Subject content
This is an extract of the full specification, which you can download from this page.
Specification
The last January exams for AS and A2 were in January 2013.
The exams are now only in June due to Changes to Alevels.
 Specification for pilot from 2009 (442.9 KB)
3.1 Algebra (USE1)
Algebra (USE1)
3.1 Algebra (USE1)
Note that Algebra is not a freestanding qualification in the pilot scheme and no separate FSMQ certificate is available for the unit outside AS and A level Use of Mathematics.Before you start this qualification  You must be able to:  This includes: 
plot by hand accurate graphs of paired variable data and linear and simple quadratic functions in all four quadrants  quadratics of the type
 
recognise and predict the general shapes of graphs of direct proportion, linear and quadratic functions  quadratics of the type  
fit linear functions to model data pairs  calculating gradient and intercept for linear functions  
rearrange basic algebraic expressions by


solve basic equations by exact methods  pairs of linear simultaneous equations  
use power notation  positive and negative integers and fractions  
solve quadratic equations  by at least one of the following methods:
 
Using calculators and computers
Using calculators and computers
When carrying out calculations, you may find the use of a standard scientific calculator sufficient.
You should learn to use your calculator effectively and efficiently. This will include learning to use:
 memory facilities
 function facilities (eg )
It is important that you are also able to carry out certain calculations without using a calculator, using both written methods and 'mental' techniques.
Whenever you use a calculator you should record your working as well as the result.
You should learn to use a graphics calculator or graph plotting software (possibly a spreadsheet) on a computer to:  This includes: 
plot graphs of paired variable data  
plot graphs of functions  
use function facilities  etc . 
trace graphs (if possible)  finding intersections of functions with other functions and axes 
use zoom facilities (if possible)  finding significant features of functions such as turning points 
Fitting functions to data
Fitting functions to data  You should:  This includes: 
be familiar with the graphs of quadratic functions of the form 
 
be familiar with the graphs of functions of powers of 
 
be familiar with the graphs of trigonometric functions of the form: 
 
be familiar with the graphs of exponential functions of the form
and (m positive or negative) 
understanding ideas of exponential growth and decay  
be familiar with graphs of natural logarithmic functions of the form  understanding the logarithmic function as the inverse of the exponential function  
understand the idea of inverse functions and be able to find graphically the inverse of a function for which you have a graph  using reflection in the line  
have an understanding of how geometric transformations can be applied to basic functions. This understanding should assist you when fitting a function to data. 
 
be able to determine parameters of nonlinear laws by plotting appropriate linear graphs  Applications only in the two cases below

Interpreting models
Interpreting models  You should learn to:  This includes: 
understand
 
find and use intercepts of functions with axes and other functions to make predictions about the real situation investigated  
find local maximum and minimum points and understand in terms of the real situation their physical significance  
calculate and understand gradient at a point on the graph of a function using tangents drawn by hand  using the zoom and trace facilities of a graphics calculator or computer software if possible  
use and understand the correct units in which to measure rates of change  
interpret and understand gradients in terms of their physical significance  
identify trends of changing gradients and their significance both for functions that you know and curves drawn to fit data 
Using algebraic techniques
Using algebraic techniques  You should learn to:  This includes: 
rearrange any quadratic function into the forms

quadratics expressed in the form
 
find maximum and minimum turning points of quadratics by completing the square i.e. expressing in the form


solve polynomial equations of the form


solve trigonometric equations of the form:


solve exponential equations of the form  
understand how logarithms can be used to represent numbers 
 
know and use the laws of logarithms 
 
use logarithms to convert equations to logarithmic form  for example
gives
 
use logarithms to solve equations 

3.2 FSMQ Data Analysis (9993)
Data Analysis (9993)
You should learn:  Including:  
Statistical diagrams 


Measures of location and spread 


Bivariate data 


Normal distribution 


3.3 FSMQ Dynamics (9995)
Prior Learning
Prior learning  Candidates will need knowledge of the following. 
Trigonometry:
Algebra:
Solution of a quadratic equation by at least one of the following methods:

Formulae
Formulae  Candidates should learn the following formulae which may be required to answer questions.  
Constant Acceleration Formulae 
 
