Teaching Level 2 Further Maths alongside GCSE Maths
I’ve taught Level 2 Further Maths for several years now in two different schools. I think it’s a great qualification to offer alongside the GCSE for those students aspiring towards A-level Maths or just wanting a bit more of a challenge.
What I especially like about the qualification is that virtually all of it can be taught alongside the normal GCSE (with the slight exception of two main topics) and the questions are really good and challenging.
In my experience, this hasn’t been too hard a sell to students or staff.
For staff, having the content sprinkled throughout the schemes of work makes it manageable and supplies really good extension material easily. There isn’t that much extra to learn how to teach, so it’s a nice professional development opportunity without adding too much to the workload.
For students, there are several selling points. It can really help them get up to the higher grades in the normal GCSE. Seeing more challenging questions throughout years 10 and 11 helps to stretch the higher attaining and keep them well engaged.
Whilst I teach the material to everyone in the higher sets, I’ve always made entrance to the final exam completely optional. Students have a lot of exams to sit at the end of year 11. I don’t want to add to potential anxiety by requiring students to sit two more exams.
However, a big selling point for the L2FM is that there isn’t an awful lot of extra content to learn, it’s often just harder questions. If students are aiming for grade 7 or above in the GCSE, then they should be able to grade relatively comfortably in L2FM. We tend to get around 75% of our top sets sitting the exam and this has increased the longer we’ve run it.
Largely, these students really enjoy doing maths so they often love having more challenging questions. They especially enjoy the new content on matrices and calculus as it feels much more A-level-like to them. The fact that matrices is only in A-level Further Maths also makes them feel especially smug!
How to start?
Most of the topics are extensions of the existing GCSE content so can fit nicely within your existing scheme of work. The matrices and calculus sections are the only new topics and these can be taught across three or four lessons for each. Everything else can be seen as extension work.
We’ve spread the teaching across years 10 and 11 (we have a two-year KS4) to make it more manageable. For most of the content, we found an extra lesson or two at the end of an appropriate section worked really well. For example, when we were expanding double and triple brackets, we added an extra lesson or two on expanding 4, 5, 6 and n brackets using Pascal’s triangle. When we did the equation of a circle, we added an additional lesson about circles with centre not at (0,0).
We left the two new topics of matrices and calculus to the end of the course. Each topic took three or four hours in total but you could do some of this by flipped learning, for example.
Because you’re teaching it to high-attaining students, they tend to get through the normal GCSE material (especially the grade 5 and 6 work) very quickly so we’ve found there’s plenty of time. They also like having a separate further maths book or section in their folder. It’s worth doing this so all specific content can be found more easily at the end of year 11.
Where should the new content go?
This table shows what I believe to be more specific, ‘additional’ content that is in L2FM. Of course, this is my view so check the specification and teaching guidance to ensure you have covered everything.
There is more content than this but much of it mirrors the GCSE very closely, eg functions or algebraic fractions. The references to the GCSE specification help you link your existing scheme of work.
There isn’t a lot of additional content, it’s just preparing your students to attempt harder, more challenging questions on a more regular basis.
Additional Further Maths content
Related GCSE Maths content (specification reference is in bold)
Number
The product rule for counting
**N5** The further maths version is slightly more formal and you could introduce factorial or N choose R for extension here.
Manipulating Surds and rationalising the denominator
**N8** The further maths content uses more complicated denominators such as
Algebra
Domain and range of functions and inverses
**A7** Functions are referred to as having inputs and outputs in the GCSE so it’s just a matter of formalising the language.
Expand using Pascal’s triangle or binomial
**A4** Fits in well with teaching the expansion of triple brackets and linking it to Pascal’s triangle.
Use the factor theorem for rational values
**A18** Link to factorising and solving quadratics then show that substituting in the roots make it equal 0.
Drawing and sketching functions including exponentials like
**A12** L2FM extends to include
Solving triple simultaneous equations
**A19** Extension to solving two simultaneous equations. Top sets have often already done this before so good chance for revision and extension.
Solving equations involving indices
**N7 and A4** Link to calculating with indices and index laws then extend.
Limiting values of sequences as n tends to infinity
**A23 to A25** Add to the sequences section, especially geometric. Top-set students will have seen nth term many times before so it’s a chance to revise and add extension.
Geometry
Equations of circles with centre (a,b)
**A13 and A16** It’s a nice opportunity to use a graphing package, eg desmos.com or geogebra.org, or link the equations of circles to graph transformations.
Use trig identities such as
and
**G20 to G22** Useful to link this to exact values and get students to derive tan values from sine divided by cosine.
Solve trigonometric equations in given intervals (ie between 0 and 360 degrees)
**A12** and **G20 to G22** Solving is an extension of solving sinx=0.5, for example, but over a given range. Link this to the graphs of sine and cosine and show them the solutions.
Calculus
Differentiate functions in the form
where n is an integer (including negatives)
Links to **A15, A16, R14, R15** especially calculating or estimating gradients. The rest is all new content.
Find maxima and minima and use the second derivative to determine which
New content to teach.
Matrices
Multiplication of matrices
Links to **G25** Vectors and 2x1 matrices. New content to teach.
Identity matrix
New content to teach.
Transformations using 2x2 matrices and combinations
New content to teach.
Resources
For teaching, AQA has produced some really good worksheets on all the topics which are great for teaching and final revision. Even if you decide to not enter students, they contain really good extension questions.
Corbett Maths has an excellent selection of worksheets and videos to go with all the topics. Mr Barton Maths has collated some excellent resources too. Be aware that some of these may relate to the previous specification, but there’s enough similarity that they’re still useful.
There are also some websites that all have useful sections within them, though some require subscriptions: