Noteful Mathematics

Vectors in 2D

• Be able to use vectors in two dimensions
• Column vectors and i,j notation
• Difference between a scalar and vector and distinguish between them when writing
• Calculate magnitude and direction of a vector
• Convert between magnitude/direction form and component form
• Calculate the modulus of a vector and interpret as magnitude
• Be able to add vectors diagrammatically
• Perform vector addition
• Multiply vectors by scalars
• Understand and be able to use position vectors
• Understand the meaning of displacement vector, component vector, resultant vector, parallel vector, equal vector and unit vector
• Calculate the distance between two points represented by position vectors
• Use vectors to solve problems in pure mathematics and in context, including forces (see forces notes)
• Use vectors to solve problems in kinematics (see kinematics notes)

Vectors in 3D

• Be able to use vectors in three dimensions
• Column vectors and i,j notation
• Extend points above to 3D (excluding the direction of a 3D vector)

• Understand and be able to use the fundamental quantities and units in the S.I. system.
• Understand that the three base quantities of length, time and mass are mutually independent.
• Understand and be able to use derived quantities and units.
• Be able to add the appropriate unit to a given quantity.

Summary

Introduction

• Understand and be able to use the language of kinematics: position, displacement, distance, distance travelled, velocity, acceleration, equation of motion.
• Understand the vector nature of dispalacement, velocity and acceleration and the scalar nature of distance travelled and speed.

Graphical Representation

• Understand, use and interpret graphs in kinematics for motion in a straight line.
• Be able to interpret displacement-time and velocity-time graphs.
• Use the interpretations of gradients and areas of these graphs as necessary.

Non-Uniform Acceleration (Calculus)

• Be able to use differentiation and integration with respect to time in 1D to solve simple problems.

Non-Uniform Acceleration (Calculus) and Vectors

• Be able to extend the application of differentiation and integration to two dimensions using vectors.
• Questions set may involve either column vector or i,j notation.

Constant Acceleration

• Understand, use and derive the formulae for constant acceleration for motion in a straight line.
• Be able to derive the formulae by integration/differentiation, using and interpreting graphs and substitution.

Constant Acceleration and Vectors

• Extend the constant acceleration formulae to motion in 2D using vectors.
• Questions set may involve either column vector or i,j notation.

Motion Under Gravity

• Be able to model motion under gravity in a vertical plane using vectors where a = -gj.

Projectiles

• Be able to model the motion of a projectile as a particle moving with constant acceleration and understand the limitation of this model.
• Use horizontal and vertical equations of motion to solve problems on the motion of projectiles.
• Find the magnitude and direction of the velocity at a given time or position.
• Find the range on a horizontal plane and the greatest height achieved.
• Derive and use the cartesian equation of the trajectory of a projectile.

Summary

Overview and Combining Forces

• Understand the concept and vector nature of a force.
• Identify the forces acting on a system and represent them in a force diagram.
• Understand and be able to use Newton's first law.
• Understand and be able to use Newton's second law for motion on a straight line of bodies of constant mass moving under the action of constant forces.
• Understand and be able to use Newton's second law in simple cases of forces given as 2D vectors.

Types of Forces and Equilibrium

• Be able to complete a diagram with the force(s) required for a given body to remain in equilibrium.
• Understand and be able to use the weight of a body to model the motion in a staight line under gravity.
• Understand the gravitational acceleration, g, and its value (may be assumed to take a constant value of 9.8ms_-2 but should be aware it is not a universal constant).
• Understand and be able to use Newton's third law.
• Use the concept that a system in which none of its components have any related motion may be modelled as a single particle.
• Understand and be able to use the concept of a normal reaction force.
• Be able to use the model of a smooth contact and understand the limitations of the model.
• Be able to use the concept of equilibrium together with 1D motion in a straight line to solve problems that involve connected particles and smooth pulleys.
• Be able to solve problems involving simple cases of equilibrium of forces on a particle in 2D using vectors, including connected particles and smooth pulleys.

Resolving Forces

• Be able to extend use of Newton's second law to situations where forces need to be resolved.
• Be able to extend use of Newton's third law to situations where forces need to be resolved (restricted to 2D).
• Understand the term resultant.
• Be able to use vector addition in solving problems involving resultants and components of forces.
• Be able to find and use perpendicular components of a force, for example to find the resultant of a system of forces or calculate the magnitude and direction of a force.
• Be able to solve problems involving the dynamics of motion for a particle moving in a plane under the action of a force or forces.

Friction, Inclined Planes, Contact Forces

• Be able to resolve forces for more advanced problems involving connected particles and smooth pulleys.
• Understand the concept of a frictional force and be able to apply it in contexts where force is given in vector or component form, or the magnitude and direction of force are given.
• Be able to represent contact forces between two rough surfaces by two components (the normal and frictional contact forces).
• Understand and be able to use the coefficient of friction and model of friction in 1D and 2D.
• Understand the concept of limiting friction.
• Understand and be able to solve problems regarding the static equilibrium of a body on a rough surface and solve problems regarding limiting equilibrium.
• Understand and be able to solve problems regarding the motion of a body on a rough surface.

Further Equilibrium

• Be able to use the principle that a particle is in equilibrium if and only if the sum of resolved parts in a given direction is zero.
• Problems may involve resolving forces including cases where it is sensible to resolve horizontally and vertically, resolve parallel and perpendicular to an inclined plane, resolve in directions to be chosen by the learner, or use a polygon of forces.

• Understand and be able to use the terms population and sample.
• Be able to use samples to make informal inferences about the population.
• Understand and be able to use sampling techniques including simple random sampling and opportunity sampling.
• Select or critique sampling technqiues in the context of solving a statistical problem.
• Understand that different samples can lead to different conclusions about the population.
• Be familiar with and be able to critique systematic, stratified, cluster and quota sampling.

• Interpret tables and diagrams for single-variable data: vertical line charts, dot plots, bar charts, stem-and-leaf diagrams, box-and-whisker plots, cumulative frequency diagrams and histograms.
• Understand, in context, the advantages and disadvantages of different statistical diagrams.
• Interpret scatter diagrams and regression lines for bivariate data.
• Recognise scatter diagram which include distinct sections of the population.
• Understand informal interpretation of correlation.
• Understand that correlation does not imply causation.
• Calculate and interpret measures of central tendency and variation including mean,median,mode,percentile,quartile,inter-quartile range, standard deviation and variance.
• Understand standard deviation is the root mean square deviation from the mean.
• Use the mean and standard deviation to compare distributions.
• Calculate mean and standard deviation from a list of data, from summary statistics or a frequency distribution using calculator statistical functions.
• Understand in the case of grouped frequency distribution the calculated mean and standard deviation are estimates.
• Recognise and be able to interpet 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.

Introduction

• Understand and be able to use mutually exclusive and independent events when calculating probabilities.
• Use appropriate diagrams to assist in the calculation of probabilities, including tree diagrams, sample space diagrams and Venn diagrams.

Set Notation

• Understand and use set notation and venn diagrams.

Conditional Probability

• Understand and be able to use conditional probabilty, including the use of tree diagrams, Venn diagrams and two-way tables.
• Understand the concept of conditial probability and calculate it from first principles in given contexts.
• Understand and be able to use the conditional probability formula.

Modelling Examples

• Be able to model with probability including critiquing asssumptions made and the likely effect of more realistic assumptions.