Further Mathematics

Year 12

Unit 1 - Complex Number 1

Key questions/concepts

  • Imaginary and complex numbers

  • Complex conjugates

  • Real and complex roots of polynomial equations

  • Argand diagrams

  • Radians as a measure of angle

  • Different forms of complex numbers (Cartesian and modulus-argument) and how to convert

  • Loci of circles and half lines

  • Arithmetic with complex numbers

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Modulus Length of the line segment drawn from the origin of the complex plane to the point representing the complex number in the Argand diagram
Argument Angle that the line segment drawn from the origin of the complex plane to the point representing the complex number in the Argand diagram makes with the real axis
Imaginary number A number that is expressed in terms of the square root of a negative number
Complex number A number made up of real and imaginary parts
Radian A measure of angle such that 2π radians is equal to a full turn (360 degrees)

 

 

 

 

 

 

 

Wider reading

Electronics and electrical engineering

BBC podcasts: 

Unit 2 - Matrices 1 

Key questions/concepts

  • Add, subtract and multiply conformable matrices

  • Calculate transpose matrices

  • Factorise matrices by a scalar

  • 2D transformations by matrices

  • Identify invariant points and/or invariant lines of matrix transformations

  • Determinant of a 2x2 matrix

  • Inverse of a non-singluar 2x2 matrix

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Conformable A matrix is conformable if its dimensions are suitable for defining a mathematical operation
Element Each number in the matrix array 
Transpose A matrix that has been flipped along its diagonal so that the rows become columns and the columns become rows
Identity matrix A matrix with the entries on the diagonal from the top left to bottom right are 1 and the other elements are 0
Determinant A property of a matrix calculated by row and column operations and essential for identifying singular matrices and calcualting inverse matrices. 

 

 

 

 

 

 

 

Wider reading

Game design and programming

Unit 3 - Algebra and Series

Key questions/concepts

  • Mathematical induction to prove statements

  • Roots of polynomials up to quartic equations

  • Understand the relationships between roots and coefficients of polynomials up to quartic equations

  • Use linear transformations of roots to form new equations

  • Sum of series formulae and notation

  • Method of differences for summing series

  • General Maclaurin series expansion using the formulae book

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Root The value at which a given function is equal to 0
Series The addition of all terms in a sequence
Transformation The process of converting an expression or object into another
Coefficient A multiplier placed in front of a variable
Polynomial An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).

 

 

 

 

 

 

Wider reading

Car manufacturing and graphic design

The Music of the Primes (Marcus Du Sautoy) 

Podcasts: 

Unit 4 - Curve Sketching 1

Key questions/concepts

  • Graphs of linear and quadratic functions

  • Finding asymptotes, turning points, intercepts and limits

  • Solving rational inequalities graphically

  • Conversion between polar and Cartesian coordinates

  • Sketching equations given as polar functions

  • Graphs of conic sections - parabolas, elipses and hyperbolae

  • Equations and graphs of linear transformations of conic sections

  • Know the exponential form of hyperbolic functions

  • Sketching graphs of hyperbolic functions

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Intercept The value at which a give function crosses the x axes
Asymptote A line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
Conic section These functions can be found by cuting into a cone. Conic sections incolude: circles, parabolas, ellipses and hyperboae
Pole When working in polar coordinates the pole is the point (0,0)
Parabola A symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity follows a curve of this formshape.

 

 

 

 

 

 

 

Wider reading

Aeronautical engineering
Cartography
Analysis of performance in sport
Birch and Swinerton-Dyer conjecture 

Unit 5 - Integration 2

Key questions/concepts

  • Definition of an improper integral 

  • Evaluating improper integral using limits

  • Differentiation of inverse trigonometric functions

  • Differentiation of hyperbolic functions

  • Integration by splitting expressions into partial fractions

  • Use of reduction formulae 

  • Integration of polar functions to find areas and arc lengths

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Improper integral A definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration
Reduction formulae An integral which is of the same type as the given integral but of a lower degree. The repeated application of the reduction formula helps us to evaluate the given integral.
Archimedean spiral The locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity
Integrate A method of adding or summing up the parts to find the whole. It is a reverse process of differentiation
Differentiate The process of finding the rate of change of a function

 

 

 

 

 

 

 

Wider reading

 

