19 January 2016

Five things you might not know about the new GCSE content #2

I recently wrote a post about five changes to the content of maths GCSE. In today's post I list five more changes that may not be widely known.

1. Invariance
When teaching shape transformations to Higher Tier students, you'll now need to ensure that students are be able to identify invariant points. This is described in the specification as follows: "Describe the changes and invariance achieved by combinations of rotations, reflections and translations".

We are given this example question from AQA:
"Write down the coordinates of a point that is invariant when it is reflected in the line y = x".
Here, students will need to know that the points (1,1), (2,2) etc lie on the line y = x and would therefore be unchanged if reflected in this line.

There's a set of resources for this topic in the Mind the Gap Maths Toolbox and Peter Mattock has shared a resource 'Invariance activity sheet'.

2. Geometric Sequences
When I saw that geometric sequences now appear on the GCSE specification I assumed I'd be making use of my AS level (C2) resources. This is not the case.

In C2 we find the nth term of a geometric sequence using the formula Un = arn-1.  This gives us a 'position-to-term' rule. For example if we have the sequence
3,  6,  12,  24,  48, ... 
then we have a = 3 (the first term) and r = 2 (the common ratio), so the nth term is Un = 3(2n-1).

Although geometric sequences do come up at GCSE, there's two points from the specification that are worth noting when teaching this content:

1. Geometric sequences will be in the form rn , for example:
2,  4,  8,  16,  ...  
3,  9,  27,  81, ...
√5,  5,  5√5,  25,  ...  (Higher Tier only)
2. Students are not required to find expressions for the nth term of these sequences. They only have to do that for linear and quadratic sequences.

Examples of new GCSE questions include:
The nth term of a sequence is (√3)n. What is the 5th term of this sequence?
and...
Un = 2Un-1     U1 = 2     Write down the first four terms of this sequence.
In both examples students are using an nth term formula but not deriving the formula themselves. So when you teach this topic I suggest that you focus on:
  • recognising geometric sequences
  • finding a common ratio and using this to continue a sequence
  • substituting into both position-to-term and term-to-term rules (including those using subscript notation). 
If you've spotted this question in OCR's Practice Paper 5 you may think that it contradicts what I've written here about geometric sequences:
Here is a sequence.
2,  2√7,  14,  14√7,  ... 
a) Work out the next term   (1)
b) Find the nth term   (3)
c) Find the value of the 21st term divided by the 17th term.   (2)
I asked OCR for clarification and they sent a helpful reply:
"...the Assessment Objectives around making deductions, inferences and problem solving mean that some questions may involve taking known elements of content and taking them that bit further, as seen in the question. Given that students at Higher tier have to know how to find the formula for the nth term of a quadratic sequence and also to 'recognise and use sequences of... geometric progressions (r^n where n is a... surd) and other sequences', the question is an example of how higher ability students might be expected to make a deduction from known content. I don't expect this to be a question we'd regularly examine in live assessment, but is the sort of thing we'd want to include in sample assessment to give an indication of how the content and Assessment Objectives are brought together in writing questions."
3. Scatter Graphs
There's a bit more to scatter graphs now, though some teachers may have already been covering these things. Students will need to know:
  • that correlation does not necessarily indicate causation
  • the difference between interpolation and extrapolation, and the dangers of extrapolation.
Although I will use the words interpolate and extrapolate in class, I doubt these words will be used in exams. Questions are likely to be in the form of this example from Edexcel:
You should not use a line of best fit to predict the number of units of electricity used for heating when the outside temperature is 30°C. Give one reason why. 
Students may also have to identify an outlier from a scatter graph (note that this is informal identification of outliers - ie by eye) and decide whether to ignore it when drawing a line of best fit.

The Scatter Graphs: True or False activity from MathsPad covers all of this content and is absolutely excellent.

Don't forget that some statistics content has been removed from the new GCSE, such as questionnaires and stratified sampling. It's worth looking at these exam questions and this OCR Check-In Test to get an idea of the new sampling content.

4. SUVAT
The early versions of the DfE’s new GCSE maths specification contained content relating to the 'suvat equations', but this content was removed from the final published draft. However the formulae were retained in the appendix (in the section entitled 'Formulae that candidates should be able to use, but need not memorise. These can be given in the exam, either in the relevant question, or in a list from which candidates select and apply as appropriate').
This means that the exam board might include these formulae in a question, but this will be no different to how a student would be expected to work with any formula or equation provided (for example, students may have to substitute into or rearrange a suvat formula).

In short: you don't need to teach this topic at GCSE in the way you would teach it in M1 at A level, but you might find it useful to use these formulae in your teaching of algebra - for example using this great resource from Christine Norledge.

5. Don't follow the textbooks... yet!
I've said before that if you plan to invest in textbooks for the new GCSE then it's best to wait for the second editions. I've already found a few inconsistencies between the specifications and first edition textbooks. For example a helpful conversation with @STABMaths confirmed that an Edexcel textbook contains graph stretches, as does a sample paper, but these won't be examined (as stated in my previous post).

Also, in my post about real life graphs, I said that I was surprised to see sketching cubics in my new GCSE textbook - including identifying roots from factorised expressions.
Cubic graphs from Edexcel GCSE (9 - 1) Mathematics: Higher Student Book 

This is what the specification says about cubic graphs:
recognise, sketch and interpret graphs of ... simple cubic functions ...
This is on both the Foundation and Higher Tier. The AQA Teaching Guidance provides further details, stating that students should be able to:
draw, sketch, recognise and interpret graphs of the form y = x3 + k where k is an integer
There's certainly no mention of the type of cubic graphs that feature in my textbook, and I think this is one of the things that will probably be removed from subsequent editions.

A few more clarifications
I asked OCR what they are frequently asked about the new GCSE. In addition to questions regarding grading and teaching time, they are often asked to clarify the following:
  • whether students will be expected to differentiate to find the gradient of a curve (no, they will have to estimate gradient from a tangent at a point, or potentially a chord between two suitable points) - see my blog post about this. 
  • whether Foundation Tier students are required to calculate turning points of quadratic graphs (no, at Foundation they’ll just need to read from the graph. At Higher they will find turning points by completing the square).
  • if Higher Tier students have to know about the equations for any circle (no, only those centred around the origin).
This is really helpful information from OCR.

So that's it - I hope you're now feeling more informed about the new GCSE content. Do let me know your thoughts and questions.







6 comments:

  1. Thank you for this information. I am telling my department to read your blog posts as I think they are invaluable, not just regarding the new GCSE but for teaching everything! When reading your comments about scattergraphs it reminded me of a question that one of my department had found in the SAMs. It involved finding the mean from a scattergraph. Not difficult but none of us had seen anything like it before so it would definitely throw the students.

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    1. Thanks! Yes, I suppose this is the challenge for students - being able to answer questions that are unlike any they've seen before. Mean from a scattergraph shouldn't be hard but it will be unfamiliar and I agree that it would throw them.

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  2. Love the blog - thanks
    Looking at the Sampling questions (Question 1b) we do not know the answer - are you able to help http://2fv5d843v9w22sxtto1ibxtu.wpengine.netdna-cdn.com/wp-content/uploads/2015/12/Statistics-H-Sampling-v2.pdf

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  3. The answer is based on the student response to the first part of the question, e.g., if they have rounded 28 to 30, then they likely have an underestimate, if they rounded 3 to 5, overestimate etc. It would also be acceptable to state that the sample is too small to decide. Darren(OCR maths team)

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  4. Nice post Jo, thanks. On the stats bit you mentioned, some.things like Frequency Polygons and stem and leaf have gone from the spec but are still present on specimen papers and published textbooks, I expect they'll still be examined.

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    1. Thanks Cav.

      The exams boards could still include a stem and leaf question but in the question they'd have to explain how to read it. They won't assume prior knowledge (ie they won't just say 'draw a stem & leaf diagram'). Same for frequency polygons - we don't need to teach these explicitly, but students will need to be able to interpret any kind of statistical graph (even if it's not in a form they've seen before).

      I'm planning to write a post about this 'unknown' element of the new GCSE.

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