Weight  
Momentum  Momentum =  
Newton's Second Law  or Force = rate of change of momentum  
Friction 
No knowledge of calculus is required in this unit. 
Mathematical Modelling
Mathematical Modelling  
Use of assumptions in simplifying reality. 
Candidates are expected to use mathematical models to solve problems by making assumptions to create a simple model of a real situation. Candidates are expected to use experimental or investigational methods to explore how the mathematical model used relates to the actual situation. 
Mathematical analysis of models.  Modelling will include the appreciation that:
it is appropriate at times to treat relatively large moving bodies as point masses; the friction law is experimental; the force of gravity can be assumed to be constant only under certain circumstances. 
Interpretation and validity of models.  Candidates should be able to comment on the modelling assumptions made when using terms such as particle, light, inextensible string, smooth surface and motion under gravity. Candidates should be familiar with the use of the words; light, smooth, rough, inextensible, thin and uniform. 
Refinement and extension of models. 
Vectors
Vectors  
Understanding of a vector; its magnitude and direction. Addition and subtraction of two vectors. Multiplication of a vector by a scalar. Addition and subtraction of quantities using vectors. Magnitude and direction of quantities represented by a vector. Candidates may work with the i, j notation or column vectors, but questions will be set using the column vector notation. 
Kinematics in One and Two Dimensions
Kinematics in One and Two Dimensions  
Displacement, speed, velocity, acceleration.  Understanding the difference between displacement and distance. Understanding the difference between velocity and speed. 

Sketching and interpreting kinematics graphs.  Use of gradients and area under graphs to solve problems The use of Calculus is NOT required for this unit. 

Knowledge and use of constant acceleration equations. 
 
Application of vectors in two dimensions to represent position, velocity or acceleration, including the use of unit vectors i and j.  Candidates may work with the i, j notation or column vectors, but questions will be set using the column vector notation.  
Vertical motion under gravity.  
Average speed and average velocity.  
Magnitude and direction of quantities represented by a vector.  
Finding position, velocity, speed and acceleration of a particle moving in two dimensions with constant acceleration. 
The solution of problems such as when a particle is at a specified position or velocity, or finding position, velocity or acceleration at a specified time
Use of constant acceleration equations in vector form, for example, 
Forces
Forces  
Drawing force diagrams, identifying forces present and clearly labelling diagrams.  Candidates should distinguish between forces and other quantities such as velocity, that they might show on a diagram. 
Force of gravity (Newton's Universal Law not required).  The acceleration due to gravity , , will be taken as . 
Friction, limiting friction, coefficient of friction and the relationship of  
Tensions in strings and rods.  
Knowledge that the resultant force is zero if a body is in equilibrium. 
Find the unknown forces on bodies that are at rest or moving with constant velocity. Candidates will not be expected to resolve forces or find the components of forces. 
Momentum
Momentum  
Concept of momentum  Momentum as a vector in one or two dimensions. (Resolving velocities is not required.) Momentum = 
The principle of conservation of momentum applied to two particles for direct impacts in one dimension.  Knowledge of Newton's law of restitution is not required. 
Newtonâ€™s Laws of Motion
Newton's Laws of Motion  
Newton's three laws of motion.  Problems may be set in one or two dimensions and may include the use of vectors. 
Simple applications of the above to the linear motion of a particle of constant mass.  
Application of Newton's second law to particles moving with constant acceleration.  Candidates will be expected to find the acceleration of a body if the forces acting are specified, or unknown forces if the acceleration is given. 
Use of as a model for dynamic friction. 
Projectiles
Projectiles  
Motion of a particle moving freely under uniform gravity in a vertical plane.  Candidates will be expected to state and use equations of the form and . Candidates should be aware of any assumptions they are making. 
Calculate range, time of flight and maximum height.  Formulae for the range, time of flight and maximum height should not be quoted in examinations. Inclined plane and problems involving resistance will not be set. The use of the identity will not be required. Candidates may be expected to find initial speeds or angles of projection. 
Modification of equations to take account of the height of release. 
3.4 FSMQ Mathematical Principles for Personal Finance (9996)
Mathematical Principles for Personal Finance (9996)
The content of this unit covers three areas: the value of money over time, indices used to measure key financial information and tables and diagrams of financial information.
The value of money over time
The value of money varies over time. Imagine you were asked if you would like to be given a £1000 now or in ten years time. What would be your response? Even if you didn’t spend the money for ten years it would be better if you had the money now: you could invest it and it would be worth more at the end of the ten years. If, for example, you were able to invest it at 4% interest per year, after 10 years it would be worth £1480. Of course, in that period due to inflation, depending on what you  spend the £1000 on it is likely to cost you more. However, some goods come down in price over time: this is often true, for example, for computer equipment. A question you need to consider then is, what is the cost of what you might want to buy likely to be at the end of the ten year period relative to what it costs now? Understanding how money varies over time is, therefore, a very important idea to consider when making all manner of financial decisions 
What you need to learn
Financial aspect  Mathematical understanding  This includes 
The key idea of present and future values 
present value, future value, 

