The General Gradient Function Coursework
The lattice thermal conductivity of was studied using ab initio density functional methods. The performance of generalized gradient approximation (GGA), GGAPBE, and PBE0 exchangecorrelation functionals was compared for various electronic and phononrelated properties. The transition metal oxides such as are known to be a challenging case for pure GGA functionals, and in comparison to the GGAPBE the PBE0 hybrid functional clearly improves the description of both electronic and phononrelated properties. The most striking difference is found in the lattice thermal conductivity, where the GGA underestimates it as much as 40% in comparison to experiments, while the difference between the experiment and the PBE0 hybrid functional is only a few percent.
3 More Received 12 May 2017
 Revised 28 June 2017
DOI:https://doi.org/10.1103/PhysRevB.96.014304
©2017 American Physical Society
Physics Subject Headings (PhySH)
Note: AQA have decided to discontinue these qualifications. The last exams will be in the June 2018 series, with a final resit opportunity in 2019.
For AQA information see the AQA website.
AS Use of Mathematics
The AQA specification comprises the compulsoryunit 'Algebra' plus 2 applications units from the following list: FSMQ Data analysis ; FSMQ Hypothesis testing ; FSMQ Dynamics ; FSMQ Mathematical principles for personal finance ; FSMQ Decision mathematics
There are many ways in which a course for AS Use of Mathematics could be organised. You may prefer to teach the optional FSMQs in parallel to the compulsory unit, after the compulsory unit or at some other point in the course. Separate work schemes for each of the above units are available from this Nuffield website.
Alevel Use of Mathematics
The AQA specification comprises an AS (as described above) plus an A2 both of which must be from the pilot scheme.
A2 Use of Mathematics comprises three compulsory units: FSMQ Calculus Mathematical applications Mathematical comprehension
(A separate work scheme for learners doing FSMQ Calculus only is available from this Nuffield website.)
The Mathematical Comprehension unit builds on the mathematical knowledge, skills and understanding developed in the compulsory Algebra and Calculus units and can be studied alongside them. The assessment in a written examination will concentrate on reading and making sense of the mathematics of other people and the processes involved when mathematics is used to solve problems.
Below are work schemes for the Algebra and Calculus units which have been extended beyond that required for AS Use of Mathematics to incorporate relevant parts of the Mathematical Comprehension unit. The rest of the content of the Mathematical Comprehension unit is given at the end. You may need to change the suggested time allocations according to the time you have available and your learners' needs. Extra time will be needed for portfolio preparation for the Mathematical Applications unit.
The tasks in the coursework portfolio for the Mathematical Applications unit can be based on any of the units in AS level or the Calculus unit and should be relevant to learners' other studies or interests. They can be done alongside work for the other units or after that work is completed.
The portfolio requirements and information about its assessment are given on pages 38–40 of the GCE Use of mathematics specification. Note that students will only achieve high marks if they show initiative in developing their own portfolio tasks. Although you may be able to adapt some of the ideas in the assignments below that were written for the legacy advanced FSMQs, it would be better if students used their own ideas as far as possible.
Algebra Scheme of Work – extended for A2
Although the compulsory topics are listed separately in this work scheme, it would often be beneficial to use a variety of skills within the same piece of work. Some techniques should be introduced as soon as possible and used throughout the course.
Before starting this unit learners should be able to:
 plot by hand accurate graphs of paired variable data & linear & simple quadratic functions (including the type y = ax^{2} + bx + c) in all 4 quadrants
 recognise and predict the general shapes of graphs of direct proportion, linear and quadratic functions (including the type y = kx^{2} + c)
 fit linear functions to model data (using gradient and intercept)
 rearrange basic algebraic expressions by collecting like terms, expanding brackets & extracting common factors
 solve basic equations by exact methods including pairs of linear simultaneous equations
 use power notation (including positive and negative integers and fractions)
 solve quadratic equations by factorising and by using at least one of : graphics calculator, the formula (must be memorised), completing the square
A suggested work scheme for this unit is given below. It includes some revision of the above as well as the other topics and methods to be covered for A2.
