Level 3 FSMQ: Additional Maths – introducing the enumeration section
19 March 2024
Steven Walker, Maths Subject Advisor
Following on from my recent Developing confidence with algebra blog, I now want to walk you through the enumeration section of the FSMQ, exploring the rationale for each item, and I’ve shared some ideas for teaching too.
The FSMQ is designed as both an enrichment course alongside the Maths GCSE and an acceleration programme to introduce Maths GCSE students to post-16 maths content.
The section on enumeration is all about finding the required outcomes from an exhaustive list of outcomes, without getting exhausted!
Binomial expansion
Students will be familiar with expanding (ax+b)2 and may also have looked at finding (ax+b)3 = (ax+b)2 (ax+b). The FSMQ extends this to look at (ax+b)n, finding full expansions, or more often finding the coefficient of one term xm.
There are direct links to solving probability problems such as “what is the probability of rolling exactly 7 fives with 10 regular dice?”. There are direct links to AS content in both pure and statistics.
Permutations and combinations
Questions such as “How many ways of awarding top three prizes in a group of ten people?” and “How many ways are there to select three people out of a group of ten?” are subtly different (the order matters in the former, but not in the latter). Both could be answered by listing all the outcomes, but the techniques covered in the FSMQ provide an efficient formula to find a solution whether it is 3 from 10 or 30 from 1000. This is more focused on enrichment of the GCSE curriculum, although does feature in some Further Maths options.
It is not unusual to see questions on this in the UKMT intermediate maths challenge, another common enrichment programme.
Representation and applications
In addition to the techniques, students are encouraged to use tables and diagrams to solve problems in a variety of contexts, including probability, where an efficient evaluation of different outcomes is required. Again, this is an enrichment of GCSE content. It also helps emphasise the use of diagrams as good mathematical practice to be encouraged for progression through to A Level Maths.
Teaching ideas
Students will be familiar with area grids being used for multiplying out pairs of brackets. This could be extended to modelling the multiplication of three bracketed terms in 3D by using multilink cubes to build up solid shapes. Cubes, or other manipulatives, could also be used to visualise arrangement problems before progressing to tabulating outcomes and subsequent use of formulae.
Team quiz competitions can be a good motivating activity, whilst also encouraging students to support each other’s learning. Allowing students the opportunity to be the “quiz master” and create their own puzzles to be solved by the rest of class is another way of ensuring deeper understanding.
Remember, students will often think of problems at the harder end of the challenge scale, so make sure they can answer their own puzzles before sharing across the class.
There is some interesting project style resources produced by the Centre for Innovation in Mathematics Teaching (CIMT) in collaboration with the Bletchley Park National Codes Centre and other related organisations, For example looking at how the design structure of Braille allows letters, numbers and symbols to be represented.
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 with your Exams Officer about access).
The Enumeration delivery guide has some teaching ideas and links to other useful third party resources.
The Enumeration Check In test, with fully worked solutions, has just been updated with extra questions.
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 or questions, you can email us at maths@ocr.org.uk, call us on 01223 553998 or message us on X @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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