Faculty of Health, Education and Society
School of Education
PGCE Secondary Mathematics Pathway
Subject Knowledge Profile
2015-16
0
1
Using ICT in Mathematics Teaching
Trainee Teachers should understand the uses of the following
types of software and be aware of what would be an
appropriate “tool kit” for a mathematics department
Interactive White Boards – Operation of the IWB and the
creation of simple resources using SMART Board software.
Presentation Software – Being able to create mathematical
resources in PowerPoint (or similar) to enhance lessons and
use as starters. Using the equation editor
Spreadsheets – Being able to create spreadsheets that can be
used on IWBs or with a projector. Also being able to create
spreadsheets for pupils to work on individually.
Graph Plotters – For example Autograph. Being able to use
packages like this and being able to design meaningful learning
activities using them.
Graphic Calculators (e.g. TI-83, TI-84) Being able to use the
basic features of these calculators to perform calculations etc
and also to design meaningful activities using them.
Use ICT packages such as Autograph and the equation editor
to create high quality resources for use in lessons.
Internet – Be able to find and use mathematical resources on
the Internet. Use data and information, download teaching
resources and select interactive resources for use in a
classroom.
2
Familiar Competent
in
with
Date
taught /
used with
a class
Use Geometry packages, such as Geogebra or Cabri to
demonstrate geometrical principles and as a tool for pupils to
use in lessons.
Trainee Teachers should know how to use ICT to find things out
Familiar
with







Date
taught
Competent
in
Date
taught
Competent
in
Date
taught
identifying sources of information and discriminating
between them, e.g. CD-ROM, Internet, Intranet; sources on
the Internet with no editorial scrutiny
planning and putting together a search strategy, e.g.
translating enquiries expressed in ordinary language into
forms required by the system; using key words and logical
operators (AND, OR and NOT)
collecting and structuring data; interpreting data and
considering the validity, reliability and reasonableness of the
results
Trainee Teachers should know how to use ICT to try things out
& make things happen

Competent
in
Familiar
with
modelling relationships and exploring alternatives, e.g.
changing the variables in a spreadsheet simulation
predicting patterns and rules, hypothesising and checking,
e.g. predicting patters in spreadsheets or relationships with
Cabri
knowing how to give instructions and sequence actions, e.g.
following a sequence of commands when using dynamic
geometry software
defining conditions e.g. putting conditions into a
spreadsheet formula
considering cause and effect, and understanding how
feedback systems work, e.g. programming a graphics
calculator
Trainee Teachers should understand the features of ICT most
useful in teaching mathematics
speed and automatic functions

recording events which would normally take very short or
long periods of time, e.g. performing rapidly repeating
calculations in a spreadsheet to illustrate patterns of
numbers

recording events which could not be gathered within a
classroom, e.g. using a sensor to collect data over the
period of a week
3
Familiar
with
capacity and range

using the range of forms in which ICT can present
information, e.g. as numbers, formula and graphs when
using a spreadsheet

accessing a range of possible sources, e.g. using realistic
data from the Internet; communicating with experts
interactivity & provisionality

making use of the ability to make rapid changes, e.g. using
a slide bar to change a variable in a spreadsheet

saving work at different stages to keep a record of the
development of ideas, e.g. in creating a spreadsheet to
solve a problem.
GCSE Mathematics Subject Knowledge
The GCSE subject knowledge section has been adapted from the
GCSE Subject Criteria for Mathematics as published by Ofqual. The full
Subject Criteria can be found at the link below.
http://www.ofqual.gov.uk/files/2009-03-gcse-maths-subject-criteria.pdf
Note that topics in bold are for the higher tier GCSE only.
