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

# 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 The links below go to pages from which you can download the resources, some recently revised. 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  Students explore the idea that the area under speed-time graphs can be used to find the distance travelled. Gradient functions (6 hours) Gradients of chords leading to the numerical approximation: gradient of tangent $\approx&space;\frac{f(a&space;+h)&space;-&space;f(a)}{h}$ where $h$ is small. Use $\frac{f(a&space;+h)&space;-&space;f(a)}{h}$ to generate gradient data and sketch graphs of gradient functions. Use of $\frac{f(x&space;+&space;h)&space;-&space;f(x)}{h}$ to generate a gradient function. Gradient of $y&space;=&space;x^n$ is $\frac{dy}{dx}&space;=&space;nx^n^-^1$ Differentiate polynomials, sums and differences, functions multiplied by a constant using notation $\frac{dy}{dx}$ and ${f}'(x\)$ Include units and interpretation of gradients and rates of change. Gradients   Presentation introducing differentiation and a worksheet and 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 resource Areas under curves (5 hours) Use of trapezium rule or mid-ordinate rule (plus Simpson’s rule if you wish). Over and under-estimates, improving accuracy by using a smaller interval. Coastal erosion A   Activity using 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  (6 hours) Find areas under curves, between x = a and x = b  using $\int&space;\begin{matrix}&space;b&space;\\&space;a&space;\end{matrix}&space;f(x)dx$ , $\left&space;(&space;f&space;(x)&space;\geq&space;0&space;\right&space;)$ 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   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. Slide presentation, notes and exercises. This extends the content of the 'Speed and distance' activity. Mean values Students use area formulae and integration to find mean values of speed, share prices and water depth.  Supported by presentation. Second derivatives (5 hours) Using gradients to identify key features - maxima, minima and points of inflexion. Finding second derivatives using notation: $\frac&space;{d^2y}{dx^2}$ and ${f}''(x)$ 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 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 resource More rules of differentiation (6 hours) Differentiating · trigonometric functions (radians needed) · exponential functions · products · functions of functions · any other functions which are appropriate. Exponential rates of change Introduction to differentiation of exponential functions. Students draw tangents to curves, then investigate gradient functions using a spreadsheet. More integration (4 hours) Integration of · trigonometric functions Asin(mx + c), Acos(mx + c) · exponential functions kemx  (m positive or negative) · any other appropriate functions. Applications of integration. 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. Differential equations (6 hours) Sketching direction-field diagrams: $\frac&space;{dy}{dx}&space;=&space;\textup{f}(x)$ , $\frac&space;{dy}{dx}&space;=&space;\textup{f}(y)$ where $\textup{f}(y)&space;=&space;k$ and $\textup{f}(y)&space;=&space;ay^n$ Using integration to find families of solutions of simple differential equations. Particular solutions of simple differential equations using boundary conditions. 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?    Students 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 (6 hours)

Page last updated on 25 January 2012