Interest rates: AER calculating the annual effective interest, r, rate given a nominal interest rate, i 


Calculating the future value of a present sum (using ideas of compound interest) 
where is the interest rate expressed as a decimal and is the number of time periods 
understanding as a geometric series
Use of recurrence relations* 
Calculating the present value of a future sum 
Continuous compounding 
understanding that the idea of continuous compounding leads to exponential functions ie considering the case where is the amount after years for an initial investment of when the interest is compounded times per year, and giving 

APR
(annual percentage rate) 
Assume no arrangement or exit fees. Use of the simplified version formula for APR in straightforward cases. ie
where is the APR expressed as a decimal, is the number identifying a particular instalment, is the amount of the instalment is the interval in years between the payment of the instalment and the start of the loan. 
For simple cases only: for example, (i) for a single loan repaid in full after a fixed period in whichcase where is the number of years between the advance of the loan and its repayment. (ii) for a loan repaid in a small number of instalments (eg 2,3 or 4). ie working with an equation of the form

In this case you will be expected to either
Applications to financial areas such as:


Personal Taxation  Complex calculations involving multiple rates  To include income tax, national insurance and value added tax. Capital gains tax, including the effect of indexation on the taxable gain. 
Indices used to measure key financial information
When you make a financial decision you need to have measures available that allow you to make sense of data. For example, as you found in section 1 when considering how the value of money varies over time, it is useful, if you are considering borrowing money and investigating which loan you should take that you make sure you have details of the APR  (annual percentage rate) for each possibility so that you can compare like with like. In this section you will learn how indices such as the retail price index and the FTSE 100 share index are developed so that you can quickly understand financial information such as how the cost of living is varying or how share prices are increasing or decreasing 
What you need to learn
Financial aspect  Mathematical understanding  This includes 
Understanding of an index as a ratio that describes the relative change in a variable (eg price) compared to a certain base period (eg one specific year). As applied in particular to measures of inflation such as the Retail Price Index (RPI), Consumer Price Index (CPI) and Average Earning Index.  The index at any time tells you what percentage the variable is of its respective value at the base time. The value of the index at the base time is 100.  Calculations using measures of inflation, including annual changes to pensions and tax allowances. 
Calculating contributions made by individual items to indices, eg calculating contributions made by the prices of commodities in different shops and regions to a consumer price index.  Weighted averages  for example, carrying out calculations such as finding the effective costs of a commodity which varies in price between shops. Eg the commodity costs £5 in shop A and £6 in shop B. 0.4 of customers buy the commodity from shop A whereas 0.6 buy it from shop B. The effective cost of the commodity to be used in calculating an index is given by 0.4 × £5 + 0.6 × £6 = £5.60 
Understanding the idea of calculating a composite index by combining indices using weighting eg in calculating a price index the index of each commodity multiplied by its weighting is totalled and this sum is divided by the sum of all the commodities weights. 

