28 May 2015

New GCSE: Inequalities

The new GCSE specification has two additions under the heading Inequalities:

a) using set notation to represent solutions

b) solving quadratic inequalities.

This post explains these changes and provides teaching support and resources.

Set Notation
Students will now be required to represent solutions to inequalities using set notation. This is in addition to representing solutions on number lines and graphs. The OCR specification gives us two examples of set notation:
If you're not familiar with set notation, it's explained here (it's commonly referred to as 'set-builder notation'). Note that either a vertical line or a colon can be used to represent 'such that'.
Image source: coolmath.com
The examples from the OCR specification imply that GCSE students will not need to know symbols representing number types (eg ℤ for integers), and therefore will not be required to express their answers like this:
Image source: mathsinsfun.com
Set notation comes up elsewhere in the new GCSE specification - under the title 'Venn Diagrams and Sets' we have this:
Source: OCR specification

So do we have any resources to practise this? There's plenty of resources relating to set notation and probability in my Data library. I've also made this simple worksheet so students can practise using set notation to represent inequalities. 

Quadratic Inequalities
Here's a question from OCR's Sample Assessment Materials Higher Paper 5 (non-calculator):

Find the range of values of x for which x2 - 3x - 10 ≤ 0

If you haven't taught AS level maths or iGCSE then you might not have taught this topic before. There's a few different methods for solving quadratic inequalities. A common method involves sketching a graph of the quadratic function then identifying the required region (see example below). To use this method, our GCSE students will need to be confident in sketching quadratics (this is covered in the new specification). I often find that my Year 12 students make mistakes in their final answer because they don't bother sketching the graph. To me, sketching the graph is essential.
Desmos is a fantastic tool for exploring quadratic inequalities - thanks to Cathal (@CGA_PGS) for sharing this example.

An alternative method for solving quadratic inequalities involves using a number line and test points. In this method students still have to find the critical values, but instead of sketching the graph they check the value of a test point in each region.
Given that this topic has been taught in iGCSE and A level courses for some time, there's surprisingly few resources available. There's a few resources in my Algebra library, including a 'Spot the Mistake' activity I made, but I need more! If you know of anything else then please share it. 


  1. I think we are all busy working out the changes Jo! I think a good plan will be for us all to use all the exam board resources we can get hold of which ever board we are doing! For inequalities my immediate thought is Desmos and WolframAlpha! I documented changes myself when trying to sort out the main differences. A tiny number of thoughts for resources in each section at the moment which I'll add to.

    1. Thanks Colleen, your page is very useful. I've not seen any exam board resources yet but look forward to lots becoming available over the coming months. Many schools will hold off buying textbooks for at least the first year so we'll need a good source of resources for student practice.

  2. If you have access to them, graphical calculators are excellent for inequalities

    1. Thank you, good idea - I hadn't thought of that.

  3. This is quite a nice interactive tool from our friends in over in Wales.


    1. Actually - ignore that - it's already in the Algebra Library :)

    2. Some nice questions at the end of this worksheet. Answers are included but a few of the questions are duplicated so needs to be slightly edited before using. https://www.saddleback.edu/faculty/lperez/algebra2go/cahsee/inequalities/42.pdf

    3. Thank you - nice to have a big set of questions to draw on.