**Steven Walker, Maths Subject Advisor**

Continuing my series of blogs looking at the FSMQ content, I will now look at the calculus section, the rationale for its inclusion and discuss teaching ideas.

Introducing calculus is an obvious acceleration into A Level Maths. However, it is also useful for those students that are looking at progressing to higher education with post-16 qualifications, especially in physics, engineering or economics.

### Differentiation

The GCSE (9-1) Maths content covers the gradient of a straight line and dips into the investigation of average and instantaneous rates of change using estimation by eye to find tangents to curves. The introduction of calculus allows accurate calculations of these tangents, leading to the evaluation of local maximum and minimum points for polynomial functions. The concept of perpendicular lines is also expanded upon to cover the normal to a curve.

### Integration

Similarly, area under a curve is found by estimation in GCSE, finding the sum of compound areas made up of rectangles, triangles and/or trapeziums. The FSMQ allows more accurate calculations to find the area enclosed by a curve and defined lines, or between two curves.

### Application to kinematics

The FSMQ does not have a mechanics section. However, the inclusion of kinematics, covering both constant acceleration and variable acceleration, provides a concrete example of the use of calculus in the real world. The kinematic formulae are given in the formulae sheet since there was never any intention that students should spend time rote learning this information; rather it was hoped that the study of functions, curves and calculus would encourage a stronger understanding of the kinematic models and help avoid the misconceptions seen at AS Level where students use *suvat* regardless of the nature of the acceleration.

### Links across the specification

There is a clear link between this section and the coordinate geometry section. There is also the link with the numerical methods section allowing students to solve problems analytically if the techniques are known, but still able to find an estimate where the function is too complex. The whole section on calculus involves practicing basic algebraic substitutions (including negative numbers, fractions and surds), manipulation and solving equations, all of which provide great practice for GCSE Maths.

### Teaching ideas

Maximum volume puzzles, such as the cuboid challenge on Nrich, can be investigated by students on GCSE programmes, with the opportunity for those on the FSMQ to extend the problem with calculus.

A visit over to the science laboratory to investigate motion with varying acceleration, using ticker-timer tape, could be used to generate experimental data to plot velocity time graphs. See an example lesson idea on ticker-timers for investigating speed from the Institute of Physics. Simulations, using graphing technology (for example, see constant versus variable acceleration by Mark Willis on Geogebra), can be a complement to physical experiments.

Encouraging students to use graphing technology to plot curves and add in their own dynamic tangents (and normal), for example see these introduction to gradient and introduction to enclosed area examples on Desmos.

AS exam style questions, limited to continuous functions with integer positive indices, make good exercise questions (see ExamBuilder).

### OCR support

In addition to the past papers, you will also find two practice papers on our secure teacher site, Teach Cambridge (login required, talk to your exams officer about access).

The Calculus delivery guide has some teaching ideas and links to third party resources.

The Calculus Check In test has been updated, providing a 15-question worksheet, with fully worked solutions.

Don’t forget to join us for the termly FSMQ teacher network and look out for professional development events throughout the academic year.

### Stay connected

Share your ideas and resource links in the comment box below.

If you have any queries, you can email us at maths@ocr.org.uk, call us on 01223 553998 or message us on X (formerly Twitter) @OCR_Maths. You can also sign up to subject updates and receive information about resources and support.

### About the author

Steven originally studied engineering before completing a PGCE in secondary mathematics. He has taught secondary maths in England and overseas. Steven joined OCR in 2014 and worked on the redevelopment of OCR’s FSMQ and the A Level Mathematics suite of qualifications. Away from the office he enjoys cooking and travel.