Unit 6 - Vectors 2

Key questions/concepts

  • Calculation of vector product

  • Use of vector product to find angles and areas

  • Conversion between equations of lines (Cartesian, vector, vector product)

  • Defining a plane in terms of a vector product

  • Calculation of the distance between two lines or a line and a plane

  • Finding the angle between a line and a plane

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Scalar product An algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number
Vector product A binary operation on two vectors in three-dimensional space
Normal A line that is perpendicular to a tangent
Plane A flat surface on which a straight line joining any two points on it would wholly lie
Orthagonal Perpendicular

 

 

 

 

 

 

 

Wider reading

Cauchy-Schwatz inequality

Unit 7 - Discrete and Continuous Random Variables

Key questions/concepts

  • Calculate key statistics including expectation from discrete distributions
  • Use of the Poisson distribution to calculate probability and understand the conditions for use
  • Calculate key statistics of continuous distributions
  • Derive probability generating functions for random variables
  • Use transformations of probability distribution functions to calculate statistics

Timeframe

September - October

Key vocabulary

Word/phrase Definition
Discrete Data that can only take certain values
Continuous Data that can take any value
Random variable A variable whose values depend on outcomes of a random phenomenon
Statistic A fact or piece of data obtained from a study of a large quantity of numerical data
Parameter A parameter is used to describe the entire population being studied and is inherent to that population

 

 

 

 

 

 

 

Wider reading

Oxford University podcasts:

Unit 8 - Hypothesis Testing and Contingency Tables

Key questions/concepts

  • Identify type I and type II errors in context
  • Calculate the probability of making a type I error
  • Null and alternative hypotheses of a chi squared test
  • Perform a chi squared test for assosciation and interpret the result in context
  • Confidence intervals and analysis in context
  • Identify critical values or the p-value of a test statistic

Timeframe

September - October

Key vocabulary

Word/phrase Definition
Type I error Also known as a false positive and occurs when a researcher incorrectly rejects a true null hypothesis
Type II error Also known as a false negative and occurs when a researcher fails to reject a null hypothesis which is really false
Test statistic A random variable that is calculated from sample data and used in a hypothesis test
Degrees of freedom The number of independent values or quantities which can be assigned to a statistical distribution
Contingency table A table showing the distribution of one variable in rows and another in columns, used to study the correlation between the two variables

 

 

 

 

 

 

Wider reading

Sally Clark case: 

More or less website and podcasts: 

Unit 9 - Forces and Energy

Key questions/concepts

  • Formulae for gravitational potential and kinetic energy 

  • Modelling with work, energy and power formulae

  • Evaluate the consistency of a formula using dimensional analysis

  • Hooke's law for elastic springs and strings

  • Model simple scenarios and conclude in context including considering limitations to models

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Work Work is done when a force acts upon an object to cause a displacement
Energy The capacity for doing work
Power He amount of energy transferred or converted per unit time
Friction The resistance to motion of one object moving relative to another
Traction The force used to generate motion between a body and a tangential surface

 

 

 

 

 

Wider reading

Women in math podcast: 

Unit 10 - Momentum

Key questions/concepts

  • Identify type I and type II errors in context

  • Calculate the probability of making a type I error

  • Null and alternative hypotheses of a chi squared test

  • Perform a chi squared test for assosciation and interpret the result in context

  • Confidence intervals and analysis in context

  • Identify critical values or the p-value of a test statistic

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Momentum A measure of the quantity of motion of a moving body, due to its mass and velocity
Impulse Change in momentum of a particle in the direction of motion
Elasticity Ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed
Coefficient of restitution Ratio of the final to initial relative velocity between two objects after they collide
Coalesce Come together to form one mass or whole

 

 

 

 

 

 

Wider reading

Unit 11 - Circular Motion 1

Key questions/concepts

  • Kinematics of horizontal circular motion

  • Angular velocity 

  • Application of Newton's third law to horizontal circular motion

  • Limiting factors in horizontal circular motion modelling

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Horizontal Parallel to the plane of the horizon
Radial velocity Rate of change of the distance between the object and the point
Tangential speed The speed of an object taken at a tangent to the path of the object
Centripetal force A force that acts on a body moving in a circular path and is directed towards the centre around which the body is moving
Angular velocity The rate of change of angular position of a rotating body

 

 

 

 

 

 

Wider reading

 

