AS Use of Maths (legacy) scheme of work
AQA AS: Use of Mathematics comprises three equally weighted assessment units:
1 FSMQ Working with algebraic and graphical techniques (compulsory)
2 either FSMQ Using and applying statistics (optional)
or FSMQ Modelling with calculus (optional)
or FSMQ Using and applying decision mathematics (optional)
3 Applying Mathematics (compulsory)
Each of these units requires 60 guided learning hours giving a total of 180 hours. There are many ways in which a course for AS Use of Mathematics could be organised. You may prefer to teach the optional unit in parallel to the compulsory units, after the compulsory units or at some other point in the course. Some teachers use the first term of a oneyear course to cover the bulk of the compulsory units, then concentrate on the optional unit during the spring term, before revising and completing the compulsory units before the examinations in the summer term.
The work scheme below combines the two compulsory units Working with Algebraic and graphical techniques and Applying Mathematics.
Before starting these units learners should be able to:
 plot by hand accurate graphs of paired variable data and linear and 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 & 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 and 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 using the formula (must be memorised).
The suggested work scheme below includes some revision of the above, as well as covering the other topics and methods needed for the two AQA compulsory units. Note that you will need to allow time for students to complete assignments for their AQA Coursework Portfolio for the 'Working with algebraic and graphical techniques' FSMQ. The AQA Coursework Portfolio requirements are listed on page 56 of the Advanced FSMQ specification and also on page 101 of the AS Use of Mathematics specification. The assignments below provide some examples of the sort of work your students could include in their portfolios, but if possible you should use work that is more relevant to their other studies or interests. Separate work schemes are available for each of the optional units. For the full AQA AS level you will need to cover the work described below and also plan another 60 hours into your course for the optional unit (statistics, calculus or decision mathematics).
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. These include:
 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 by hand and 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.
Topic area 
Content 
Nuffield resource 
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. Use error bounds to consider a range of possible functions to model data. 
Linear graphs 
Graphs of functions in Excel 

Interactive graphs 

Graphic calculators 

Using the CASIO fx7400G PLUS 

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. 
Linear inequalities 
Linear programming 
Topic area 
Content 
Nuffield resource 
Quadratic functions 
Draw graphs of quadratic functions of the form:
· y = ax^{2} + bx + c relating the shape, orientation and position of the graph to the constants, relating zeros of the function f(x) to roots of the equation f(x) = 0 and developing an appreciation of the symmetry of graphs of quadratic functions. – Fit quadratic functions to real data. Rearrange any quadratic function into the forms y = ax^{2} + bx + c and y = a(x + b)^{2} + c Find maximum and minimum points of quadratics by completing the square. 
Interactive graphs 
Broadband A, B, C Presentation shows the algebra version. 

Two on a line and three on a parabola 

Test run 

Completing the square 

Factor cards 

Methods of solving equations 
Solve 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 f(a) is of a different sign from f(b) there is at least one solution of f(x) = 0 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 


Topic area 
Content 
Nuffield resource 
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. 
Tin can 
Maximum and minimum problems 

Power functions 
Draw graphs of functions of powers of x ,
and Learn the general shape and position of such functions and investigate their symmetries. Develop an understanding of the nature of discontinuities (including the occurrence of 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


Topic area 
Content 
Nuffield resource 
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 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 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 = aln(bx) and understand the logarithmic function as the inverse of the exponential function.
Solve exponential equations of the form Learn and use the laws of logarithms: log(ab) = log a + log b, log = log a – log b , and log(a^{n}) = n log a to convert equations involving powers to logarithmic form and solve them (using both base 10 and natural logarithms).

Climate prediction A and B 
Cup of coffee 
Simulations 
Random events, probability and discrete probability distributions. Use tables and graphic calculators to find random numbers (being aware that graphic calculators generate pseudorandom numbers). Use random numbers to simulate discrete random events. Interpret simulation models being aware of the limitations due to simplifying assumptions and simulations of small numbers of occurrences. 
Queues 
Topic area 
Content 
Nuffield resource 
Transformations of graphs 
Use: · translation of y = f(x) parallel to the y axis to give y = f(x) + a · translation of y = f(x) parallel to the x axis to give y = f(x + a) · stretch of y = f(x) parallel to the y axis to give y = af(x) · stretch of y = f(x) parallel to the x axis to give y = f(ax) Include a study of the nature of discontinuities of functions of the form and and limiting values of functions of the form and .
Use geometric transformations to assist in fitting functions to real data. 
Water flow 
Sea defence wall (assignment) 

Coughs and sneezes 

Trigonometric functions 
Draw graphs of Learn the general shape and position of trigonometric functions and use the terms amplitude, frequency, wavelength, period and phase shift correctly. Fit trigonometric functions to real data and use the symmetry of trigonometric graphs to solve problems. Solve trigonometric equations of the form Asin(mx + c) = k and Acos(mx + c) + k 
SARS A and B (assignment) 
Sunrise and sunset times (assignment) 

Tides (assignment) 
Topic area 
Content 
Nuffield resource 
Linearising data 
Determine parameters of nonlinear laws (in real contexts) by plotting appropriate linear graphs, for example: · y = ax^{2} + b by plotting y against x^{2} · y = + b by plotting y against · y = ax^{3} + b by plotting y against x^{3} · y = a sin(x) + b by plotting y against sin(x)
· y = ax^{b} and y = a^{x} using base 10 or natural logarithms

Earthquakes  Log graphs 
Earthquakes 

Gas guzzlers 

Smoke strata 

Recurrence relations 
Investigate discrete models using recurrence relations in applications such as population growth (including birth and death rates) and investment. Include the use of subscript notation, finding a sequence of values using an initial value, x_{0} and a relation between x_{n + 1 }and the previous term x_{n}. Plot a graph of x_{n }against n to illustrate the results. Understand the difference between discrete and continuous models and between recurrence relations such as a_{n + 1 }= ka_{n} + b and closed forms such as = u_{n} = a + kn. 
Credit cards 
Chaotic population 
Making sense of mathematics 
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 

Page last updated on 17 February 2012