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

# 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 one-year 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 = ax2 + bx + c) in all 4 quadrants
• recognise and predict the general shapes of graphs of direct proportion, linear & 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 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 $\frac{-&space;b&space;\pm&space;\sqrt&space;{b^2&space;-&space;4ac}}{2a}$  (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 $\therefore$ , equals =, approximately equals $\approx$, inequalities $<$, $>$, $\leq$, $\geq$and implies $\Rightarrow$,
• 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 (4 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. Use error bounds to consider a range of possible functions to model data. 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 three parts.) Graphic calculators   Presentation introducing students to the CASIO fx-7400G PLUS calculator. Using the CASIO fx-7400G PLUS Includes how to draw the graph of a function, investigate how well a model fits data and how to find a model. Car bonnet  Students are asked to consider linear approximations to temperature data. Match linear functions and graphs   12 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. Linear inequalities Slide presentation and activity to introduce linear inequalities. Linear programming  Slide presentation and activity in which linear inequalities are used to solve problems in real contexts.

 Topic area Content Nuffield resource Quadratic functions (6 hours) Draw graphs of quadratic functions of the form: · y = ax2 + bx + c  · y = (rx – s) (x + t)        · y = (x + n)2 + p 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 = ax2 + bx + c  and y = a(x + b)2 + c Find maximum and minimum points of quadratics by completing the square. Interactive graphs   Uses interactive spreadsheet graphs to introduce the shape and main features of proportional, linear, quadratic and power graphs.  (Can be split into three separate parts.) 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. 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. Completing the square    Presentation shows how to complete the square and use this form to sketch graphs.  Card-matching activity using a selection from 24 sets each of three cards showing a quadratic graph, the corresponding function, and its completed square form. 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. Methods of solving equations (applied in real situations wherever possible) (10 hours) Solve quadratic equations by: · factorising · completing the square · using the formula $\frac{-&space;b&space;\pm&space;\sqrt&space;{b^2&space;-&space;4ac}}{2a}$ 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 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   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.

 Topic area Content Nuffield resource Gradients of curves, maxima and minima (6 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. Tin can   Students design a tin can, using algebraic and graphical techniques.  Optional use of the internet. Maximum and minimum problems Slide presentation and practice questions using a spreadsheet or graphic calculator to solve problems involving maximum and minimum values. Power functions and Inverse functions (8 hours) Draw graphs of functions of powers of x including y = kxn  where n is a positive integer, $y&space;=&space;kx^-^1&space;=&space;\frac{k}{x}$,  $y&space;=&space;kx^-^2&space;=&space;\frac{k}{x^2}$ and $y&space;=&space;kx^\frac{1}{2}&space;=&space;k\sqrt&space;x$ 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 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 three separate parts.)

 Topic area Content Nuffield resource Growth and decay (6 hours) Draw graphs of exponential functions of the form y = kamx and  y = kemx (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 $\times$ 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 = aln(bx) and understand the logarithmic function as the inverse of the exponential function. Solve exponential equations of the form  Aexp(mx + c) = k Learn and use the laws of logarithms: log(ab) = log a + log b, log $\left&space;(&space;\frac{a}{b}&space;\right&space;)$ = log a – log b , and   log(an) = 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  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).

 Simulations (required for Applying Mathematics only) (8 hours) Random events, probability and discrete probability distributions. Use tables and graphic calculators to find random numbers (being aware that graphic calculators generate pseudo-random 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   Use of random numbers to simulate the queue that builds up in a newsagent’s shop.

 Topic area Content Nuffield resource Transformations of graphs (8 hours) 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 $f(x)=&space;\frac&space;{k}{x}$ and $g(x)=&space;\frac&space;{k}{x&space;-&space;a}$ and limiting values of functions of the form $\textup{P}(t)&space;=&space;Ae^k^t$ and $g(x)&space;=&space;K&space;-&space;Ae&space;^k^x$ . Consider the nature of horizontal asymptotes and discuss the way vertical asymptotes may be displayed incorrectly on graphic calculators. Use geometric transformations to assist in fitting functions to real data. Water flow Includes data about the velocity of water as it flows along an open channel and sample examination question. Data could also be used to give practice for portfolio requirements or form the basis for an assignment. 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. 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. Trigonometric functions (8 hours) Draw graphs of · y = Asin(mx + c)    · y = Acos(mx + c)  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) Data set giving the number of deaths from SARS.  Students choose, draw and evaluate functions to model the data. 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. Tides (assignment)  Data set giving the water depth each hour during a day. Students choose, draw and evaluate functions to model the data.

 Topic area Content Nuffield resource Linearising data (8 hours) Determine parameters of non-linear laws (in real contexts) by plotting appropriate linear graphs, for example: · y = ax2 + b by plotting y against x2 · y = $\frac&space;{a}{x}$ + b by plotting y against $\frac&space;{1}{x}$ · y = ax3 + b by plotting y against x3 · y = a sin(x) + b by plotting y against sin(x)    · y = axb and y = ax using base 10 or natural logarithms Earthquakes - Log graphs Examples (involving earthquakes and planetary motion) that can be used to introduce log graphs. Ideas for experiments and other situations that can be used for portfolio work. Earthquakes    Activity using natural or base 10 logarithms to find an equation connecting the energy released by an earthquake and its Richter value. 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 for portfolio requirements or form the basis for an assignment. Recurrence relations (required for Applying Mathematics only)  (6 hours) 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, x0 and a relation between xn + 1 and the previous term xn. Plot a graph of xn against n to illustrate the results.   Understand the difference between discrete and continuous models and between recurrence relations such as an + 1 = kan + b and closed forms such as = un = a + kn. Credit cards Students use recurrence relations to work out how long it takes to pay off credit card debts. Includes the use of both a graphic calculator and spreadsheet. Chaotic population       Students use a graphic calculator, then a spreadsheet, to investigate a recurrence relation which simulates population changes, some of which exhibit chaotic behaviour.

 Making sense of mathematics (required particularly for Applying Mathematics) (6 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)

Page last updated on 17 February 2012