Year 13

Unit 1 - Complex Number 2

Key questions/concepts

  • Complex numbers in exponential form 

  • Use exponential form of complex numbers to solve trigonometric equations

  • De Moivre's theorem

  • Calculate roots of complex numbers including roots of unity

  • Use of De Moivre's theorem to solve trigonometric equations

  • Use of multiplication by a complex number to produce a rotation

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Expansion A method for calculating a function that cannot be expressed by just elementary operators
Root  of unity Any complex number that yields 1 when raised to some positive integer power n
Exponential Form  Way of representing repeated multiplications of the same number
Root Where a function equals zero
Trigonometric Equation An equation that includes a trigonometric function

 

 

 

 

 

Wider reading

Electronics and electrical engineering
Riemann Hypothesis (Clay Mathematics Institute's Millennium Prize Problems)
Fractal geometry

Podcast about imaginary numbers: 

Unit 2 - Matrices 2

Key questions/concepts

  • Determinant of a 3x3 matrix

  • Inverse of a 3x3 matrix

  • Solve systems of equations using matrices

  • Eigenvalues and eigenvectors of matrices

  • Understand the relationship between eigenvalues, eigenvectors and invariant lines

  • Matrix diagonalisation and its use in calculating with powers of matrices

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Minor The determinant of the square matrix formed by deleting one row and one column from some larger square matrix
Cofactor The number you get when you remove the column and row of a designated element in a matrix
Eigenvalue The special set of scalar values that is associated with the set of linear equations most probably in the matrix equations
Eigenvector A nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it
Sheaf of planes The set of all planes that have the same common line

 

 

 

 

 

 

Wider reading

Game design and programming
Eigenfaces (face classification and human identification)

Unit 3 - Series

Key questions/concepts

  • Sum series by splitting into partial fractions 
  • Maclaurin's series with differentiation for series expansion
  • L'Hopital's rule and when to use it
  • Evaluate limits using L'Hopital's rule

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Summation The addition of a sequence of any kind of numbers
Convergent When a sequence or series tends to a limit
Divergent When a sequence or series doesn't tend to a limit
Limit The value the sequence approaches as the number of terms goes to infinity
Indeterminate Not definitively or precisely determined

 

 

 

 

 

Wider reading

Car manufacturing and graphic design

Unit 4 - Curve Sketching 2

Key questions/concepts

  • Sketching reciprocal and modulus graphs
  • Solving problems with reciprocal and modulus functions
  • Transformation of graphs of conic sections
  • Sketching graphs of inverse hyperbolic functions
  • Sketching graphs of reciprocal hyperbolic functions
  • Sketching graphs with oblique asymptotes

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Modulus The non-negative value of x without regard to its sign
Hyperbolic function Analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle
Inverse function A function that undoes the action of the another function
Reciprocal function 1 divided by the function
Oblique Slanting

 

 

 

 

 

Wider reading

Aeronautical engineering
catenary curves
Cartography
Analysis of performance in sport

Unit 5 - Intergration 2

Key questions/concepts

  • Definition of an improper integral 
  • Evaluating improper integral using limits
  • Differentiation of inverse trigonometric functions
  • Differentiation of hyperbolic functions
  • Integration by splitting expressions into partial fractions
  • Use of reduction formulae 
  • Integration of polar functions to find areas and arc lengths

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Improper integral A definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration
Reduction formulae An integral which is of the same type as the given integral but of a lower degree. The repeated application of the reduction formula helps us to evaluate the given integral.
Archimedean spiral The locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity
Integrate A method of adding or summing up the parts to find the whole. It is a reverse process of differentiation
Differentiate The process of finding the rate of change of a function

 

 

 

 

 

 

 

Wider reading

Unit 6 - Vectors 2

Key questions/concepts

  • Calculation of vector product
  • Use of vector product to find angles and areas
  • Conversion between equations of lines (Cartesian, vector, vector product)
  • Defining a plane in terms of a vector product
  • Calculation of the distance between two lines or a line and a plane
  • Finding the angle between a line and a plane

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Scalar product An algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number
Vector product A binary operation on two vectors in three-dimensional space
Normal A line that is perpendicular to a tangent
Plane A flat surface on which a straight line joining any two points on it would wholly lie
Orthagonal perpendicular

 

 

 

 

 

 

Wider reading

Cauchy-Schwatz inequality

 

Unit 7 - Differential Equations

Key questions/concepts

  • Strategies for solving first order differential equations
  • Strategies for solving homogenous second order differential equations
  • Strategies for solving non-homogenous second order differential equations
  • Identifying simple harmonic motion
  • Using differential equations to model simple and damped harmonic motion
  • Identifying the degree of damping that occurs with damped harmonic motion
  • Using coupled differential equations to model predator-prey and other bi-variable scenarios

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Differential equation An equation that relates one or more functions and their derivatives
homogenous equation All terms depend linearly on the dependent variable or its derivatives
General solution Solution to an equation that still involves unknown constants
Particular solution Solution to an equation that has taken into account boundary conditions and has no unknown constants remaining
Damped harmonic motion Vibrating systems for which the amplitude of vibration decreases over time

 

 

 

 

 

 

Wider reading

Navier Stokes Equation

Unit 8 - Numerical Methods

Key questions/concepts

  • Simpson's rule for estimating area under a curve
  • Euler's method for estimating area under a curve
  • Improved Euler's method for estimating area under a curve
  • Relative error of an estimate

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Ordinate The y-coordinate of a point
Iteration The repeated application of a function or process in which the output of each step is used as the input for the next iteration
Algorithm A description of a set of steps that can be used to solve a mathematical computation
Estimate An approximation to the solution
Relative error The ratio of the absolute error of a measurement to the measurement being taken

 

 

 

 

 

 

Wider reading

Unit 9 - Radom Processes

Key questions/concepts

  • Calculating key statistics of mixed random variables
  • Rectangular and exponential probability models to calculate probabilities
  • Understand and use the relationship between the probability distribution function and the cummulative distribution function
  • Derive the formulae for mean and variance of rectangular and exponential random variables

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Cummulative Increasing or increased in quantity, degree, or force by successive additions
Dummy variable A variable used as a place holder when the primary variable is utilised in several ways throughout the calculation. The dummy variable enables the calculation to occur without misunderstanding of the multiple use. 
Probability density The density of probability rather than the probability mass
Variance A measure of how spread out numbers are
Mean A calculated "central" value of a set of numbers

 

 

 

 

 

 

Wider reading

Unit 10 - Hypothesis testing and the t-test

Key questions/concepts

  • Probability of making a type I or type II error
  • Power of a hypothesis test
  • T-test to evaluate a possible population mean for a Normal distribution
  • Identify when a t-test is the appropriate hypothesis test to use. 
  • Evaluate conclusion of t-test in context

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
hypothesis
A statement that might be true, which might then be tested
Normal distribution A probability function that describes continuous data that forms a bell shaped curve with mean approximately equal to the mean and the median. 
confidence interval A range of values for an unknown parameter
unbiased estimator An estimator is unbiased if it produces parameter estimates that are on average correct
Sample A selection taken from a larger group

 

 

 

 

 

 

Wider reading

Unit 11 - Circular Motion 2

Key questions/concepts

  • Analysis of motion in a vertical circle
  • Analysis of motion in a conical pendulum
  • Application of energy equations to vertical circular motion

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Conical pendulum
A weight (or bob) fixed on the end of a string or rod suspended from a pivot. Its construction is similar to an ordinary pendulum; however, instead of swinging back and forth, the bob of a conical pendulum moves at a constant speed in a circle with the string (or rod) 
Radial components The aspects of motion moving radially in an outward direction from point O
Transverse components The aspects of motion moving tangentially to the direction of movement
Energy equation The mathematical formulation of the law of conservation of energy.
Inextensible As a fixed length- it is impossible to stretch

 

 

 

 

 

 

 

Wider reading

Unit 12 - Centres of mass and stability

Key questions/concepts

  • Centre of mass for point particles, uniform laminas and planes
  • Centre of mass for a solid
  • Calculating moments for couples
  • Identifying equilibrium and stability of solids

Timeframe

  • September - October

Key vocabulary

Word/phrase Definition
Lamina

A two-dimensional planar closed surface which has a mass and a surface density
Torque A measure of the force that can cause an object to rotate about an axis
Pivot The point about which rotation occurs
Couple Two parallel forces that are equal in magnitude, opposite in sense and do not share a line of action.
Centroid Geometric center of a plane figure is the arithmetic mean position of all the points in the figure

 

 

 

 

 

 

 

Wider reading

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