Nuffield Mathematics teaching resources are for use in secondary and further education

AS and A-level Use of Maths (pilot) - scheme of work

For AQA information see the AQA website.

AS Use of Mathematics 

The AQA specification comprises the compulsory unit '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.

A-level 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 = ax2 + 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 = kx2 + 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 \frac{- b \pm \sqrt {b^2 - 4ac}}{2a} (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 \therefore , equals =, approximately equals \approx , inequalities, <, >, , \leq, \geq and implies \Rightarrow,
  • 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
The links below go to pages from which you can download the resources, some recently revised  (originally written for Working with algebraic and graphical techniques (WAG) and Applying Mathematics (AM)

 

Linear functions
(3 hours)

 

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 
Presentation and activity to introduce linear graphs.

Graphs of functions in Excel   
This activity shows students how to draw graphs of algebraic functions in Excel.

Interactive graphs   
Uses interactive spreadsheet graphs to introduce the shape and main features of proportional, linear, quadratic and power graphs. (Can be split into 3 separate parts.)

Graphic calculators 
Presentation introducing students to the CASIO fx-7400G PLUS calculator.

Using the CASIO fx-7400G PLUS  
Notes on how to use this calculator - includes how to draw the graph of a function, investigate how well a model fits data and how to find a model.

Graphic calculator – Equations   
A variety of equations to provide practice in using a graphic calculator. 

Car bonnet  
Students are asked to consider linear approximations to temperature data.

Match linear functions and graphs  
Twelve sets of cards, each containing a linear graph, its equation and the real situation it represents – for students to match.

Simultaneous linear equations and inequalities
(6 hours)

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   
Instructions for using the CASIO fx-7400G PLUS calculator to solve simultaneous equations.

Linear inequalities 
Presentation and activity to introduce linear inequalities.

Linear programming  
Presentation and activity in which linear inequalities are used to solve problems in real contexts. 

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Quadratic functions
(7 hours)

Draw graphs of quadratic functions:

· y = ax2 + 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 = ax2 + bx + c and
y = a(x + b)2 + c

Find maximum and minimum points of quadratics by completing the square.

 

Test run   
Students interpret a speed-time graph and fit both linear and quadratic models.  The performance data is also given in an Excel spreadsheet for comparison with models.

Model the path of a golf ball  
Students consider linear and quadratic models for the path of a golf ball.

Broadband A, B, C  
Instructions showing how to use Excel, a graphic calculator and algebra to find a quadratic model for the growth in broadband connections in recent years.

Presentation shows the algebra version.

Two on a line and three on a parabola  
Spreadsheets giving a linear or quadratic function that passes through particular points.

Factor cards  
Nearly 100 pairs of cards showing a wide variety of quadratic expressions and their factors.  Pairing will give students practice in expanding brackets or factorising.

Water flow  (assignment)  
Includes data about the velocity of water as it flows along an open channel. Data could also be used to give practice in finding a quadratic model, but is too structured to be used as a Mathematical Applications portfolio task.

Completing the square 
Presentation shows how to complete the square and use this form to sketch graphs.  Card-matching activity with 24 sets each of three cards showing a quadratic graph, the corresponding function and completed square form.

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Methods of solving equations
(applied in real situations wherever possible)
(9 hours)

Revise solving quadratic equations by:
· factorising
· using the formula  \frac{- b \pm \sqrt {b^2 - 4ac}}{2a}

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  
Instructions for using the CASIO fx-7400G PLUS calculator to solve simultaneous equations.

Graphic calculator equations   
A variety of equations to provide practice in using a graphic calculator.

Gradients of curves, maxima and minima
(5 hours)

 

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   
Students design a tin can, using algebraic and graphical techniques. Optional use of the internet.

Maximum and minimum problems  
Presentation and practice questions using a spreadsheet or graphic calculator to solve problems involving maximum and minimum values. 

 

Topic area

Content

Nuffield resource
The links below go to pages from which you can download the resources, some recently revised.

Power functions and inverse functions
(6 hours)

Draw graphs of functions of powers of x including y = kxn where n is a positive integer, y = kx-1 = \frac{k}{x}}     

y = kx-2  =   \frac{k}{x^2},

and y = kx^\frac{1}{2} = k\sqrt{x}     


Learn the general shape and position of such functions.

Develop an understanding of the nature of discontinuities in functions such as f(x) = \frac{k}{x}  and g(x) = \frac{k}{x - a}   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 axn = b.

Interactive graphs    
Uses interactive spreadsheet graphs to introduce the shape and main features of proportional, linear,quadratic and power graphs. 
(Can be split into 3 separate parts.)