Calculating and using different index formulae 
For the indices below the following apply: is the price of commodity at time is the quantity of commodity at time represents the base period so for example represents the quantity of commodity at the base period 
Laspeyres index formula (weighted by quantities in the base period) 
Paasche index formula (weighted by quantities in the calculation period)  
Fischer index formula  The geometric mean of the Laspeyres index formula and the Paasche index formula 

using indices to understand change  
understanding ideas of fixed base indices and chain indices  fixed base index
chain index 
understanding that for a fixedbase index quantities at time t are compared with the base period ( t = 0 ) understanding that in a chain index comparisons are always made between subsequent points and therefore take account of changes between the start and end points. 
percentage change  
calculating average changes  using the arithmetic mean  
eg quarterly change figures from monthly figures  
understanding and working with basic principles as well as applying to areas of finance such as:

Making sense of data over time
Data you may want to use to make financial decisions is often presented as timeseries data, that is a particular measure is given every month, quarter or annually. Sometimes, particularly when the data fluctuates a lot, this may have been processed so that you can identify trends over time. For example,  share prices can fluctuate from day to day, as can the FTSE 100 share index. To understand the underlying trend over time it useful to average the data before considering this. Other issues you may need to consider include seasonal variation and cyclical patterns. 
What you need to learn
Financial aspect  Mathematical understanding  This includes 
Considering data, either primary or secondary (such as financial indices), over time. Interpreting trends. The types of data considered should relate to data or indicators likely to be met when making personal financial decisions: for example, prices of stocks and shares (including 100 share index), interest rates, exchange rates and so on.  Time series data in unprocessed form and understanding variability and how this may be random, seasonal or cyclical in nature. Representation graphically and identifying linear trends 
Inspection of data tables and graphs. Data over different time intervals, for example daily, weekly, quarterly etc. Finding linear equations to model data using gradient and intercept and algebraic substitution. 
To smooth shortterm fluctuations, timeseries data can be averaged so that longerterm trends can be identified. For example, indices such as the 100 share index can fluctuate from daytoday, but over a month or two there may be a distinct trend. Seasonal and cyclical variations may also be more easily detected by such smoothing.  Moving averages:
for data points the simple moving average, at interval m takes account of n data points Calculating successive values of the simple moving average using 
Interpretation of situations which may include seasonal and cyclical variations. 
Understanding that the simple weighted average based on a relatively large number of data points can be considered to lag behind the trend of the data  The problem of lag in moving averages can be addressed using weighting. The linear weighted moving average (over n intervals) weights the current data with weight n, the previous day with weight (n – 1) and so on. 
recognising the denominator as a triangular number with sum 
Tables and diagrams of financial information
Much basic financial information is presented in as simple a form as possible, for example using indices such as the Retail Price Index and FTSE 100 index which you learned about in section 2. Other information is often quoted in tabular or diagrammatic form giving simplified data and measures so that you can quickly compare like with like.  In this section you will learn how to make sense of a range of information presented in tables and diagram relating to personal finance. For example, you will learn to interpret information about how an investment might perform or how to compare financial products. 
You need to learn:
 to be able to extract and understand data from tables and diagrams
 to work with the data carrying out calculations using basic mathematics, such as calculating with percentages
 to interpret the original data and results of your calculations in terms of the financial situation
3.5 FSMQ Hypothesis Testing (9994)
Hypothesis Testing (9994)
You should learn:  Including:  
Binomial Distribution 





 

 

 
Sampling 



 

 
Hypothesis Testing 



 




Opinion Polls  
Food Tasting 
 
Clinical Trials 
 
NonParametric Tests 



3.6 FSMQ Decision Mathematics (9997)
Decision Mathematics (9997)
What you need to learn  Throughout your work you need to develop a critical and questioning approach to your own use of decision mathematics diagrams and techniques and also learn how these can be used to draw conclusions and summarise findings.
You will carry out work that involves you in: selecting appropriate data to use drawing appropriate network(s) carrying out an analysis using an algorithmic approach drawing conclusions and summarising findings. The key ideas that you will meet and some specific techniques that you need to be able to use are set out below. 
Using networks to model real world situations  You should be able to represent a situation so that some of the relationships are clarified by the use of appropriate networks. In drawing networks you should consider and understand: terminology such as vertices, edges, edge weights, paths and cycles connectedness directed and undirected edges and graphs You should: be able to store graphs as matrices e.g. adjacency/distance matrices understand the degree of a vertex and be aware of odd and even vertices 
Trees and spanning trees  You should understand that a tree is a connected graph with no cycles and that every connected graph contains at least one tree connecting all the vertices of the original graph.  

Shortest Paths  In developing ideas about shortest paths you will need to appreciate that problems of finding paths of minimum time and cost can both be considered to be shortest path problems  

Route Inspection Problem  In developing ideas about route inspection you will need to appreciate the connection with the classical problem of finding an Eulerian trail.  

