# The circle

This set of resources contains activities in which students will be encouraged to recognise and use and interpret the equation of a circle. Students will also find the equation of a tangent to a circle at a given point.

Visit the secondary mathematics webpage to access all lists.

### Circles with Whole Number Points

‘Circles with Whole Number Points’ provides teacher inspiration for investigating equations of a circle and Pythagoras’ theorem. The student task sheet refers to the use of Geogebra. The task consists of drawing a circle, centre (0,0), radius 5 and finding all the points with integer coordinates through which the circle passes. The activity can be extended by altering the radius of the circle or moving the centre of the circle from the origin.

The resource includes initial explanation and suggested teaching strategies and provides a good opportunity to use the freely available Geogebra software.

### Circle Geometry

These three lesson ideas are designed to explore the properties of circles and their equations. Each activity is accompanied by teacher notes suggesting how the activity could be delivered and possible extension ideas. Extension ideas introduced in transformations of graphs include investigation of circles whose centre is not the origin.

The first activity requires students to match the equations of circles to statements cards. There is not a unique solution to this problem thus requiring students to explain the mathematics used to justify their solution.

The second activity asks students to determine whether each of a number of statements is true or false. Once again students are required to justify their answer showing the mathematics they have used. In this exercise students are required to complete the square to rearrange the equation of the circle into a form that can be used to determine whether the statement is true or false.

The third activity contains a miscellany of probing questions which can be used in a variety of ways in the classroom in order to assess student understanding.

### What's the Question?

‘What’s the Question?’ is an activity designed to act as stimuli for work on the geometry of straight lines and circles. The activity consists of two mathematical diagrams. Students are asked to generate questions that could be asked related to each diagram. The teacher notes give examples of the kinds of questions expected, provide teaching tips and suggest how the activity could be adapted. The second stimulus picture relates to lines and circle. The image could be adapted to vary difficulty, such as a circle not centred on the origin.

### Higher 2

This resource is designed for students following the higher tier in mathematics. Each chapter begins with learning outcomes and contains reference boxes which explain techniques or concepts to be mastered, as well as discussion points and key facts. The second part of the book is a comprehensive revision guide which provides notes and relevant questions for the whole syllabus. Chapter 8 includes tangents to circle.

### Circles - Equations and Graphs

This Virtual Textbook excel program contains a series of interactive spreadsheets related to the equation of the circle. The first sheets enable the student to alter the radius and centre of the circles centred at the origin and elsewhere. The program allows the student to view the graphs of the circles as they transform.

The penultimate sheet generates circles and allows students to find and verify the centre, radius and the equation of the circle in the form

(x+a)^{2} + (y+b)^{2} = r^{2} and the alternate form x2+y2+2fx+2gy+c=0.

The final sheet generates circles and lines so that students can find and verify the points of intersection.