29 July 2015

New GCSE: Sequences

I've been busy getting my head around the additional sequences content in the new 9 - 1 Maths GCSE and thought it would be helpful to share my findings. Quite a lot has been added to the higher tier. Previously students covered all the GCSE sequences content at Key Stage 3 so there was nothing new to teach at Key Stage 4. But now we need to include a few weeks' worth of sequences lessons in our GCSE schemes of work.

We've gone from this (from the current Pearson Edexcel Specification - GCSE 2012 Maths A):
...to this (from the new Pearson Edexcel GCSE (9-1) in Mathematics):
All the linear sequences content remains (see my previous post about methods for finding an nth term). The terms arithmetic progression and geometric progression are now used in both the specification and sample assessment materials - terms I normally don't introduce to students until Year 12.

Let's take a look at the new content in detail.
1. "Recognise and use sequences of triangular, square and cube numbers"
There's no specific mention of triangular numbers in the current Key Stage 3 and 4 curricula. I've always covered triangular numbers as an enrichment topic at Key Stage 3 anyway, but now students could be examined on this. It's likely that square, cube and triangular numbers will come up in Venn Diagram questions like in this activity from Flash Maths:
If you're looking for resources for square, cube and triangular number sequences then there's plenty available, like this worksheet from Channel 4 Learning and this lovely activity from Great Maths Teaching Ideas. Stacy Brookes (@Stacy_Maths) has written a very helpful post Special Numbers featuring many more resources.

2. "Recognise and use quadratic sequences... Deduce expressions to calculate the nth term of quadratic sequences."
It wasn't long ago that quadratic sequences were removed from GCSE - now they're back. I'm aware of three approaches for teaching students how to find the nth term of a quadratic sequence:

Pattern Spotting
In a fairly simple sequence, students might be able to make comparisons to known sequences (eg square numbers) to derive the nth term. This is a nice introduction to the topic - I have a worksheet on it here.

Finding the Linear Sequence in the Remainder
For more complicated quadratic sequences, I've always used the method outlined below (source: mathshelper.co.uk).
Finding the Coefficients
Huge thanks to Matt Dunbar (@MathsDunbar) of Trinity Maths for introducing me to a new method. In this method you find the value of the coefficients in the nth term an2 + bn + c, where a is half the second difference, c is the zeroth term and b can be calculated using these values.
I've always wondered why we halve the second difference when finding the nth term of a quadratic sequence. Matt Dunbar kindly explained it to me (I feel like I should have worked this out myself!):

There's no shortage of resources for quadratic sequences - big thanks to Dawn (@mrsdenyer) for gathering a large collection here.

3. "Recognise and use geometric sequences"
Foundation and higher tier students will be required to recognise simple geometric progressions - for higher tier students these sequences may involve surds (for example 1, √2, 2, 2√2, …). The sample assessment materials provide two examples of geometric progression questions. This question appeared on AQA Paper 2 for both foundation and higher tier:
This question was from Edexcel Higher Paper 3:
We're fortunate that there's a decent amount of resources available for geometric sequences. Key Stage 5 textbooks and worksheets cover this topic, and a number of resource makers have already started producing activities including Don Steward and Ed Southall. Note that, unlike at A level, students won't have to find the nth term of geometric sequences.

4. "Recognise and use Fibonacci type sequences"
I'm pleased to see Fibonacci on the new GCSE specifications because, like triangular numbers, it should definitely feature in every child's mathematics education. Until now it's always been something we've taught for enrichment.

I was a bit confused by the meaning of 'Fibonacci type sequences' because I thought that there was only one Fibonacci sequence and by definition it is 1, 1, 2, 3, 5, 8, 13, ... . Thankfully Edexcel's sample assessment materials clarify this for us - this question appears on both their foundation and higher sample papers:
5. Recurrence Relations
The OCR specification mentions the use of subscript notation, and we are shown a recurrence relation. The AQA specification also mentions recursive sequences.
I'm a bit worried about how my GCSE students will cope with recurrence relations because my Year 12s often struggle with the notation.

In the OCR sample assessment materials we have this question:
There's not many recurrence relation resources suitable for GCSE students, but A level textbooks may be useful and there's some great stuff from CIMT MEP. In fact the CIMT chapter on sequences covers pretty much everything we need for sequences in the new GCSE. Also, do have a look at AQA's resources for this topic - their recurrence relations activities are great.

6. "And other sequences...!"
I don't like this vagueness in the specification but I believe this refers to, for example, sequences like this:
where the numerator and denominator are separate linear sequences.

So, as you can see, sequences has suddenly become a much bigger topic at GCSE. We already have some resources, but I'd love to share more so please let me know about anything you create. I'm starting to build a collection in my algebra library.

I hope this post has helped you figure out how this topic has changed under the new GCSE specifications. Let me know what you think.

1. This has been really helpful. Thank you for sharing :-)

3. Thank you this is brilliant

4. What is your understanding of nth term of arithmetic sequences. I'm seeing a lot of text books even at foundation using a+(n-1)d. Do pupils now have to be able to find the nth term using this formula?

1. They don't *have* to use that formula in the new GCSE but their teacher may choose to teach it to them. I normally save it for A level. It has pros and cons... I'm surprised to hear that this formula is in GCSE textbooks - I didn't know it was commonly used at this level. There are a number of methods for finding the nth term of an arithmetic sequence - featured in this blog post - teachers can choose any approach they like (or teach multiple methods and let students choose).

2. Thanks. I've never seen it in an old gcse text book but it's common place in all the new text books I've seen for 9-1 even foundation.

5. thank you - this has been really helpful and made me more aware of what is expected from students on a topic that previously had very little airtime and relevance. I tend to steer away from the formula when teaching quadratics as I feel that finding the nth term in the linear sequence in the remainder is an extension of what they tend to know really well; also I feel it's too much information for some higher students to absorb. I really appreciate the links to all the resources - brilliant!

6. Thanks so much I have found a wealth of information regarding resources for the new gcse topics. Hopefully my students will be able to assess the materials.