Travelling Salesperson Problem  In developing ideas about the Travelling Salesperson Problem you will need to appreciate the connection with the classical problem of finding a Hamiltonian cycle.  

Critical Path Analysis  In developing ideas about Critical Path Analysis you will need to understand both how to construct and how to interpret activity networks with vertices representing activities.  

Mathematical modelling  You should be able to apply mathematical modelling to situation relating to the topics covered in this module. You will need to interpret results in contexts. 
Using calculators and computers  The use of a standard scientific calculator is sufficient for this unit.
However, software for the construction of networks or for the carrying out of algorithms is available commercially. 
3.7 FSMQ Calculus (9998)
Calculus (9998)
This qualification has been developed to allow you to demonstrate your ability to use
 differentiation
 integration
 differential equations
to analyse, make sense of and describe real world situations and to solve problems. You will also investigate the use of numerical methods to find gradients and evaluate integrals and compare these with analytic methods.
Before you start this qualification  You must:  This includes: 
be able to use algebraic methods to rearrange and solve linear and quadratic equations  Solution of a quadratic equation by at least one of the following methods:
 
have knowledge of basic functions and how geometric transformations can be applied to them using
 being familiar with graphs and functions of:

Using calculators and computers
Using calculators and computers
When carrying out calculations, you may find the use of a standard scientific calculator sufficient.
You should learn to use your calculator effectively and efficiently. This will include learning to use:
 memory facilities
 function facilities (e.g., )
It is important that you are also able to carry out certain calculations without using a calculator, using both written methods and 'mental' techniques.
Whenever you use a calculator you should record your working as well as the result.
Understanding and using differentiation
Understanding and using differentiation  You should learn to:  This includes: 
understand and calculate gradient at a point,
, on a function
using the numerical approximation:
where is small  understanding how to improve the calculation of gradient at a point by using a smaller interval,  
understand and interpret gradients in terms of their physical significance  
understand and use the correct units with which to measure gradients /rates of change  
sketch graphs of gradient functions 
 
identify the key features of gradient functions in terms of the gradient of the original function 
 
understand how can be used to generate a gradient function  
differentiate functions 
 
Differentiate
 
find the second derivatives of functions  using notations and  
identify the key features of a second derivative 
 
Applications of differentiation to gradients, maxima and minima and stationary points, increasing and decreasing functions 

Understanding and using integration
Understanding and using integration  You should learn to:  This includes: 
estimate areas under graphs of functions using numerical methods 
 
understand and find areas under curves, between and using ,  
understand integration as the reverse process of differentiation  
understand and determine indefinite integrals of functions 
 
Integration by inspection and by one use of integration by parts 
 
understand the idea of constant of integration and be able to calculate this in known situations  
be able to determine definite integrals for functions 

Understanding and using differential equations
Understanding and using differential equations  You should learn to:  This includes: 
find families of solutions to first order differential equations with separable variables 

3.8 Mathematical Applications (USE2)
Mathematical Applications (USE2)
Before you start this qualification  You must: Have completed or be following courses for two of the following units:
 
Portfolio requirements 
Candidates are required to produce two pieces of work in their Coursework Portfolios.
 