Note that the Nuffield assignments below will not contribute to the AQA assessment of this unit which is by examination only, but it is possible that ideas from of the assignments could be adapted for the A2 Mathematical Applications unit. In this case it is essential that you change the content where necessary to enable students to work independently as far as possible.
The following techniques should be introduced as soon as possible and used throughout the course:
 using a calculator effectively and efficiently, recording the working as well as the result and deciding on an appropriate degree of accuracy
 doing calculations without a calculator using written methods and mental techniques
 showing all working by writing clear and unambiguous mathematical statements, including the correct use of brackets
 using notation correctly, including therefore , equals =, approximately equals , inequalities, , , , , and implies ,
 graph plotting using either computer software or a graphical calculator
 checking calculations using estimation, inverse operations and different methods and questioning whether solutions are reasonable/valid.
Throughout the course the emphasis should be on the use of algebraic functions to model real situations. Students need to appreciate the main stages in developing a model, understanding that simplifying assumptions are often necessary but may limit the usefulness of solutions. They should interpret the main features of models and consider the validity of any models used. They should understand that a general mathematical model can be used to solve a variety of related problems and use models to predict unknown values.
For further information and examples consult the AQA specification (page 43).
Topic area  Content  Nuffield resources 
Linear functions
 Revise the main features of graphs of direct proportional (y = mx) and linear (y = mx + c) functions. Fit such functions to real data using gradients and intercepts. Understand whether it is appropriate or not to use a particular function to model data by consideration of intercepts, long term behaviour (etc.) in real world terms. Solve linear simultaneous equations using graphical and algebraic methods.  Linear graphs 
Graphs of functions in Excel  
Interactive graphs  
Graphic calculators  
Using the CASIO fx7400G PLUS  
Graphic calculator – Equations  
Car bonnet  
Match linear functions and graphs  
Simultaneous linear equations and inequalities  Use algebraic and graphical methods to solve real problems involving linear simultaneous equations and inequalities (on graphs using dashed lines when boundaries are not included, full lines when boundaries are included and shading to indicate regions not included). Use substitution of numerical values in equations and inequalities to verify that solutions are valid.  Simultaneous equations on a graphic calculator 
Linear inequalities  
Linear programming 
Topic area  Content  Nuffield resources 
Quadratic functions  Draw graphs of quadratic functions: · y = ax^{2} + bx + c · y = (rx  s)(x  t) · y = m(x + n)^{2} + p Relate the shape, orientation and position of the graph to the constants and zeros of the function f(x) to roots of the equation f(x) = 0. Fit quadratic functions to real data. Rearrange any quadratic function into the forms y = ax^{2} + bx + c and Find maximum and minimum points of quadratics by completing the square.
 Test run 
Model the path of a golf ball  
Broadband A, B, C Presentation shows the algebra version.  
Two on a line and three on a parabola  
Factor cards  
Water flow (assignment)  
Completing the square 
Topic area  Content  Nuffield resources 
Methods of solving equations  Revise solving quadratic equations by: Use a graphic calculator to solve quadratic and other polynomial equations and simultaneous equations. · Find values of x where y = f(x) crosses the x axis to solve f(x) = 0. · Appreciate that when f(x) is continuous and is of a different sign to f(b) there is at least one solution of f(x) between a and b. · Find points of intersection of y = f(x) and y = g(x) to solve f(x) = g(x) and develop a graphical understanding of when systems of equations have one or more solutions, no unique solution or no solution. Use algebra to solve simultaneous equations where one is linear and the other quadratic. Understand that in general a system of n equations is needed to find n unknowns. Compare algebraic, graphical and numerical methods of solving equations to develop an appreciation of when a method is appropriate, inappropriate or possibly unsound.  Simultaneous equations on a graphic calculator Graphic calculator equations 
Gradients of curves, maxima and minima
 Calculate and understand gradient at a point on a graph using tangents drawn by hand (and also using zoom and trace facilities on a graphic calculator or computer if possible). Use and understand the correct units for rates of change. Interpret and understand gradients in terms of their physical significance. Identify trends of changing gradients and their significance both for known functions and curves drawn to fit data. Find local maximum and minimum points and understand their significance in terms of the real situation.  Tin can 
Maximum and minimum problems 
Topic area  Content  Nuffield resource 
Power functions and inverse functions  Draw graphs of functions of powers of x including y = kx^{n} where n is a positive integer, y = kx^{1} = y = kx^{2 }=^{ }, and
Develop an understanding of the nature of discontinuities in functions such as and and horizontal and vertical asymptotes. Fit power functions to real data. Find the graph of an inverse function using reflection in the line y = x. Solve polynomial equations of the form ax^{n} = b.  Interactive graphs 
Growth and decay  Draw graphs of exponential functions of the form y = ka^{mx} and y = ke^{mx} (m positive or negative) and understand ideas of exponential growth and decay. Fit exponential models to real data. Recognise how a general mathematical model enables the solution of a variety of problems (such as the use of a = B x c^{t} to model radioactive decay where the values of B and c depend on the substance).
 Growth and decay 
Population growth  
Calculator table  
Ozone hole  
Logarithmic functions  Draw graphs of natural logarithmic functions of the form y = a ln(bx) and understand the logarithmic function as the inverse of the exponential function. Understand how logarithms can be used to represent numbers. Solve exponential equations of the form Learn and use the laws of logarithms: Convert equations involving powers to logarithmic form (such as gives ) Use natural logarithms to solve equations such as a^{x} = b.  Climate prediction A and B 
Cup of coffee 
Topic area  Content  Nuffield resource 
Transformations of graphs  Use the following to transform graphs of basic functions: · translation of y = f(x) by vector to give y = f(x) + a · translation of y = f(x) by vector to give y = f(x + a) · stretch of y = f(x) scale factor a, invariant line x = 0 to give · stretch of y = f(x) scale factor , invariant line y = 0 to give Include a study of the nature of discontinuities of functions of the form and and limiting values of functions of the form and .
Describe geometric transformations fully. Use transformations to fit a function to data.  Sea defence wall (assignment) This could be used as a classroom activity or for homework, but should not be used in its present form as a Mathematical Applications portfolio task. 
Trigonometric functions  Draw graphs of · y = A sin(mx + c) · y = A cos(mx + c) Learn the general shape and position of trigonometric functions and use the terms amplitude, frequency and period correctly. Fit trigonometric functions to real data. Use the symmetry of trigonometric graphs to solve problems. Solve trigonometric equations of the form A sin(mx + c) = k and  Coughs and sneezes 
SARS A and B(assignments) This could be used as a classroom activity or for homework, but should not be used in its present form as a Mathematical Applications portfolio task.  
Sunrise and sunset times (assignment) This could be used as a classroom activity or for homework, but should not be used in its present form as a Mathematical Applications portfolio task.  
Tides (assignment) 
Topic area  Content  Nuffield resource 
Linearising data  Determine parameters of nonlinear laws by plotting appropriate linear graphs in applications of the cases below: · y = ax^{2} + b by plotting y against x^{2} · y = ax^{b} and y = a^{x} using natural logarithms  Earthquakes  Log graphs 
Gas guzzlers  
Smoke strata  
Earthquakes (ln version)  
Revision  Revise topics. Work through revision questions and practice Algebra papers. Discuss the Data Sheet  make up and work through questions based on it. 

Calculus Scheme of Work – extended for A2
A suggested work scheme showing topics and methods to be covered is given below but the order and time allocations can be varied to suit different groups of students.
Note that the Nuffield assignments below will not contribute to the AQA assessment of this unit which is by examination only, but it is possible that ideas from of the assignments could be adapted for the A2 Mathematical Applications unit. In this case it is essential that you change the content where necessary to enable students to work independently as far as possible.
Students should continue to:
use a calculator effectively and efficiently, including the use of memory and function facilities (recording the working as well as the result)
do calculations without a calculator using written methods and mental techniques
check calculations using estimation, inverse operations and different methods.
Topic area  Content  Nuffield resources 
Introduction to calculus  What is calculus? Brief revision of gradients of straight lines and curves (including real contexts). Positive, negative and zero gradients and their interpretation. Sketch graphs of gradient functions. Find and interpret area under graphs (eg speed/time, acceleration/time) using areas of triangles, rectangles and trapezia. Discuss the assumptions made and the limitations of results found in this way.
