Level 3 FSMQ: Additional Maths – using numerical methods to solve complex problems
06 June 2024
Steven Walker, Maths Subject Advisor
Continuing my series of FSMQ blogs, I will take a look at the numerical methods section of the FSMQ, the rationale for its inclusion and support ideas.
The FSMQ is designed as both a GCSE enrichment course and an acceleration programme to introduce students to A Level study. This section is all about going into a bit more depth with the content introduced in the GCSE (9–1) Maths specification.
Solving equations
Students have progressed from filling in boxes to complete maths sentences at KS1 through to guessing exact solutions to equations by inspection at the end of primary KS2. KS3 then introduces the techniques to solve problems analytically by algebraic manipulation.
GCSE Maths mainly focuses on finding exact solutions to linear and quadratic equations, but there is a small section on informal trial and improvement. The FSMQ includes more formal algorithms to obtain increasingly better approximations for solutions to complex problems.
Gradients of curves
GCSE Maths introduces the idea of estimating the rate of change for a non-linear relationship by visual inspection, which is developed in the FSMQ to finding an exact gradient by calculus for simple functions. The formal introduction of the use of chords to estimate a gradient again provides students with a method for estimation when the relationship is too complex for their current knowledge of calculus techniques.
Area under a curve
Similarly, the GCSE does include finding areas under graphs, and interpret the result in contexts such as distance-time graphs, velocity-time graphs and financial graphs. Students will be familiar with using triangles and rectangles (or perhaps even trapezium), but, as with gradient, the FSMQ introduces the use of thin rectangular strips, or trapezia, for a more accurate method of estimating the area in situations where they do not yet have the required calculus techniques to solve more complex functions.
Problems set in context
There is a focus on modelling throughout the FSMQ, and this item allows questions to be set in a variety of different contexts, reaffirming the facilitating nature of maths across real-life scenarios.
Teaching ideas
In the classroom, with spreadsheets and graphing software, it is relatively easy to highlight both the techniques to find an estimated result and to apply the principles needed to improve that estimate. This practical experience will set your students up well to answer the typical questions set in the exam, where the repetitive nature of the calculations can be a bit limiting.
The link between using chords to find the gradient of a function and differential calculus allows a neat exploration of differentiation from first principles (see An Introduction to Differentiation on Nrich). The link between increasing the number of strips and reducing the strip thickness provides a nice segway into formal integration as summation.
Care is needed if using A Level Maths resources since there is a tendency to include functions involving the natural logarithm or trigonometric ratios in radians for graph problems used for numerical methods. Focus on functions involving surds or reciprocals which are straight forward to generate using graphing technology.
Supporting GCSE Maths
Numerical methods are all about approximations, so this topic sits nicely with the work on accuracy and rounding. Much of the work involves graphed functions so students will be practicing substitutions, algebraic manipulation and linking equations to graphs. Much of this work is accessible to higher tier GCSE students providing valuable additional practice, even if they don’t go on to sit the FSMQ exam at the end of year 11.
OCR support
In addition to the past papers, you will also find two practice papers on Teach Cambridge.
The numerical methods delivery guide outlines some of the common misconceptions and has some teaching ideas with links to third party resources.
The numerical methods Check In test has been updated, providing a 15-question worksheet, with fully worked solutions.
OCR sponsorship allows your centre free subscription to FSMQ resources hosted on the MEI learning platform for teachers and students.
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 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 to travel.
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