Number and algebra
Familiar
with
Add, subtract, multiply and divide any number
Order rational numbers
Use the concepts and vocabulary of factor
(divisor), multiple, common factor, highest
common factor, least common multiple, prime
number and prime factor decomposition
Use the terms square, positive and negative
square root, cube and cube root
4
Competent
in
Date taught
Use index notation for squares, cubes and
powers of 10
Use index laws for multiplication and division
of integer, fractional and negative powers
Interpret, order and calculate with numbers
written in standard index form
Understand equivalent fractions, simplifying a
fraction by cancelling all common factors
Add and subtract fractions
Use decimal notation and recognise that each
terminating decimal is a fraction
Recognise that recurring decimals are exact
fractions, and that some exact fractions are
recurring decimals
5
Familiar
with
Understand that ‘percentage’ means ‘number
of parts per 100’ and use this to compare
proportions
Use percentage, repeated proportional
change
Understand and use direct and indirect
proportion
Interpret fractions, decimals and percentages
as operators
Use ratio notation, including reduction to its
simplest form and its various links to fraction
notation
Understand and use number operations and
the relationships between them, including
inverse operations and hierarchy of operations
Use surds and π in exact calculations
Calculate upper and lower bounds
Divide a quantity in a given ratio
Approximate to specified or appropriate
degrees of accuracy including a given power
of ten, number of decimal places and
significant figures
Use calculators effectively and efficiently,
including statistical and trigonometrical
functions
Distinguish the different roles played by letter
symbols in algebra, using the correct notation
Distinguish in meaning between the words
equation, formula, identity and expression
6
Competent
in
Date taught
Familiar
with
Manipulate algebraic expressions by collecting
like terms, by multiplying a single term over a
bracket, and by taking out common factors,
multiplying two linear expressions,
factorising quadratic expressions including
the difference of two squares, and
simplifying rational expressions
Set up and solve simple equations including
simultaneous equations in two unknowns
Solve quadratic equations
Derive a formula, substitute numbers into a
formula and change the subject of a formula
Solve linear inequalities in one or two
variables, and represent the solution set on a
number line or suitable diagram
Use systematic trial and improvement to find
approximate solutions of equations where
there is no simple analytical method of solving
them
Generate terms of a sequence using term-toterm and position-to-term definitions of the
sequence
Use linear expressions to describe the nth
term of an arithmetic sequence
Use the conventions for coordinates in the
plane and plot points in all four quadrants,
including using geometric information
Recognise and plot equations that correspond
to straight-line graphs in the coordinate plane,
including finding gradients
Understand that the form y = mx + c
represents a straight line and that m is the
gradient of the line and c is the value of the
y-intercept
Understand the gradients of parallel lines
7
Competent
in
Date taught
Familiar
with
Competent
in
Date taught
Competent
in
Date taught
Find the intersection points of the graphs
of a linear and quadratic function, knowing
that these are the approximate solutions of
the corresponding simultaneous equations
representing the linear and quadratic
functions
Draw, sketch, recognise graphs of simple
cubic functions, the reciprocal function
y
1
with x  0 , the function y  k x for
x
integer values of x and simple positive
values of k, the trigonometric functions y =
sin x and y = cos x
Construct the graphs of simple loci
Construct linear, quadratic and other
functions from real-life problems and plot their
corresponding graphs
Discuss, plot and interpret graphs (which may
be non-linear) modelling real situations
Generate points and plot graphs of simple
quadratic functions, and use these to find
approximate solutions.