Growth and decay
(7 hours)

Draw graphs of exponential functions of the form y = kamx  and y = kemx (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 ct  to model radioactive decay where the values of B and c depend on the substance).

 

Growth and decay    
Presentation that uses compound interest and radioactive decay to introduce exponential growth and decay.

Population growth  
Students use a given exponential function to model population data, then consider predictions made by the model.

Calculator table  
Students use the calculator’s table function to complete tables for population models then draw and use the corresponding graphs.

Ozone hole
Data concerning depletion of ozone levels and the increase in the area of the Antarctic ozone hole over the last twenty years.  Students investigate possible linear, quadratic and exponential models.  Optional use of spreadsheet.

Logarithmic functions
(6 hours)

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
A exp(mx + c) = k

Learn and use the laws of logarithms:

\textup{log}(ab) = \textup{log}~a + \textup{log}~b

\textup{log}(\frac{a}{b}) = \textup{log}~a + \textup{log}~b

log (a^n) = n \textup{log}~a

Convert equations involving powers to logarithmic form

(such as y = ka^m^x  gives   \textup{log }k + mx \textup{log }a))

Use natural logarithms to solve equations such as ax = b.

Climate prediction A and B  
Students use an Excel spreadsheet and /or graphic calculator to find polynomial functions to model temperature change and compare with exponential models.

Cup of coffee 
Data sheet gives the amount of caffeine remaining in the bodies of a group of people at intervals of 1 hour after they have drunk a cup of coffee or cola.  Students are asked to model the data (exponential and linear functions).

 

Topic area

Content

Nuffield resource
The links below go to pages from which you can download the resources, some recently revised.

Transformations of graphs
(7 hours)

Use the following to transform graphs of basic functions:

· translation of y = f(x)  by vector  \begin{pmatrix} 0 \\ a \end{pmatrix}  to give y = f(x) + a

· translation of y = f(x)  by vector \begin{pmatrix} -a \\ 0 \end{pmatrix} to give y = f(x + a)

· stretch of  y = f(x)  scale factor a, invariant line x = 0 to give
y = af(x)

· stretch of y = f(x)  scale factor \frac{1}{a} , invariant line y = 0 to give
y = f(ax)

Include a study of the nature of discontinuities of functions of the form f(x) = \frac{k}{h}   and   g(x) = \frac{k}{x - a}

and limiting values of functions of the form P(t) = Ae^k^t  and g(x) = K - Ae^k^x .


Consider the nature of horizontal asymptotes and discuss the way vertical asymptotes may be displayed incorrectly on graphic calculators.

Describe geometric transformations fully.

Use transformations to fit a function to data.

Sea defence wall  (assignment)  
Two versions of an assignment in which students find functions to model the outline of a sea defence wall. The first version encourages students to work independently, the second is more structured for less able students.

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
(10 hours)

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
A cos(mx + c) = k

Coughs and sneezes   
Includes data about the way in which an outbreak of the common cold spreads.  Students are asked to model the data using trigonometric and polynomial functions.

SARS A and B (assignments)  
Data set giving the number of deaths from SARS.  Students choose, draw and evaluate functions to model the data.

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)  
Students find and evaluate trigonometric functions to model how the amount of daylight varies with the day of the year.  Includes data for Adelaide, Brisbane and London.

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)
Data set giving the water depth each hour during a day.  Students choose, draw and evaluate functions to model the data. 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.

 

Topic area

Content

Nuffield resource

Linearising data
(6 hours)

Determine parameters of non-linear laws by plotting appropriate linear graphs in applications of the cases below:

· y = ax2 + b by plotting y against x2

· y = axb and y = ax using natural logarithms

Earthquakes - Log graphs   
Examples involving earthquakes and planetary motion which can be used to introduce log graphs. Ideas of experiments and other situations that can be used for practice. (Includes logs to base 10.)

Gas guzzlers
Presentation and activity in which students use a log graph to find an exponential function to model real data.

Smoke strata
Includes data about the height of smoke layers due to a fire in a tall building and sample examination question. Data could also be used to give practice in linearising data.

Earthquakes (ln version)   
Activity using natural logarithms to find an equation connecting the energy released by an earthquake and its Richter value. 

Revision
(8 hours)

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
The links below go to pages from which you can download the resources, some recently revised
(originally written for the 'Modelling calculus' FSMQ)

Introduction to calculus
(4 hours)

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  
Students explore the idea that the area under speed-time graphs can be used to find the distance travelled. 

Gradient functions

(7 hours)

Calculate the gradient at a point a on a function y = f(x) using the numerical approximation:


                gradient of tangent ~ \frac{f(a + h) - f(a)}{h}  where the interval  h is small


Improve the result by using smaller intervals.

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 \frac{f(x + h) - f(x)}{h}  to generate gradient functions leading to gradient of

y = x^n   is   \frac{dy}{dx} = nx^n^-^1 .

Differentiate polynomials and sums and differences of other powers of x using notation \frac{dy}{dx} and f'(x)

Gradients   
Slide presentation introducing differentiation and a worksheet and Excel spreadsheet which students can use to calculate gradients from increments.

Derivative matching    
Twelve sets of cards, each consisting of a polynomial function, its graph, its gradient function and the graph of the gradient function – for students to match. 

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Areas under curves
(4 hours)

Estimate the areas under graphs of functions using numerical methods (including the trapezium rule).  Include discussion of the assumptions made, consideration of over and under-estimates and improving accuracy by using a smaller interval.

Coastal erosion A    
This activity uses the context of coastal erosion to introduce the trapezium rule for estimating the area under a curve.  (Could be combined with 'Coastal erosion B'.)

Integration
(7 hours)

Find areas under curves, between x = a  and x = b  using \int \frac{b}{a}\left ( x \right )dx\left ( f\left ( x \right ) \geq 0\right )

Integration as the reverse of differentiation for xn (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   
Students use integration to estimate loss of land due to coastal erosion.  (Could be combined with 'Coastal erosion A'.)

Area under a graph     
Introduces integration using the area under velocity-time graphs. This extends the content of the 'Speed and distance' activity. Slide presentation, notes and exercises.

Mean values  
Students use area formulae and integration to find mean values of speed, share prices and water depth.  Supported by slide presentation.

Second derivatives
(6 hours)

Find second derivatives using notation: \frac{d^2y}{dx^2}and {f}''\left ( x \right )

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  
Slide presentation, examples and practice questions on sketching graphs.

Maxima and minima 
Slide presentation and practice questions using differentiation to solve maxima and minima problems.

Containers (assignment)  
Assignment in which students use differentiation to minimise surface area.

Maximising and minimising
Slide presentation including worked examples for use in discussion about maximum and minimum problems that students could investigate.

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

More rules of differentiation
(8 hours)

Differentiate

· trigonometric functions A \textup{sin}\left ( mx + c \right ), A \textup{cos}\left ( mx + c \right )   (using radians)

· exponential functions ke^m^x  (m positive or negative)

· sums and differences of functions

· products of functions.

Exponential rates of change
Introduction to differentiation of exponential functions. Students draw tangents to curves, then investigate gradient functions using a spreadsheet.

More integration

(8 hours)

Integrate

· trigonometric functions A~\textup{sin}\left ( mx + c \right ), A~\textup{cos}\left ( mx + c \right )   ,

· exponential functions ke^m^x (m positive or negative)

· x^-^1

· sums and differences of functions

Include integration by inspection such as \int e^-^5^xdx , \int \textup{sin }6x~dx

Integrate by one use of integration by parts, such as \int xe^-^5^xdx , \int x \textup{cos }4x~dx, \int~x~\textup{ln}~x~dx

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   
Data sheet shows how drug clearance after taking a painkiller can be modelled by exponential decay. Students investigate further (including the clearance of caffeine after a variety of drinks).

What’s it worth?    
Investigate a variety of suggested models for the depreciation of a car. Involves solving differential equations, drawing sketch graphs and comparison with real data.

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.

Coursework Portfolio tasks

Mathematical applications – Dynamics   
Gives a range of suggestions for coursework tasks based on the AQA Advanced Dynamics FSMQ. Includes teacher notes about marking coursework tasks.

Mathematical applications – Finances  
Gives a range of suggestions for coursework tasks based on the advanced Mathematical Principles for Personal Finance FSMQ. Includes teacher notes about marking coursework tasks.

Mathematical applications – Hypothesis testing   
Gives a range of suggestions for coursework tasks based on the advanced Hypothesis Testing FSMQ. Includes teacher notes about marking coursework tasks.

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
The links below go to pages from which you can download the resources, some recently revised.

Making sense of mathematics

(8 hours)

Read and understand mathematical work done by somebody else.  Explain steps in mathematical working, developing sub-steps 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 
Article involving the historical development of the theory of mortality used by insurance companies and questions based on the article.

Power to the people
Article explaining how electricity is distributed as alternating current using trigonometric functions to model the alternating voltages involved.

Revision (12 hours)

Revise topics.  Work through revision questions and practice Comprehension papers.

 

 

 

Page last updated on 26 January 2012