 Speed and distance 
Gradient functions (7 hours)  Calculate the gradient at a point a on a function y = f(x) using the numerical approximation:
Interpret gradients in terms of their physical significance and use the correct units to measure gradients/rates of change. Sketch graphs of gradient functions (including curves not given as functions as well as curves defined as functions). Identify the key features of gradient functions in terms of gradients of the original functions (including zeros of gradient functions linking to local turning points). Use to generate gradient functions leading to gradient of is . Differentiate polynomials and sums and differences of other powers of x using notation and  Gradients 
Derivative matching 
Topic area  Content  Nuffield resources 
Areas under curves  Estimate the areas under graphs of functions using numerical methods (including the trapezium rule). Include discussion of the assumptions made, consideration of over and underestimates and improving accuracy by using a smaller interval.  Coastal erosion A 
Integration  Find areas under curves, between x = a and x = b using , Integration as the reverse of differentiation for x^{n}(excludingn =– 1) and constants. Simple integration rules including sums, differences and multiplication of powers of x by a constant. Include correct notation and constant of integration. Definite integration  Coastal erosion B 
Area under a graph  
Mean values  
Second derivatives  Find second derivatives using notation: and Identify key features of a second derivative – linking positive values to increasing gradient, negative values to decreasing gradient and zeros to points of inflexion. Apply second derivatives to gradients, maxima, minima and stationary points, increasing and decreasing functions. Include examples where zero values of second derivatives occur at maximum and minimum points as well as points of inflexion.  Stationary points 
Maxima and minima  
Containers(assignment)  
Maximising and minimising 
Topic area  Content  Nuffield resources 
More rules of differentiation  Differentiate · trigonometric functions , (using radians) · exponential functions (m positive or negative) · sums and differences of functions · products of functions.  Exponential rates of change 
More integration (8 hours)  Integrate · trigonometric functions , , · exponential functions (m positive or negative) · · sums and differences of functions Include integration by inspection such as , Integrate by one use of integration by parts, such as , , Include the constant of integration and calculate it in known situation. Find definite integrals. Apply integration in real contexts.  That’s a lot of rock! (assignment) Students fit a curve to the cross section of a tunnel, then use integration and numerical methods to estimate the volume of rock removed and the time taken. This could be used as a classroom activity or for homework, but should not be used in its present form as a Mathematical Applications portfolio task. 
Differential equations (8 hours)  Use integration to find families of solutions to first order differential equations with separable variables. Find particular solutions when boundary conditions are given.  Drug clearance 
What’s it worth?  
Revision (8 hours)  Revise topics. Work through revision questions and practice Calculus papers. Discuss the Data Sheet  make up & work through questions based on it. 

Mathematical applications
AQA requires candidates to produce two pieces of work in an AQA Coursework Portfolio. The portfolio requirements and information about its assessment are given on pages 38–40 of the AQA GCE Use of mathematics specification. Note that students will only achieve high marks if they show initiative in developing their own portfolio tasks. Although some suggestions for portfolio tasks are given below, it would be better if students used their own ideas as far as possible. For more information see the AQA website.
Suggestions for Portfolio tasks
Ideally the tasks used for portfolios should be relevant to learners' other studies or interests. They could be done alongside work for the other units or after that work is completed.
The following resources give suggestions based on three of the AQA pilot advanced FSMQs.
Mathematical comprehension
As stated previously, the assessment of the AQA Mathematical Comprehension unit will include reading and making sense of the mathematics of other people and the processes involved when mathematics is used to solve problems. The following resources aim to give students practice in these aspects of the course. It is intended that they are used towards the end of the course before the Mathematical Comprehension examination.
Topic area  Content  Nuffield resources 
Making sense of mathematics(8 hours)  Read and understand mathematical work done by somebody else. Explain steps in mathematical working, developing substeps where necessary. Relate mathematics in new situations to mathematics in familiar situations. Develop strategies such as considering boundary conditions, extreme values and simple values to help make sense of mathematics. Develop alternative representations (algebraic, graphical or numerical) to help explain the mathematics.  Mortality 
Power to the people  
Revision (12 hours)  Revise topics. Work through revision questions and practice Comprehension papers. 
