FSMQ Level 3 (legacy) Modelling with calculus scheme of work
This FSMQ requires a total of 60 guided learning hours that could be used in a variety of ways such as 2 hours per week for 30 weeks, 4 hours per week for 15 weeks, or 5 hours per week for 12 weeks. 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 you will also need to allow time for students to complete a Coursework Portfolio. The AQA Coursework Portfolio requirements are listed in the Advanced FSMQ specification for Modelling with Calculus and in 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.
Note that before starting the course students are expected to:
 be able to use algebraic methods to rearrange and solve linear and quadratic equations
 be familiar with the graphs of basic functions (powers of x, quadratic, trigonometric, exponential and logarithmic functions) and how to transform them using translations and stretches parallel to the x and y axes.
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 and recording the working as well as the result
 doing calculations without a calculator using written methods and mental techniques
 graph plotting by hand and using either computer software or a graphical calculator (including zoom and trace facilities if possible)
 checking calculations using estimation, inverse operations and different methods.
Topic area 
Content 
Nuffield resources 
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. Sketching graphs of gradient functions. Finding and interpreting area under graphs (such as speed/time, acceleration/time) using areas of triangles, rectangles and trapezia. 
Speed and distance 
Gradient functions (6 hours) 
Gradients of chords leading to the numerical approximation: Use to generate gradient data and sketch graphs of gradient functions. Use of to generate a gradient function. Gradient of is Differentiate polynomials, sums and differences, functions multiplied by a constant using notation and Include units and interpretation of gradients and rates of change. 
Gradients 
Derivative matching 
Topic area 
Content 
Nuffield resource 
Areas under curves 
Use of trapezium rule or midordinate rule (plus Simpson’s rule if you wish). Over and underestimates, improving accuracy by using a smaller interval. 
Coastal erosion A 
Integration 
Find areas under curves, between x = a and x = b using , Simple integration rules including sums, differences and multiplication by a constant including the use of correct notation and constant of integration. Definite integration 
Coastal erosion B 
Area under a graph 

Mean values 

Second derivatives 
Using gradients to identify key features  maxima, minima and points of inflexion. Finding second derivatives using notation: and Interpreting second derivatives and using them to find stationary points: maxima, minima and points of inflexion. Include fact that zero values of second derivatives can occur at maxima and minima as well as points of inflexion. 
Stationary points 
Maxima and minima 

Containers (assignment) 

Maximising and minimising 
Topic area 
Content 
Nuffield resource 
More rules of differentiation 
Differentiating 
Exponential rates of change 
More integration 
Integration of Applications of integration. 
That’s a lot of rock! (assignment) 
Differential equations 
Sketching directionfield diagrams: , where and Using integration to find families of solutions of simple differential equations. Particular solutions of simple differential equations using boundary conditions. 
Drug clearance 
What’s it worth? 

Revision (6 hours) 


Page last updated on 25 January 2012