Geometry and measures
Familiar
with
Recall and use properties of angles at a point,
angles at a point on a straight line (including
right angles), perpendicular lines, and opposite
angles at a vertex
Understand and use the angle properties of
parallel and intersecting lines, triangles and
quadrilaterals
Calculate and use the sums of the interior and
exterior angles of polygons
8
Familiar
with
Recall the properties and definitions of special
types of quadrilateral, including square,
rectangle, parallelogram, trapezium, kite and
rhombus
Recognise reflection and rotation symmetry of
2D shapes
Understand congruence and similarity
Use Pythagoras’ theorem in 2D and 3D
Use the trigonometrical ratios and the sine
and cosine rules to solve 2D and 3D
problems
Distinguish between centre, radius, chord,
diameter, circumference, tangent, arc, sector
and segment
Understand and construct geometrical
proofs using circle theorems
Use 2D representations of 3D shapes
Describe and transform 2D shapes using
single or combined rotations, reflections,
translations, or enlargements by a positive
scale factor then use positive fractional and
negative scale factors and distinguish
properties that are preserved under particular
transformations
Use and interpret maps and scale drawings
Understand and use the effect of enlargement
for perimeter, area and volume of shapes and
solids
Interpret scales on a range of measuring
instruments and recognise the inaccuracy of
measurements
Convert measurements from one unit to
another
Make sensible estimates of a range of
measures
Understand and use bearings
9
Competent
in
Date taught
Familiar
with
Competent
in
Date taught
Competent
in
Date taught
Understand and use compound measures
Measure and draw lines and angles
Draw triangles and other 2D shapes using a
ruler and protractor
Use straight edge and a pair of compasses to
do constructions
Construct loci
Calculate perimeters and areas of shapes
made from triangles and rectangles and other
shapes
Calculate the area of a triangle using
1
ab sin C
2
Find circumferences and areas of circles
Calculate volumes of right prisms and of
shapes made from cubes and cuboids
Solve mensuration problems involving
more complex shapes and solids.
Statistics and probability
Familiar
with
Understand and use statistical problem solving
process/handling data cycle
Identify possible sources of bias
Design an experiment or survey
Design data-collection sheets distinguishing
between different types of data
Extract data from printed tables and lists
Design and use two-way tables for discrete
and grouped data
10
Familiar
with
Produce charts and diagrams for various data
types
Calculate median, mean, range, quartiles and
inter-quartile range, mode and modal class
Interpret a wide range of graphs and diagrams
and draw conclusions
Look at data to find patterns and exceptions
Recognise correlation and draw and/or use
lines of best fit by eye, understanding what
these represent
Compare distributions and make inferences
Understand and use the vocabulary of
probability and the probability scale
Understand and use estimates or measures of
probability from theoretical models (including
equally likely outcomes), or from relative
frequency
List all outcomes for single events, and for two
successive events, in a systematic way and
derive related probabilities
Identify different mutually exclusive outcomes
and know that the sum of the probabilities of
all these outcomes is 1
Know when to add or multiply two
probabilities: if A and B are mutually
exclusive, then the probability of A or B
occurring is P(A) + P(B), whereas if A and B
are independent events, the probability of A
and B occurring is P(A) × P(B)
Use tree diagrams to represent outcomes
of compound events, recognising when
events are independent
Compare experimental data and theoretical
probabilities
11
Competent
in
Date taught
Familiar
with
Understand that if they repeat an experiment,
they may - and usually will - get different
outcomes, and that increasing sample size
generally leads to better estimates of
probability and population characteristics
12
Competent
in
Date taught
Plymouth University
Secondary PGCE Programme
Mathematics Pathway
A-Level Subject Knowledge Profile
13
Introduction
The subject knowledge profile is organised in 5 sections. These are:
1. AS Core Mathematics
2. A2 Core Mathematics
3. AS Statistics
4. AS Mechanics
5. AS Decision Mathematics
Sections 1 and 2 are based on the QCA Subject Criteria. This document defines all the
mathematics that should be included in A-level Mathematics and is used by the examination
boards. Every A-level in mathematics must conform to the requirements of this QCA
document. The full version of this document can be found at:
http://www.qca.org.uk/downloads/5660_maths_revised_subject_criteria_as_a_level.pdf
Working through sections 1 and 2 will help you to ensure that you are familiar with the core
content for A-level mathematics. When you took your A-level, the core mathematics would
probably have been known as pure mathematics.
Section 3, 4 and 5 relate to the application areas of A-level mathematics. In this profile we
have included a selection of content that is typical of a first module in that application area.