Assessing the Coursework Portfolio 
The two tasks must be marked separately (using the same grid) and the two marks totalled to produce one final mark for the unit.
For each of the two tasks, the candidate will be given a mark, from 0 to 7, for each of three themes:
The marking grid gives a description under each of these themes for work at various marks. 
Structuring and presenting work  Using appropriate mathematics (and technology) and working accurately  Interpreting mathematics  
01  The portfolio task has substantial omissions and is poorly presented.  There is little evidence of using mathematics accurately at the appropriate level.  There is little evidence of relating mathematics to the situation(s) investigated or there are substantial errors in interpretation. 
23
 The portfolio task has been completed with only a little advice and is well presented so that it is easy to follow.  A significant proportion of the work is beyond GCSE and is substantially correct.  The candidate has interpreted the main mathematical findings in terms of the situation(s) investigated. 
45  The candidate has worked independently and produced a portfolio task that is wellstructured and reported with clarity.  A significant proportion of the work is beyond Higher Level GCSE and is substantially correct, using relevant mathematical techniques and ICT where appropriate  The candidate has used mathematics to correctly summarise and draw conclusions about the situation(s) investigated. 
67  The candidate has shown initiative in developing their portfolio task and has structured it logically and has reported their work fluently.  The candidate has used appropriate, efficient and concise methods of working.  The candidate has considered, how their initial data, and assumptions where appropriate, affect their findings. 
Evidence to Support the Award of Marks 
Portfolios must be presented in a clear and helpful way for the moderator. An indication must also be given at the appropriate point in the work, or in accompanying information, of any further guidance given by the teacher (or other person) which has significant assessment implications.
When the assessment of the portfolio is complete, the mark awarded for each theme and the total mark awarded must be entered on the Candidate Record Form, with supporting information given in the spaces provided. 
3.9 Mathematical Comprehension (USE3)
Mathematical Comprehension (USE3)
In their study of this unit candidates will build on the mathematical knowledge, skills and understanding they develop in working towards the compulsory units Algebra (studied as part of AS Use of Mathematics) and FSMQ Calculus. This unit can be studied alongside these other units.
In particular, this assessment will concentrate on:
 reading and making sense of the mathematics of other people
 the processes involved when mathematics is used to solve problems developing clarity in the communication of mathematics
Written Examination  The comprehension paper will be one and a half hours in duration. It encourages candidates to communicate their results, working and reasoning clearly by allowing time for them to present their solutions with care.
The paper will have 45 marks allocated in the ratio 2:1 between sections A and B. The questions of Section A will be based in contexts using data from a Data Sheet that will be made available to candidates up to 14 days, and at least 7 days, prior to the examination. Candidates will be expected to familiarise themselves with the contexts outlined in the Data Sheet and should discuss these with others, including teachers. Section B will require candidates to read a brief unseen article, describing the application of mathematical principles in context, upon which a number of questions will be based. Both sections will include questions that test the mathematical principles developed in the AS unit Algebra and the FSMQ unit Calculus together with the mathematical processes and principles set out later in section 3.9. Candidates will be required to answer all questions. The use of a graphics calculator will be expected. Candidates will be expected to remember all appropriate formulae as no formulae sheet will be provided. 
Prior learning  This unit extends and develops the mathematics of the Algebra and Calculus units. You are therefore expected to be familiar with the content as outlined in their specifications. It is possible to study for this unit either alongside one or both of these units or to complete your studies for them prior to starting study of this unit.  
 
You should be able to sketch graphs you have plotted with your graphics calculator showing clearly all significant features such as intercepts with axes, turning points and asymptotes. 
What you need to learn
What you need to learn  This unit emphasises 'process' skills  the skills you need to develop to understand mathematical models of real situations. These skills are identified and described in the five learning objectives below. 
Learning Objectives
Learning Objectives  The five Learning Objectives define the skills, techniques and understanding that you need to develop during your study of this unit. 
You will learn to:  
LO1
understand how mathematics can be used to model different situations  
LO2
extend, develop and use a range of knowledge, skills and understanding in the areas of algebra, graphs and calculus when modelling situations  
LO3
appreciate that general mathematical principles may be applicable in a range of different contexts  
LO4
make sense of mathematics  
LO5
work accurately, structure mathematical arguments carefully and communicate mathematics clearly 
LO1 Understanding how mathematics can be used to model different situations
LO1 Understanding how mathematics can be used to model different situations  Throughout the course leading to this qualification you will use mathematical models of a range of real life contexts. It is important that you appreciate how these models relate to the situation and are aware of their limitations.  
You should:  Notes / Examples  
LO1a  appreciate the main stages involved in developing a mathematical model of a real situation  you may find a diagram useful in assisting your understanding of the stages of the process
for example: 
LO1b  understand that simplifying assumptions will be made when a mathematical model is developed and that these may introduce limitations into the usefulness of the model  for example:
when modelling the growth of savings in a bank account a simplifying assumption may be that interest is added twice a year. In fact interest is often calculated daily, which can have the effect of increasing or decreasing the amount of interest added to the account over a period when using numerical integration methods, such as the trapezium rule, you effectively assume that the variable follows a mathematical rule between the known data points (for example, a linear rule in the case of the trapezium rule). Consequently this can lead to an under or over estimate of the true value. 
LO1c  be able to interpret the main features of mathematical models in terms of the real situations which they model and use these models to predict what will happen in situations for which you have no data  for example:

LO1d  consider the validity of mathematical models  for example:
by considering the predicted behaviour in extreme or simple cases 
LO1e  appreciate that a general mathematical model allows you to solve a variety of related problems  for example:
the model gives the rate of radioactive decay of all radioactive substances where the coefficients and depend on the substance a situation that can be modelled by a differential equation of the form has a general solution and consequently exhibits features of exponential growth or decay. 
LO2 Extending, developing and using a range of numerical, algebraic, graphical and calculus techniques when modelling situations
LO2 Extending, developing and using a range of numerical, algebraic, graphical and calculus techniques when modelling situations  
LO2a  You should learn to:  Notes/Examples 
appreciate when algebraic, graphical, numerical or calculus techniques (or combinations of these) are most appropriate, when they are inappropriate and when possibly unsound  for example
 
establish links between different ways of understanding mathematical ideas 
 
LO2b  find solutions to equations show that particular solutions to equations are valid know when each of these methods is appropriate  you should be able to:

LO2c  use algebra to solve systems of:
 you should:

LO2d  be able to solve linear inequalities  you should be able to:

LO2e  use algebraic, graphical and calculus techniques to investigate continuous models in a wide variety of contexts such as population modelling, drug absorption, traffic flow and so on.  This includes the use of any of the functions specified in the AS unit Algebra and any of the methods and techniques specified in this unit and in the FSMQ Calculus. 
LO3 Appreciating that general mathematical principles may be applicable in a range of different contexts
LO3 Appreciating that general mathematical principles may be applicable in a range of different contexts  If you are to appreciate the power of mathematics as a tool that can be used to analyse a wide range of different situations, you need to be able to extract and appreciate general mathematical principles that underpin the work that you have been doing. 
You should learn to:  Notes/Examples  
LO3a  be able to identify the use of particular ideas in mathematics across a range of situations or contexts  you should aim to be able to transfer your use of mathematics into your work in other areas, for example you should:

LO3b  understand when mathematical methods will lead to solutions  for example:

LO3c  have an appreciation of general features of graphs of functions, including intercepts with axes and asymptotes  including understanding:

LO3d  develop your understanding of how geometric transformations can be applied to the graphs of basic functions and be able to use these when working with graphs that model real situations  including:

LO4 Making sense of mathematics
LO4 Making sense of mathematics  Making sense of mathematics is an important skill to develop since at some stage in your studies or future work you can expect that you may need to do this in a range of situations. When you come to explore the mathematics developed by someone else, particularly in workplace situations, you may find that it looks unfamiliar. Although the work will often be based on the same mathematical principles that you know, the context or setting of the mathematics and perhaps its development over time may result in it looking different. 
You should:  Notes / Examples  
LO4a  learn to explain steps in mathematical working by developing substeps if necessary  for example, you should be able to set out the mathematical working that explains the statement. 
LO4b  learn to relate mathematics in new situations to mathematics in situations with which you are familiar  for example, when calculating the possible percentage error due to rounding to the nearest year in a population model such as , you may relate this to earlier experience of calculating percentage errors in finding the area of a rectangle when you rounded lengths to the nearest centimetre 
LO4c  develop strategies to assist you in making sense of mathematics  for example, this could involve you in considering:

LO5 Working accurately, structuring mathematical arguments carefully and communicating mathematics clearly
LO5 Working accurately, structuring mathematical arguments carefully and communicating mathematics clearly  When applying mathematics to analyse a situation or solve a problem you will need to work accurately and structure your work carefully. You should aim to present your mathematics so that the logic of your thinking and argument can be followed easily by others. 
You should learn to:  Notes / Examples  
LO5a  identify possible errors in your working by:  For example: 

 

 
LO5b  decide on the appropriate degree of accuracy of values you calculate  for example:

LO5c  ensure that you  
 this includes correct use of brackets
your mathematics should read correctly (as an English sentence), e.g. when solving write: or NOT:
 
 notation you use should include:
 
LO5d  use algebraic, graphical, numerical and calculus techniques effectively to communicate your mathematics  for example:
when working with a population that may be modelled by the function you could:
