FSMQ Level 3 (pilot) Calculus scheme of work
Before starting this A2 level FSMQ students are expected to:
 Be able to use algebraic methods to rearrange and solve linear and quadratic equations including use of at least one of: graphic calculator, the formula (must be memorised) and completing the square. (Factorisation may be used where appropriate.)
 Be familiar with the graphs of basic functions (powers of x, quadratic, trigonometric, exponential and logarithmic functions) and how geometric transformations can be applied to them using transformations by the vectors and and stretches of scale factor a with the invariant lines x = 0 and y = 0.
 Be able to do graph plotting by hand and using either computer software or a graphical calculator (including zoom and trace facilities)
Students who have completed the AS level Use of Mathematics Algebra unit will have covered these topics.
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 AQA assessment of this FSMQ is by examination only.
The following techniques should be introduced as soon as possible and used throughout the course:
 Using a calculator effectively and efficiently, including the use of memory and function facilities (recording the working as well as the result)
 Doing calculations without a calculator using written methods and mental techniques
 Checking calculations using estimation, inverse operations, and different methods.
Topic area 
Content 
Nuffield resources 
Introduction to calculus 
What is calculus? Brief revision of gradients of straight lines and curves (including real contexts). Positive, negative and zero gradients and their interpretation. Sketch graphs of gradient functions. Find and interpret area under graphs (eg speed/time, acceleration/time) using areas of triangles, rectangles and trapezia. 
Speed and distance 
Topic area 
Content 
Nuffield resources 
Gradient functions 
Calculate the gradient at a point a on a function y = f(x) using the numerical approximation:
gradient of tangent where the interval h is small. Interpret gradients in terms of their physical significance and use the correct units to measure gradients/rates of change. Sketch graphs of gradient functions (including curves not given as functions as well as curves defined as functions). Identify the key features of gradient functions in terms of gradients of the original functions (including zeros of gradient functions linking to local turning points). Use to generate gradient functions leading to gradient of being Differentiate polynomials and sums and differences of other powers of x using notation and . 
Gradients 
Derivative matching 

Areas under curves 
Estimate the areas under graphs of functions using numerical methods (including the trapezium rule). Consider over and underestimates and improving accuracy by using a smaller interval. 
Coastal erosion A 
Integration 
Find areas under curves, between x = a and x = b using , Integration as the reverse of differentiation for x^{n} (excluding n = – 1) and constants. Simple integration rules including sums, differences and multiplication of powers of x by a constant. Include correct notation and constant of integration. Definite integration 
Coastal erosion B 
Area under a graph 

Mean values 
Topic area 
Content 
Nuffield resources 
Second derivatives 
Find second derivatives using notation: and Identify key features of a second derivative – linking positive values to increasing gradient, negative values to decreasing gradient and zeros to points of inflexion. Apply second derivatives to gradients, maxima, minima and stationary points, increasing and decreasing functions. Include examples where zero values of second derivatives occur at maximum and minimum points as well as points of inflexion. 
Stationary points 
Maxima and minima 

Containers 

Maximising and minimising 

More rules of differentiation 
Differentiate: 
Exponential rates of change 
More integration 
Integrate: Include integration by inspection such as , Integrate by one use of integration by parts, such as , , Include the constant of integration and calculate it in known situation. Find definite integrals. Apply integration in real contexts. 
That’s a lot of rock! 
Differential equations 
Use integration to find families of solutions to first order differential equations with separable variables. Find particular solutions when boundary conditions are given. 
Drug clearance 
What's it worth? 

Revision (8 hours) 
Revise topics. Work through revision questions and practice papers. Discuss the data Sheet  make up and work through questions based on it. 

Page last updated on 25 January 2012