As QCA do not define the content of these modules there is quite a bit of variation between
the examination boards. In order to overcome this problem, the profile includes more
material than would be found in a single AS module, but the list does cover most things that
you might be expected to teach in a first module. These sections have been produced with
help from the examination boards’ websites. (See below)
If you are interested in looking at the content of A2 application modules or further
mathematics, it is recommended that you visit one of the examination boards’ websites
using one of the links below:
http://www.aqa.org.uk/qual/gceasa/mathematics.php
http://www.ocr.org.uk/qualifications/AS_ALevelGCEMathematics.html
http://www.ocr.org.uk/qualifications/AS_ALevelGCEMathematics(MEI)-NewGCE.html
http://www.edexcel.org.uk/quals/gce/maths/adv04/maths/
14
Section 1 – AS Core Mathematics
Familiar
with topic
Proof
Construction and presentation of
mathematical arguments through
appropriate use of logical deduction and
precise statements involving correct use
of symbols and appropriate connecting
language.
Correct understanding and use of
mathematical language and grammar in
respect of terms such as ‘equals’,
‘identically equals’, ‘therefore’, ‘because’,
‘implies’, ‘is implied by’, ‘necessary’,
‘sufficient’, and notation such as , ,
 and .
Algebra and functions
Laws of indices for all rational
exponents.
Use and manipulation of surds.
Quadratic functions and their graphs.
The discriminant of a quadratic function.
Completing the square. Solution of
quadratic equations.
Simultaneous equations: analytical
solution by substitution, e.g. of one
linear and one quadratic equation.
Solution of linear and quadratic
inequalities.
Algebraic manipulation of polynomials,
including expanding brackets and
collecting like terms, factorisation and
simple algebraic division; use of the
Factor Theorem and the Remainder
Theorem;
Graphs of functions; sketching curves
defined by simple equations.
Geometrical interpretation of algebraic
solution of equations. Use of intersection
points of graphs of functions to solve
equations.
Knowledge of the effect of simple
transformations on the graph of
y  f (x) as represented by y  af (x) ,
y  f ( x)  a , y  f ( x  a ) and
y  f (ax)
Coordinate geometry in the (x,y)
plane
Equation of a straight line, including the
forms y  y1  m( x  x1 ) and
ax  by  c  0 .
Conditions for two straight lines to be
15
Competent
in topic
Date Taught
parallel or perpendicular to each other.
Co-ordinate geometry of the circle using
the equation of a circle in the form
( x  a) 2  ( y  b) 2  r 2 and including
use of the following circle properties:
(i) the angle in a semicircle is a right
angle;
(ii) the perpendicular from the centre to a
chord bisects the chord;
(iii) the perpendicularity of radius and
tangent.
Sequences and series
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 ) .
Arithmetic series, including the formula
for the sum of the first n natural
numbers.
The sum of a finite geometric series; the
sum to infinity of a convergent geometric
series, including the use of r  1 .
Binomial expansion of (1  x) n for
positive integer n. The notations n! and
n
  .
r
Trigonometry
The sine and cosine rules, and the area
of a triangle in the form
1
ab sin C .
2
Radian measure, including use for arc
length and area of sector.
Sine, cosine and tangent functions.
Their graphs, symmetries and
periodicity.
Knowledge and use of tan  
sin 
cos 
and sin 2   cos 2   1 .
Solution of simple trigonometric
equations in a given interval.
Exponentials and logarithms
Functions of the form y  a x and its
graph;
Laws of logarithms:
log a x  log a y  log a ( xy)
x
log a x  log a y  log a  
 y
k
k log a x  log a ( x )
16
The solution of equations of the form
ax  b.
Differentiation
The derivative of f(x) as the gradient of
the tangent to the graph of y = f (x) at a
point; the gradient of the tangent as a
limit; interpretation as a rate of change.
Second order derivatives.
Differentiation of x n , and related sums
and differences.
Applications of differentiation to
gradients, tangents and normals.
Applications of differentiation to maxima,
minima and stationary points.
Applications of differentiation to
increasing and decreasing functions.
Integration
Indefinite integration as the reverse of
differentiation.
Integration of x n
Approximation of area under a curve
using the trapezium rule.
Interpretation of the definite integral as
the area under a curve.
Evaluation of definite integrals.
17
Section 2 – A2 Core Mathematics
Proof
Familiar
with topic
Methods of proof, including proof by
contradiction and disproof by counterexample.
