- The use and manipulation of surds
- Rationalising the denominator of a fraction when it is surd

### Quadratic functions

- Plotting the graphs of quadratic functions
- Solving quadratic equations by factorisation
- Completing the square
- Solving quadratic equations by completing the square
- Solving quadratic equations by using the formula
- Sketching graphs of quadratic equations

### Equations and inequalities

- Solving simultaneous linear equations by elimination
- Solving simultaneous linear equations by substitution
- Using substitution when one equation is linear and the other is quadratic
- Solving lineare inequalities
- Solving quadratic inequalities

### Sketching curves

- Sketching the graph of cubic functions
- Interpreting graphs of cubic functions
- Sketching reciprocal functions
- Using the intersection points of graphs of functions to solve equations
- The effect of the transformations f(x + a), f(x – a), and f(x)+a
- The effect of the transformations f(ax) and af(x)
- Performing transformations on the sketch of curves

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

- The equation of a straight line in the form y=mx+b or ax+by+c=0
- The gradient of a straight line
- The equation of a straight line of the form y-y1=m(x-x1)
- The formula for finding the equation the equation of a straight line
- The conditions for two straight lines to be parallel or perpendicular

### Sequences and series

- Introduction to sequence
- The nth term of a sequence
- Sequence generated by a recurrence relationship
- Arithmetic sequences
- Arithmetic series
- The sum to n of an arithmetic series
- Using Sigma notation

### Differentiation

- The derivate of f(x) as the gradient of the tangent to the graph y=f(x)
- finding the formula for the gradient of x to the power of n
- Finding the gradient formula of simple functions
- The gradient formula for a function where the power of x are real numbers
- Expanding or simplifying functions to make them easier to differentiate
- Finding second order derivates
- Finding the rate of change of a function at a particular point
- Finding the equation of the tangent and normal to a curve at a point

### Integration

- Integrating x to the power of n
- Integrating simple expressions
- Using the integral sign
- Simplifying expressions before integrating
- Finding the constant of integration

### Core 2 Mathematics

### Algebra and functions

- Simplifying algebraic functions by division
- Dividing a polynomial by (x±p)
- Factorising a polynomial using the factor theorem
- Using the remainder theorem

### The sine and cosine rule

- Using the sine rule to find missing sides
- Using the sine rule to find unknown angels
- The rule and finding two solutions for a missing angle
- Using the cosine rule to find an unknown side
- Using the cosine rule to find a missing angle
- Using the sine rule, the cosine rule and Pythagora´s Theorem
- Calculating the area of a triangle using sine

### Exponentials and logarithms

- The function y= a to the power of x
- Writing expressions as a logarithm
- Calculating using logarithm to base 10
- Laws of logarithms
- Solving equations of the form a to the power of x = b
- Changing the base of logarithms

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

- The mid-point of a line
- The distance between two points on a line
- The equation of a circle

### The binomial expansion

- Pascal´s triangle
- Combinations and factorial notation
- Using ncr in the binomial expansion
- Expanding (a+bx) to the power of n using the binomial expansion

### Radian measure and its applications

- Using radians to measure angels
- The length of the arc of a circle
- The area of a sector of a circle
- The area of a segment of a circle

### Geometric sequences and series

- Geometric sequences
- Geometric progressions and the nth term
- Using geometric sequences to solve problems
- The sum of a geometric series
- The sum to infinity of a geometric series

### Graphs of trigonometric functions

- Sine, cosine and tangent functions
- The values of trigonometric functions in the four quadrants
- Exact values and surds for trigonometrical functions
- Graphs of sine x, cos x and tan x
- Simple transformations of sine x, cos x and tan x

### Differentiation

- Increasing and decreasing functions
- Stationary points, maximum, minimum and points of inflexion
- Using turning points to solve problems

### Trigonometrical identities and simple equations

- Simple trigonometrical identities
- Solving simple trigonometrical equations
- Solving equations of the form sin (nx+a), cos (nx+a) and tan (nx+a) = k
- Solving quadratic trigonometrical equations

### Integration

- Simple definite integration
- Area under a curve
- Area under a curve that gives negative values
- Area between a straight line and a curve
- The trapezium rule

### Core 3 Mathematics

### Algebraic fractions

- Simplify algebraic fractions by cancelling common factors
- Multiplying and dividing algebraic fractions
- Adding and subtracting algebraic fractions
- Dividing algebraic fractions and the remainder theorem

### Functions

- Mapping diagrams and graphs of operations
- Functions and function notation
- Range, mapping diagrams, graphs and definitions of functions
- Using composite functions
- Finding and using inverse functions

### The exponential and log functions

- Introducing exponential functions of the form y=a to the power of x
- Graphs of exponential functions and modelling using y=e to the power of x
- Using e to the power of x and the inverse of the exponential function ln x

### Numerical methods

- finding approximate roots of f(x)=0 graphically
- Using iterative and algebraic methods to find approximate roots of f(x)=0

### Transformation graphs and functions

- Sketching graphs of the modulus function y=|f(x)|
- Sketching graphs of the function y=f(|x|)
- Solving equations involving a modulus
- Applying a combination of transformations to sketch curves
- Sketching transformations and labelling the coordinates of given point

### Trigonometry

- The functions secant x, cosecant x and cotangent x
- The graphs of secant x, cosecant x and cotangent x
- Simplifying expressions, proving identities and solving equations using sec x, cosec x and cot x
- Using the identities 1+tan
^{2 }x =sec^{2 }x and 1+cot^{2 }x =cosec^{2 }x - Using inverse trigonometrical functions and theirs graphs

### Further trigonometric identities and their applications

- Using addition trigonometric formulae
- Using double angle trigonometrical formulae
- Solving equations and providing identities using double angle formulae
- Using the form a cos x + b sin x in solving trigonometrical problems
- The factor formulae

### Differentiation

- Differentiating using the chain rule
- Differentiating using the product rule
- Differentiating using the quotient rule
- Differentiating the exponential function
- Finding the differential of the logarithmic function
- Differentiating sin x
- Differentiating cos x
- Differentiating tan x
- Differentiating further trigonometrical functions
- Differentiating functions formed by combining trigonometrical, exponential, logarithmic and polynomial functions

### Core 4 Mathematics

### Partial fractions

- Adding and subtracting algebraic fractions
- Partial fractions with two linear factors in the denominator
- Partial fractions with three or more linear factors in the denominator
- Partial fractions with repeated linear factors in the denominator
- Improper fractions into partial fractions

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

- Parametric equations used to define the coordinates of a point
- Using parametric equations in coordinate geometry
- Converting parametric equations into Cartesian equations
- Finding the area under a curve given by parametric euations

### The binomial expansion

- The binomial expansion for a positive integral index
- Using the binomial expansion to expand (a + bx)
^{n} - Using partial fractions with the binomial expansion

### Differentiation

- Differentiating functions given parametrically
- Differentiating relations which are implicit
- Differentiating rate of change
- Simple differential equations

### Vectors

- Vector definition and vector diagrams
- Vector arithmetic and the unit vector
- Using vectors to describe points in 2 or 3 dimensions
- Cartesian components of a vector in 2 dimensions
- Cartesian components of a vector in 3 dimensions
- Extending 2 dimensional vector result to 3 dimensions
- The scalar product of two vectors
- The vector equation of a straight line
- Intersecting straight line vector equation
- The angle between two straight lines

### Integration

- Integrating standard functions
- Integrating using the reverse chain rule
- Using trigonometric identities in integration
- Using partial fractions to integrate expressions
- Using standard patterns to integrate expressions
- Integration by substitution
- Integration by parts
- Numerical Integration
- Integration to find areas and volumes
- Using integration to solve differential equations
- Differential equations in context

### Mathematics S1

### Mathematical models in probability and statistic

- What are mathematical models?
- Designing a mathematical model

### Representation and summary of data - location

- Classification of variables
- Classification of quantitative variables into continuous or discrete variables
- Frequency tables and grouping data
- Measures of location
- Which is the correct measure of location
- Calculating measures of location from frequency distribution table
- Calculating measures of location for grouped data
- Coding

### Representation and summary of data - measure of dispersion

- Range and quartiles
- Percentiles
- Standard deviation and variance for discrete data
- Variance and standard for a frequency table and grouped frequency
- coding

### Representing of data

- Stem and leaf diagrams
- Outliers
- Box plot
- Using box plots to compare sets of data
- Representing data on a histrogram
- Skewness
- Comparing the distributions of data sets

### Probability

- Probability vocabulary
- Solving probability problems by drawing Venn diagrams
- Using formulae to solve problems
- Solving problems using conditional probability
- Conditional probabilities on tree diagrams
- Mutually exclusive and independent events
- Solving probability problems

### Correlation

- Scatter diagrams
- Calculating measures for the variability of bivariate data
- Using the product moment correlation coefficient, r
- Determining the strength of a linear relationship
- The limits of the product moment correlation coefficient
- Coding

### Regression

- Connecting variables
- Independent and dependent variables
- Minimising the sum of the squares of the residuals
- Coding
- The regression equation

### Discrete and random variables

- Discrete random variables and random variables
- Specifying a variable
- Sum of probabilities
- Finding the probability that X is less than or greater than a value, or between two values
- The cumulative distribution function
- The mean or expected value of a discrete random variable
- The expected value for X to the power of 2
- The variance of a random variable
- The expected value and variance of a function of X
- Finding the mean and standard deviation of the original data given the mean and standard deviation of the coded data
- A model for the probability distribution

### The normal distribution

- Use tables to find probabilities in the standard normal distribution Z
- Use tables to find the value of z given a probability
- transform any normal distribution into Z and use tables
- Use normal tables to find µ and sigma
- Use the normal distribution to answer questions in context

### Mathematics S2

### Binomial distribution

- Factorial notation and the number of arrangements
- The binomial distribution
- Conditions for a binomial distribution
- Binomial cumulative distribution function
- Mean and variance of a binomial distribution
- Harder problems

### Poisson distribution

- Exponential series and the Poisson distribution
- The mean and variance of a Poisson distribution
- The Poisson cumulative distribution function
- Conditions for a Poisson distribution
- the Poisson approximation of a binomial distribution
- Choosing a binomial or Poisson distribution

### Continuous random variables

- Continuous random variables and their probability density function
- The cumulative distribution function
- The mean and variance of a probability density function
- Finding the mode, median and quartiles of a continuous random variable

### Continuous uniform distribution

- Continuous uniform/rectangular distribution
- Properties of a continuous uniform distribution
- Choosing the right model

### Normal approximations

- Continuity correction
- Approximating a binomial distribution by a normal distribution
- Approximating a Poisson distribution by a normal distribution
- Choosing the appropriate approximation

### Population and samples

- Populations, censuses and samples
- The advantages and disadvantages of taking a census
- Sampling
- The advantages and disadvantages of sampling
- Simple random sampling
- The concept of a statistic
- The sampling distribution of a statistic

### Hypothesis testing

- The concept of an hypothesis test
- The significance level of an hypothesis test
- One- and two-tailed test
- Hypothesis tests for the proportion p of a binomial distribution and hypothesis tests for the mean lambda of a Poisson distribution
- Hypothesis tests for the proportion p of a binomial distribution and hypothesis tests for the rate lambda of a Poisson distribution using critical regions