TES Maths: Inspect the spec - equations of circles
Support students as they start to make sense of equations of circles as part of the new GCSE specification
Everyone is talking about functions and frequency trees, but what else has changed with the advent of the new specification? And what resources are available to help? Throughout this series, TES Maths aims to find out.
What does the specification say?
The expectation is that:
- All students will develop confidence and competence with the content identified by standard type
- All students will be assessed on the content identified by the standard and the underlined type; more highly attaining students will develop confidence and competence with all of this content
- Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content.
A16h: Recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.
What's the same?
Higher tier students had to get to grips with equations of circles as part of the previous specification. In fact, there was a noticeable increase in the frequency of such questions over the last few years, perhaps in anticipation of their increased prominence now.
While students may still be required to sketch graphs of circles with their centre at the origin, there are a few additional things they will need to be able to do as well.
What has changed?
Firstly, the use of the word "recognise" in the new specification suggests that higher students may now have to work backwards from a sketch to an equation.
Secondly, and potentially more importantly, they will be expected to be able to find the equation of a tangent to a circle at a given point. This should not be confused with the new GCSE concept of estimating the gradient of a curve, which will be covered in more detail later on in this series.
In order to do this, students will need to find the gradient of the radius and, using a pair of coordinates and their knowledge of the fact that the tangent is perpendicular to the radius, calculate the equation of the tangent. It's tough enough to make even AS-level students shudder!
It remains to be seen how far the examiners will take this, but it is important that students are as prepared as possible.
How can TES Maths can help?
As ever, the wonderfully talented authors of the TES Maths community have stepped up to the mark to lend a hand. Here is a selection of my favourite resources to help support the teaching of this topic:
- Introduction to equations of a circle
Help students to understand where the formula for the equation of a circle comes from using Pythagoras in this exploratory lesson, including a plenary to extend thinking even further.
- Complete equation of a circle lesson
Starting with plotting circles and progressing into the calculation of the equation of a tangent, this lesson offers plenty of worked examples and student challenges.
- Finding equations of tangents
This well-designed activity, which comes with answers, allows students to practise sketching circles and find the equations of tangents.
- Curved graphs practice
Use this original lesson to help students to recognise equations of circles among those of cubics, quadratics, exponentials and trigonometric functions.
- Equations of circles codebreaker
Set up your students for A-level with this engaging task, which investigates circles that do not have their centre at the origin.
- Extension worksheets
Challenge more-able learners with these superhero-themed activities, originally designed for older classes.
Craig Barton, TES Maths adviser
Craig is a secondary maths teacher in the North of England.
Find more resources to support the changes to the GCSE maths specification by taking a look at the rest of the series.