## 15 August 2015

### New GCSE: Tangents and Areas At first glance it appears that calculus features in the new GCSE specification. On closer inspection it turns out that our students will find the gradient of a curve by drawing a suitable tangent rather than by differentiating. And instead of integrating, students will use the trapezium rule (or similar) to find the area under a curve. So calculus remains reserved for Key Stage 5, but our students will now be better prepared for calculus when they first meet it. GCSE will have given them a conceptual understanding of rate of change and an ability to interpret this contextually.

This post talks you through this new GCSE topic - it tells you what you need to teach and provides links to resources.

Specification and Exam Questions
Here's the relevant extracts from the new GCSE specification:

This all becomes clearer when we look at example exam questions. Huge thanks to Tom Bennison (@DrBennison) for doing the hard work for me here - he's been through the sample assessment materials for all exam boards and collated the relevant questions in this Subject Knowledge Check. In a typical question, students are given a velocity-time graph and asked to find the total distance travelled and/or the acceleration at a specific time.

AQA helpfully provides additional clarification about the specification in their teacher guide. The extract below is from GCSE Mathematics (8300) Teaching Guidance (available to All About Maths members) which provides a number of additional example questions.
Methods
The diagrams below are taken from this extract from a new GCSE textbook which sets out the standard methods that our students will use.

Students will already know how to find the gradient of a straight line (ie 'rise over run' or equivalent). To estimate the gradient of a curve they will have to draw a tangent, as shown here:
They'll also need to determine whether the gradient is positive or negative.

To estimate the area under a graph, students will have to split the area into sections. The AQA Teaching Guidance says 'the trapezium rule need not be known but it is recommended as the most efficient means of calculating the area under a curve'. Unlike at A level, they won't be given the formula in the exam.
The alternative to using the trapezium rule is to split the area into a number of triangles and rectangles.

Motion Graphs
The methods described above are fairly straightforward. I think interpretation might prove trickier (eg understanding what motion graphs are showing). Students will need to know that speed, acceleration and deceleration are rates of change, and that the area under a velocity-time graph represents distance. There's a lot of new concepts and vocabulary here.
Bear in mind that motion graphs come up in Physics GCSE too. As shown in this extract from the AQA Physics GCSE Specification, it's exactly the same content:
Other Contexts
I was initially confused by the mention of 'graphs in financial contexts' in this section of the specification - I've seen questions in which students have to interpret financial graphs but nothing involving tangents or areas. Thanks to @DJUdall for sharing the picture below (taken from this new GCSE textbook) which shows an example of estimating a rate of change in a financial context.
Could exam questions cover other contexts, besides motion and finance? It's possible. In the example below from CIMT students are asked to find the volume of water represented by the shaded area. To understand why the area under the graph represents volume we can consider the units - the units on the horizontal axis are seconds, and the units on the vertical axis are m3/s, so when we multiply the two together we get m3.

Instantaneous Rate of Change
Students will need to understand the difference between an average rate of change over a period of time and an instantaneous rate of change. In my post 'Introducing Differentiation' I talked about how to give students an intuitive understanding of the gradient of a curve at a point. It's worth reading the section entitled 'An Instantaneous Rate of Change' for ideas, videos and resources (including my worksheet 'Thinking About Gradient' which was designed for A level but is now suitable for GCSE).

Resources
Resources for this topic are listed below. I'll add these to my library and will continue to add new resources as I find them.
• @DJUdall has produced this excellent graphing activity which covers both tangents and areas under graphs.
• @jase_wanner has written this super hero activity on distance-time and speed-time graphs to develop students' understanding of how to interpret these graphs.
• Gradient on a curved graph by Owen134866 on TES gives students the opportunity to practise drawing graphs and tangents. My worksheet Finding the gradient of a curve using a tangent is similar, but students aren't required to draw the graphs themselves.
• Using Graphs from CIMT covers a wide range of graph topics. Section 17.2 covers areas under graphs using the trapezium rule and Section 17.3 covers tangents to curves. The full range of resources for this module are available on TES.
• Nuffield Mathematics provides a Free-Standing Mathematics Activity 'Speed and Distance' which is about finding the area under a speed-distance graph. Resources include slides, a student sheet and teacher notes.
I hope this post has been useful in helping you prepare to teach this new GCSE topic. Please let me know if you have any resources to share.

You might also find my other posts about new GCSE topics helpful: Sequences, Inequalities and Quadratic Graphs.

See my New GCSE Support Page for resources and links for all new GCSE content.

1. Desmos can be used to illustrate tangents to a curve see https://colleenyoung.wordpress.com/desmos-graphing-calculator/tangents-to-a-curve/.

1. Thanks Colleen. I love Desmos, and didn't know how to do this.

2. this is in science gcse too - edexcel specimen assessment materials have an example in science - paper 2 q7b page 151 http://qualifications.pearson.com/content/dam/pdf/GCSE/Science/2016/Specification/SAMs_GCSE_L1-L2_in_Chemistry.pdf the context is rates of reaction - measuring the volume of gas given off when marble chips are reacted with acid.

1. Interesting! Thank you.

3. This is fantastic, thank you! It is my first time teaching this topic so I am incredibly grateful for all of the information and resources.

4. Is this in the EdExcel IGSCE Spec? Its not in our text book or SoW so I hope not!!!!

1. Sorry, I don't know anything about iGCSE! If it's not in your textbook then probably not, but I guess you should check the spec.

2. It's in the Edexcel IGCSE specification A syllabus certainly.. Hidden away as a note in Graphs 3.3 section D "Find the gradients of non linear graphs..by drawing a tangent" But it is not new to IGCSE for the 9-1 so if you have not seen it before then maybe you teach a spec that doesn't include it?