Algebra and functions
Simplification of rational expressions
including factorising and cancelling, and
algebraic division.
Definition of a function.
Domain and range of functions.
Composition of functions.
Inverse functions and their graphs.
The modulus function.
Combinations of the transformations
y  af (x) , y  f ( x)  a , y  f ( x  a )
and y  f (ax) .
Rational functions. Partial fractions
(denominators not more complicated
than repeated linear terms).
Coordinate geometry in the (x,y)
plane
Parametric equations of curves and
conversion between Cartesian and
parametric forms.
Sequences and series
Binomial series for any rational n.
Trigonometry
Knowledge of secant, cosecant and
cotangent and of arcsin, arccos and
arctan. Their relationships to sine,
cosine and tangent. Understanding of
their graphs and appropriate restricted
domains.
Knowledge and use of
sec 2   1  tan 2  and
cosec 2   1  cot 2  .
Knowledge and use of double angle
formulae; use of formulae for
sin( A  B ) , cos( A  B) and tan( A  B ) .
18
Competent
in topic
Date Taught
Use of expressions for
a cos  b sin  in the equivalent
forms of r cos(   ) and r sin(    ) .
Exponentials and logarithms
The function e x and its graph.
The function ln x and its graph.
The function ln x as the inverse of
ex .
Exponential growth and decay.
Differentiation
Differentiation of e x , ln x , sin x ,
cos x , tan x and their sums and
differences.
Differentiation using the product rule,
the quotient rule, the chain rule and
dy
1

by the use of
.
dx  dx 
 
 dy 
Differentiation of simple functions
defined implicitly or parametrically.
Formation of simple differential
equations.
Integration
Integration of e x , ln x , sin x , cos x .
Evaluation of volume of revolution.
Simple cases of integration by
substitution and integration by parts.
These methods as the reverse
processes of the chain and product
rules respectively.
Simple cases of integration using
partial fractions.
Analytical solution of simple first
order differential equations with
separable variables.
Numerical methods
Location of roots of f(x) = 0 by
considering changes of sign of f(x) in
an interval of x in which f(x) is
continuous.
19
Approximate solution of equations
using simple iterative methods,
including recurrence relations of the
form xn1  f ( xn ) .
Numerical integration of functions.
Vectors
Vectors in two and three dimensions.
Magnitude of a vector.
Algebraic operations of vector
addition and multiplication by scalars,
and their geometrical interpretations.
Position vectors. The distance
between two points.
Vector equations of lines.
The scalar product. Its use for
calculating the angle between two
lines.
20
Section 3 - AS Statistics
Familiar
with topic
Representation of Data
Histograms, stem and leaf diagrams,
box plots.
Dealing with outliers.
Numerical Measures
Measures of location − mean, median,
mode.
Measures of dispersion − variance,
standard deviation, range and
interpercentile ranges.
Use of skewness.
Choice of numerical measures.
Probability
Elementary probability; the concept of a
random event and its probability.
Addition law of probability.
Multiplication law of probability and
conditional probability.
Independent events.
Application of probability laws.
Discrete Random Variables
The concept of a discrete random
variable.
The probability function and the
cumulative distribution function for a
discrete random variable.
Mean and variance of a discrete random
variable.
The discrete uniform distribution.
Binomial Distribution
Conditions for application of a binomial
distribution.
Calculation of probabilities using
formula.
Calculation of probabilities using tables
or a graphics calculator.
Mean, variance and standard deviation
of a binomial distribution.
Hypothesis testing using the Binomial
distribution.
Normal Distribution
Properties of normal distributions.
Calculation of probabilities.
Mean, variance and standard deviation
of a normal distribution.
Estimation
Population and sample.
Unbiased estimates of a population
mean and variance.
21
Competent
in topic
Date Taught
The sampling distribution of the mean of
a random sample from a normal
distribution.
A normal distribution as an
approximation to the sampling
distribution of the mean of a large
sample from any distribution.
Confidence intervals for the mean of a
normal distribution with known variance.
Correlation and Regression
Calculation and interpretation of the
product moment correlation coefficient.
Identification of response (dependent)
and explanatory (independent) variables
in regression.
Calculation of least squares regression
lines with one explanatory variable.
Scatter diagrams and drawing a
regression line thereon.
Calculation of residuals.
22
Section 4 - AS Mechanics
Familiar
with topic
Mathematical Modelling
Use of assumptions in simplifying reality.
Mathematical analysis of models.
Interpretation and validity of models.
Refinement and extension of models.
Vectors
Use of unit vectors i and j.
Magnitude and direction of quantities
represented by a vector.
Kinematics in One and Two
Dimensions
Displacement, speed, velocity,
acceleration.
Sketching and interpreting kinematics
graphs.
Use of constant acceleration equations.
Average speed and average velocity.
Application of vectors in two dimensions
to represent position, velocity or
acceleration.
Finding position, velocity, speed and
acceleration of a particle moving in two
dimensions with constant acceleration.
Problems involving resultant velocities.
Forces
Drawing force diagrams, identifying
forces present and clearly labelling
diagrams.
Force of gravity.
Friction, limiting friction, coefficient of
friction and the relationship F  R .
Normal reaction forces. Tensions in
strings and rods, thrusts in rods.
Finding the resultant of a number of
forces acting at a point or on a particle.
Knowledge that the resultant force is
zero if a body is in equilibrium.
Newton’s Laws of Motion.
Newton’s three laws of motion.
Simple applications of the Newton’s
Laws to the motion of a particle of
constant mass.
Connected particle problems.
Momentum
Concept of momentum in one or two
dimensions.
The principle of conservation of
momentum applied to two particles.
Impulse, change in momentum and the
relationship I  Ft .
23
Competent
in topic
Date Taught
Projectiles
Motion of a particle under gravity in two
dimensions.
Calculate range, time of flight and
maximum height.
Modification of equations to take
account of the height of release.
Moments
Moment of a force.
Simple problems involving coplanar
parallel forces acting on a body and
conditions for equilibrium in such
situations.
Calculus in Mechanics
Be able to find velocities and
accelerations by using differentiation.
Be able to find displacements and
velocities using integration.
24
Section 5 - AS Decision Mathematics
Familiar
with topic
Algorithms
Correctness, finiteness and generality.
Stopping conditions.
Bubble, shuttle, shell, quicksort
algorithms.
Bin packing algorithm.
Binary search algorithm.
Graphs and Networks
Vertices, edges, edge weights, paths,
cycles, simple graphs.
Adjacency/distance matrices.
Connectedness.
Directed and undirected graphs.
Degree of a vertex, odd and even
vertices, Eulerian trails and Hamiltonian
cycles.
Planar and non-planar graphs.
Trees.
Bipartite graphs.
Spanning Tree Problems
Prim’s and Kruskal’s algorithms to find
minimum spanning trees. Relative
advantage of the two algorithms.
Matchings
Use of bipartite graphs.
Improvement of matching using an
algorithm.
Shortest Paths in Networks
Dijkstra’s algorithm.
Route Inspection Problems
Chinese Postman problem.
Travelling Salesperson Problem
Conversion of a practical problem into
the classical problem of finding a
Hamiltonian cycle.
Determination of upper bounds by
nearest neighbour algorithm.
Determination of lower bounds on route
lengths using minimum spanning trees.
Linear Programming
Formulation of problems as linear
programs.
Graphical solution of two variable
problems using ruler and vertex
methods.
The Simplex algorithm and tableau for
maximising problems.
The use and meaning of slack variables.
Critical Path Analysis
Modelling of a project by an activity
25
Competent
in topic
Date Taught
network, including the use of dummies.
Algorithm for finding the critical path.
Earliest and latest event times. Earliest
and latest start and finish times for
activities. Total float. Gantt (cascade)
charts. Scheduling.
Flows in Networks
Algorithm for finding a maximum flow.
Cuts and their capacity.
Use of max flow − min cut theorem to
verify that a flow is a maximum flow.
Multiple sources and sinks.
Simulation
Set up and use a simulation.
26